Two metal disks, one with radius R1 = 2.2 cm and mass, M1 = 0.774 kg and the other with radius R2 = 4.28 cm and mass, M2 = 1.73 kg are welded together and mounted on a frictionless axle through their common center. A light string is wrapped around the edge of the smaller disk, and a m = 1.44 kg block is suspended from the free end of the string. 4 a) What is the total moment of inertia of the two disks? b) If the hanging mass starts at 1.23 m above the floor, what is its speed just before it strikes the floor? Answer: Check Question 11 Not complete c) What is the angular velocity of the disks at this instant?
Consider the railgun system below. The bar has width w, mass m, and current I0 provided by the DC current source, and starts at rest at y = 0. There is an external magnetic field B, and the rails are of length l measured from the bar's initial position. Assume no gravity or air resistance effects. Think of the xy-plane as a flat tabletop. (a) Draw the position and kinetic energy of the bar over time, including what you expect happens just outside the railgun. Provide the expressions. (b) Suppose the same oscillatory signal is applied to both I0 and B, with the same phase. Would the projectile manage to escape the barrel of the railgun? Justify your answer.
The figure gives angular speed versus time for a thin rod that rotates around one end. (a) What is the magnitude of the rod's angular acceleration? (b) At t = 4.0 s, the rod has a rotational kinetic energy of 1.73 J. What is its kinetic energy at t = 0? (a) Number Units (b) Number Units
Ball a, of mass ma, is connected to ball b, of mass mb, by a massless rod of length L. (Figure 1 )The two vertical dashed lines in the figure, one through each ball, represent two different axes of rotation, axes a and b. These axes are parallel to each other and perpendicular to the rod. The moment of inertia of the two-mass system about axis a is Ia, and the moment of inertia of the system about axis b is Ib. It is observed that the ratio of Ia to Ib is equal to 3: Ia Ib = 3 Assume that both balls are pointlike; that is, neither has any moment of inertia about its own center of mass. Figure 1 of 1 Find the ratio of the masses of the two balls. Express your answer numerically. View Available Hint(s) ma mb = Find da, the distance from ball a to the system's center of mass. Express your answer in terms of L, the length of the rod. View Available Hint(s) da =
Potential energy is given by PE = mgh Kinetic energy is given by KE = mv2 2 Figure 7: Trade in joules between potential and kinetic energy Consider the situation in figure 7, the total mechanical energy is 6 J. There is no loss or gain of mechanical energy, only a transformation from kinetic energy to potential energy (and vice versa). The ball, of mass m = 1 kg is released from rest. Answer the following questions: Question #:1 At point A:What is the potential energy of the ball? What is the height of the ball? What is the velocity of the ball at that point? -What is the kinetic energy? Question #:2 At point B, if the potential energy is equal to the kinetic energy (PE = KE = 3 J)What is the height H of the ball? What is the velocity of the ball at that height? Question #:3 At point C, the potential energy is zero. What is the kinetic energy of the ball? What is the velocity of the ball at that point? How would you describe the velocity at this particular point? Explain Question #:4 At point D, the ball is at the same height as in point A, it reaches a maximum height then stops before reversing direction. What is the potential energy? What is the kinetic energy of the ball? What happened to kinetic energy at that position? Question #:5 While the pendulum is oscillating between positions, use the words increase, decrease, or constant to describe the trend of the different forms of energies. Complete table 2 Table 2: Trend of the different forms of energy of a swinging pendulum Question #:6 Using figure 7, identify the corresponding positions A, B and C in figure 8 Discuss the different energies, and speeds of the glider on those points based on your study of the pendulum.
You toss a ball with a mass of 0.583 kg upward. After it leaves your hand, it continues to rise while slowing down (before eventually reversing direction and falling again). Let's say it rises a distance of 1.25 m above your hand (its original height). T maximum height initial height Does the gravitational potential energy of the ball-Earth system increase, decrease, or stay the same? increases decreases stays the same Correct. The ball's height above the Earth's surface increases, therefore the potential energy of the system increases. Calculate the change in gravitational potential energy (in J). (Be sure to include the correct sign.)
A simple pendulum consists of a small ball of mass m = 193 g swinging on a massless, inextensible string of length R = 7 m. At time t = 0 the pendulum is released from the horizontal position, as shown below. A short time later, the pendulum reaches the bottom of its swing and encounters a nail sticking out of the page a distance L = 5.7 m below the pivot point. The pendulum begins to wrap around the nail, and at time t = ttop it is at the top of its first rotation. (e) Find the magnitude and direction of the net force on the ball at t = ttop. ΣF = N (±0.2 N) (f) Find the tension in the string at t = ttop. T = N (±0.2 N)
A plot of potential energy versus position is shown for a 0.27 kg particle that can move along an x axis as it is acted upon by a conservative force. In the graph, UA = 9.9 J, UC = 20 J and UD = 31 J (corresponding to x > 8 ). The particle is observed at the point where the potential energy U forms a curved "hill" to have kinetic energy of 4.22 J (At this point UB = 14 J). What is the difference in particle speed when it is located at x = 6.5 m compared to x = 3.5 m ? Assume mechanical energy is conserved. [Find the speed for x = 6.5 m and subtract the speed for x = 3.5 m] Δv = m/s What is the range of possible particle motion? [Find the turnaround location for the left and right side. The particle turns around when v = 0. Take the difference in these (right value minus left value) to get the range of the particle's motion] |Δx| = m
Electromagnetic waves eject electrons moving with kinetic energy of 11.7×1025 eV from a metal surface that has a work-function energy of 6.9×1025 eV. Based on this scenario, what is the momentum the electromagnetic waves have? Round to 3 sig figs.
opposite end of the rod with an initial velocity of 22.0 m/s. The disk strikes the rod and sticks to it. After the collision, the rod rotates about the pivot point. pivot (a) What is the angular velocity of the two after the collision? rad/s (b) What is the kinetic energy before and after the collision? KEi = J KEf = J
A system of point particles is shown in the following figure. Each particle has mass 0.43 kg, and they all lie in the same plane. (a) What is the moment of inertia (in kg⋅m2 ) of the system about the given axis? kg⋅m2 (b) If the system rotates at 3.1 rev/s, what is its rotational kinetic energy (in J)? J
In the figure below, the mass of block A is 25 kg and mass of block B is 40 kg. The coefficients of static friction are 0.6 between blocks A and B, 0.2 between block B and the floor, and 0.5 between the rope and the fixed drum. Determine the minimum mass, m, of the weight suspended from the rope right at the point where first motion occurs. You may assume that the rope is aligned parallel and perpendicular to the floor, and that neither of the blocks will tip. Hint: You need to consider two sliding cases, calculate minimum m for each, and pick the smaller m of the two.
A bullet of mass m1 = 0.01 kg is shot at a stationary target of mass m2 = 1.5 kg. The bullet strikes the stationary target and lodges itsself into the target. After the collision, m1 and m2 rotate down, and collide elasitcally with another stationary mass m3 = 2.3 kg. The target mass then rotates back up and lands on the catcher ending with zero speed at the top of the rotation. If the coefficient of friction is μ = 0.25, how far does the block slide into the rough patch?
Determine the x-and y-coordinates of the mass center of the plate of uniform thickness t = 0.40 in. The plate has a density that varies according to ρ = ρ0(1 + x 2b). Answers: x¯ = in. y¯ = in.
Problem #5: Collision A pendulum consists of a mass M = 5m (treat as a point-particle) hanging at the bottom end of a rod of mass 2m and length L, which has a frictionless pivot at its top end. A mass m, moving with velocity v, impacts M and becomes embedded. What is the smallest value of v sufficient to cause the pendulum with embedded mass m to swing clear over the top of its arc? Give answer in terms of g and L. (The moment of inertia of a rod of mass m and length L about the end is Irod-end = (1/3)mL2). ν > 560gL 3
In the figure a small, nonconducting ball of mass m = 0.93 mg and charge q = 1.6×10−8 C (distributed uniformly through its volume) hangs from an insulating thread that makes an angle θ = 24∘ with a vertical, uniformly charged nonconducting sheet (shown in cross section). Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density σ of the sheet. Number Units
A bullet of mass 5.00 g is fired into a block of mass M = 750 g that is initially at rest on the edge of a table with height h = 1.50 m. The bullet remains in the block, and after the impact the block lands d = 2.11 m from the bottom of the table. Determine the initial speed of the bullet.
Figure 3 As shown in Figure 3, Mat and Tam carry a canoe with a mass of 50 kg, both exerting upward tangential forces to support the canoe while at the same time keeping it level. What is the ratio FMat/FTam of the force exerted by Mat to that exerted by Tam? Keep in mind that in this situation mechanical equilibrium must be maintained. A. 1.00 B. 2.00 C. 3.00 D. 4.00 E. 5.00
The figure a shows a horizontal uniform beam of mass mb and length L that is supported on the left by a hinge attached to a wall and on the right by a cable at angle θ with the horizontal. A package of mass mp is positioned on the beam at a distance x from the left end. The total mass is mb + mp = 61.8 kg. The figure b gives the tension T in the cable as a function of the package's position given as a fraction x/L of the beam length. The scale of the T axis is set by Ta = 500 N and Tb = 700 N. Evaluate (a) angle θ, (b) mass mb, and (c) mass mp. (a) (b) (a) Number Units (b) Number Units (c) Number Units
A bucket crane consists of a uniform boom of mass M = 201 kg and length L = 59.69 ft that pivots at a point on a truck bed. The boom supports an elevated bucket with a worker inside at the other end of the boom, as shown in the figure. Model the bucket and the worker together as a point mass of weight 203 Ib, located at the end point of the boom. Suppose that when the boom makes an angle of 66.3∘ with the horizontal truck bed, the bucket crane suddenly loses power, causing the bucket and boom to rotate freely toward the ground. Take the rotation axis to be at the point where the boom pivots on the truck bed. Use R = 9.81 m/s2 for the acceleration due to gravity. For unit conversions, assume that 1 m = 3.28 ft and 1 Ib = 4.45 N. Find the magnitude of the angular acceleration a of the system just after the crane loses power. Express your answer Io at least two decimal places. a = rad s2
In a collision of particles in two dimensions, mass m1( = 0.27 kg) is initially moving to the right along the x-axis at velocity 2.9 m/s while mass m2( = 0.47 kg) is at rest (prior to the collision). The angles given in the figure are: θ1 = 52 deg and θ2 = 39 deg. What is the magnitude of the velocity of m2 after the collision? Express your answer in m/s but enter only the numerical part.
Two masses are connected by a string. Mass mA rests on a frictionless inclined plane, while mB is initially held at a height of h above the floor. (a) If mB is allowed to fall, what will be the resulting acceleration of the masses? (b) If the masses were initially at rest, use the kinematic equations to find their velocity just before mass b hits the floor, (c) Use conservation of energy to find the velocity of the masses just before mB hits the floor. You should get the same answer as in part (b). EXPRESS YOUR ANSWER IN TERMS OF h, θ, g, mA, and mB.
The figure below shows the schematic for a mass spectrometer which consists of a velocity selector and a deflection chamber. The magnitude of the magnetic field in both the velocity selector and the deflection chamber is 0.0110 T, and the electric field between the plates of the velocity selector is 1600 V/m. If a singly charged ion with a mass of 6.70×10−27 kg travels through the velocity selector and into the deflection chamber, determine the radius of its trajectory in the deflection chamber. m
The figure below shows a schematic diagram of a mass spectrometer. A charged particle enters a velocity selector, in which a uniform electric field E (created by charged plates) points to the right, and a magnetic field Bin points into the page. Upon exiting the velocity selector, the particle enters a deflection chamber with a magnetic field B→0,in. The particle travels a semicircular path until it hits a photographic plate at point P: Assume the particle is a singly charged ion with mass m = 2.12×10−26 kg. The magnitude of the electric field is 970 V/m, and the magnitude of both the magnetic field in the velocity selector and in the deflection chamber is 0.900 T. What is the radius r (in m) of the particle's path in the deflection chamber? m
Recognizing Energy and collisions If a bullet of mass 5g going at 250 m/s goes into a block of mass 2 kg, how high will they go?
The diagram shows a thin rod of uniform mass distribution pivoted about one end by a pin passing through that point. The mass of the rod is 0.700 kg and its length is 1.41 m. When the rod is released from its horizontal position, it swings down to the vertical position as shown. What is the angular momentum of the center of mass of the rod about the pin when the rod has rotated through an angle θ = 22.5∘? We use the standard rectangular coordinate system with +x axis to the right, +y axis vertically up, and +z axis out toward you. (Express your answer in vector form.) L→ = kg⋅m2/s
Two children are fighting over a toy box of mass m = 13.0 kg in a sand box. Child B pulls horizontally with force FB = 43.0 N while Child A pulls on the other side at a downward angle of θ = 46.0∘ with force FA = 97.0 N. If the box just begins to move in child A's direction, what is the coefficient of static friction between the box and the sandbox? Retain your answer to two decimal places.
13.22 Two identical blocks are released from rest. Neglecting the mass of the pulleys and the effect of friction, determine (a) the velocity of block B after it has moved 2 m, (b) the tension in the cable. 13.23 Two identical blocks are released from rest. Neglecting the mass of the pulleys and knowing that the coefficients of static and kinetic friction are μs = 0.30 and μk = 0.20, determine (a) the velocity of block B after it has moved 2 m, (b) the tension in the cable. Fig. P13.22 and P13.23
The rock now is now at the bottom of the cliff, just before it touches the ground. a. What is the rock's gravitational potential energy, just before it touches the ground? J b. What is the rock's kinetic energy, just before it touches the ground? J
A proton (e) appronches a short fixed electric dipole (p) moving along the dipole axis as shown in the figure. At a large distance from the dipole, the kinetic energy of the proton was K0 = 400 eV. The graph below shows the variation of kinetic energy (K) of the proton at points close to the dipole. Find the value of ro0 ( ro0≫ length of the dipole). In the graph r is the distance from the centre of the dipole.
A sledder effortlessly glides from position A across the snow to position B (as shown in the diagram at the right). Resistance forces are negligible. At position B, the kinetic energy of the sledder is Joules and the total mechanical energy is Joules. In moving from position A to position B, the sledder lost J of potential energy and gained J of kinetic energy.
A student sits on a freely rotating stool holding two dumbbells, each of mass 3.08 kg (see figure below). When his arms are extended horizontally (Figure a), the dumbbells are 0.90 m from the axis of rotation and the student rotates with an angular speed of 0.753 rad/s. The moment of inertia of the student plus stool is 2.62 kg⋅m2 and is assumed to be constant. The student pulls the dumbbells inward horizontally to a position 0.291 m from the rotation axis (Figure b). a b (a) Find the new angular speed of the student. rad/s (b) Find the kinetic energy of the rotating system before and after he pulls the dumbbells inward. Kbefore = J Kafter = J
The winch A hoists the 50−kg load up the 30∘ inclined surface at a constant speed of 2 m/s. a) If the power output of the winch is 1 kW, determine coefficient of the kinetic friction between the load and the inclined surface. b) If the power is suddenly increased to 1.2 kW, what is the corresponding acceleration of the load?
The kinetic energy E of a rocket of mass m, traveling at velocity v is given by the formula E = 12 mv2 Suppose a rocket having mass m = 1.8 is changing velocity over time, and velocity at time t is v(t) = 3.9t2 − 3t. Assuming the mass of the rocket remains constant, what is the kinetic energy when t = 1.7? Enter the answer as a decimal number. Round to two decimal places (as needed).
Let us assume a stream of non-relativistic electrons of kinetic energy E travelling along the x-axis experiences an abrupt change in potential from 0 to V1 at x = 0, where E > V1. At x = a, the electrons experience another step change in potential from V1 to V2, as shown in the sketch below. (a) Starting from the "time independent" Schrödinger equation for the particle wavefunctions in different regions (i. e. x < 0, 0 < x < a and x > a), write down the general solutions for the allowed eigenfunctions. (b) Draw a suitably labelled diagram to appropriately represent the eigenfunctions in all regions. (c) Using appropriate boundary conditions to ensure the eigenfunctions are well behaved, show that there will be 100% transmission of electrons to the region x > a when V2 = 0 and that the wavelength of the eigenfunction in the region 0 < x < a is 2a.
The figure shows a stream of water flowing through a hole at depth h = 6.95 cm in a tank holding water to height H = 30.8 cm. (a) At what distance x does the stream strike the floor? (b) At what depth should a second hole be made to give the same value of x ? (c) At what depth should a hole be made to maximize x ?
A ball is thrown from an initial height of 3.2 m above the ground. The initial speed is 19.7 m/s at an angle of 51∘ above the horizontal. Find the horizontal range Δx. Give your answer with 1 digit precision. Hint: Find the time taken for the ball to hit the ground first
A 49.0-g golf ball is driven from the tee with an initial speed of 46.5 m/s and rises to a height of 33.2 m. (a) Neglect air resistance and determine the kinetic energy of the ball at its highest point. (b) What is its speed when it is 8.56 m below its highest point? (a) Number Units (b) Number Units
A man on a motorcycle plans to make a jump as shown in the figure. If he leaves the ramp with a speed of 31.5 m/s and has a speed of 29.5 m/s at the top of his trajectory, determine his maximum height (h) (in m) above the end of the ramp. Ignore friction and air resistance.
A cylinder with moment of inertia I1 rotates about a vertical, frictionless axle with angular velocity ωi. A second cylinder, this one having moment of inertia I2 and initially not rotating, drops onto the first cylinder. Because of friction between the surfaces, the two eventually reach the same angular velocity ωf. a. Calculate ωf. b. Show that the kinetic energy of the system decreases in this interaction and calculate the ratio of the final to the initial rotational energy.
In a particular photoelectric effect experiment, photons with an energy of 4.50 eV are incident on a metal surface, producing photoelectrons with a maximum kinetic energy of 2.40 eV. (a) What is the work function of the metal? eV (b) If the photon energy is adjusted to 6.80 eV, what will be the maximum kinetic energy of the photoelectrons? eV (c) What is the wavelength, in nanometers, of a 6.80 eV photon? nm
Two blocks A(m = 3.00 Kg) and B(m = 1.00 Kg) are tied by a string via a pulley as shown in the figure. Object B is initially 1.00 m above the floor surface. Assuming the system was released from rest and the table surface is smooth, what would be the kinetic energy of block B just before impact? 29.4 J 2.45 J 8.90 J 4.90 J
As shown in the figure, a block of mass m is at rest on a ramp inclined 30∘ to the horizontal, and another block of mass 23 m at rest on a ramp inclined 60∘. Two blocks are connected by a light, inextensible cord, and the cord passes on a light, frictionless pulley. The other end of the block of mass m is connected to an spring with a force constant k. The spring is neither compressed nor extended. (a) What is the maximum distance the blocks will move when the inclines are frictionless? Express the answer in terms of m, k, and the gravitational acceleration g. (b) Repeat (a) when the coefficients of kinetic friction are 1/3 for both blocks. (c) What is the change in the internal energy of the system composed of the blocks, the inclines, and the spring?
The figure below shows three points in the operation of the ballistic pendulum. The projectile approaches the pendulum in part (a) of the figure. Part (b) of the figure shows the situation just after the projectile is captured in the pendulum. In part (c) of the figure, the pendulum arm has swung upward and come to rest at a height h above its initial position. (a) Prove that the ratio of the kinetic energy of the projectile-pendulum system immediately after the collision to the kinetic energy immediately before is m1/(m1+m2). (Submit a file with a maximum size of 1 MB.) Choose File No file chosen
The 20 KN weight car shown is moving at 12 m/s on a 15∘ inclined road. The coefficient of kinetic friction between the wheels and the road is μk = 0.3 If the driver applies the brakes, causing the wheels to lock, how far does the driver slide before stopping? a) Find it using Conservation of Energy [30 P] b) Find it using Newton's laws of motion [30 P]
Two charges q = +4.8 μC are fixed a distance d = 6.2 cm apart (see the figure). (a) With V = 0 at infinity, what is the electric potential at point C ? (b) You bring a third charge q = +4.8 μC from infinity to C. How much work must you do? (c) What is the potential energy U of the three-charge configuration when the third charge is in place?
A 24 kg block is pushed 10.2 m along a level floor at constant speed with an applied force F directed 29 degrees below the horizontal (meaning it is directed somewhat downward). If the coefficient of kinetic friction between the block and the floor is μk = 0.19, what is the magnitude of the applied force F? |F→appl| = N By how much does the thermal energy of the block-floor system increase? ΔEth = J
A block is in SHM on the end of a spring, with position given by x = xmcos(ωt + ϕ). If ϕ = 1.18 rad, then at t = 0 what percentage of the total mechanical energy is potential energy? Number Units
A golfer hits a ball onto a green at the top of a hill and it follows the trajectory shown in the image below. Note: point A is after the ball has been hit. At what point does the ball have the highest potential energy? At what point does the ball have the highest kinetic energy? At what point does the ball have the lowest potential energy? At what point does the ball have the lowest kinetic energy? The total energy of the golf ball at point B is the total energy of the golf ball at point C. (Assume no energy loss due to air resistance.)
FIGURE EX25.10 shows the potential energy of an electric dipole. Consider a dipole that oscillates between ±60∘. a. What is the dipole's mechanical energy? b. What is the dipole's kinetic energy when it is aligned with the electric field? FIGURE EX25.10
Spring Potential energy of spring vs change in lengthFor each position (e. g. , A, B, C, . . . etc. ) sketch the force on the spring due to the potential energy. Don't worry about the absolute length of the force vectors, but try to make the relative lengths consistent. Forces have directions: Be sure to carefully indicate the direction of the force. For point B, estimate the force quantitatively (e. g., compute a number from the graph). Do these answers depend the 'history' (the time trajectory) of the spring? E. g. , is the force on the spring when it is of the length at point B the same or different if stretched from point A to point B or compressed from point C to point B ?
What is the change in the electric potential energy of a 3.0 nC point charge when it is moved from point A to point B in the figure below? 4.5×10−5 J 1.04×10−5 J 1.04×105 J 4.5×105 J
Determine the electric potential energy for the array of three charges in the drawing, relative to its value when the charges are infinitely far away and infinitely far apart. (q1 = +8.6 μC, q2 = −15.4 μC, and q3 = +20 μC) (k = 9.00×109 N.m2/C2)
The figure shows a plot of potential energy U versus position x of a 0.230 kg particle that can travel only along an x axis under the influence of a conservative force. The graph has these values: UA = 9.00 J, UC = 20.0 J and UD = 24.0 J. The particle is released at the point where U forms a "potential hill" of "height" UB = 12.0 J, with kinetic energy 7.00 J. What is the speed of the particle at (a)x = 3.50 m and (b)x = 6.50 m ? What is the position of the turning point on (c) the right side and (d) the left side?
If the skateboard and skateboarder have total mass of 30.0 kg, its potential energy at the top (point 1) is and kinetic energy is, while at the bottom of the ramp (point 2), its potential energy is , and its kinetic energy is minimum, minimum, maximum, maximum minimum, maximum; minimum, maximum maximum, minimum; minimum, maximum maximum, minimum; maximum, minimum
The figure gives the potential energy U of a magnetic dipole in an external magnetic field B→, as a function of angle φ between the directions of B→ and the dipole moment. The vertical axis scale is set by Us = 1.9×10−4 J. The dipole can be rotated about an axle with negligible friction so as to change φ. Counterclockwise rotation from φ = 0 yields positive values of φ, and clockwise rotations yield negative values. The dipole is to be released at angle φ = 0 with a rotational kinetic energy of 5.9×10−4 J, so that it rotates counterclockwise. To what maximum value of φ will it rotate? Number Units
The potential energy (in Joules) of a particle of mass 9 kg is given by the function U(x) = 1500 − 70x2 + x4. The figure shows a plot of U(x) versus the particle position x. The particle can travel only along the x axis and is under the influence of a conservative force. The particle is released at 5 m with a speed of 11 m/s. (a) Determine the total mechanical energy of the particle. Etot = J
The potential energy (in Joules) of a particle of mass 9 kg is given by the function U(x) = 1500 − 70x2 + x4. The figure shows a plot of U(x) versus the particle position 2 . The particle can travel only along the x axis and is under the influence of a conservative force. The particle is released at 5 m with a speed of 11 m/s. (a) Determine the total mechanical energy of the particle. Etot = (b) What is the speed of the particle at x = 7 m? v = m/s (c) Determine the two turning points of the motion of the particle. Enter your answer such that x1 represents the turning point with larger numerical value (ie, x1 > x2 ). x1 = m x2 = m
Examine the following potential energy diagram and answer the questions that follow. ( provide quantitative values where possible) a) What is the free energy for this reaction? b) What is the Ea for the forward reaction c) What is the Ea for the reverse reaction d) What area on the graph would we find the activated complex
Label the diagram with the correct values for kinetic energy (KE) and potential energy (PE) along the path of the roller coaster. Assume that the roller coaster starts from rest. Answer Bank KE: 0 J KE: 5500.0 J KE: 4125.0 J KE: 1375.0 J KE: 2750.0 J PE: 5500.0 J PE: 0 J PE: 1375.0 J PE: 4125.0 J PE: 2750.0 J
Kinetic and spring potential energy problem: A block (m = 5.0 kg) is pushed against a spring that is compressed a distance of x = 10.0 cm. The block is then released and travels up a frictionless inclined surface see the figure. a. How much potential energy is stored in the spring? b. After the block leaves the spring, what is the kinetic energy of the spring, c. Find the velocity of the block right after the block leaves the spring d. Find the maximum height of h of the block if the spring constant k = 100 N/m.
A large mass m1 = 5.30 kg is attached to a smaller mass m2 = 3.52 kg by a string. The string is hung over a pulley as shown in the figure, and the mass of the pulley and string are negligible compared to the other two masses. Mass m1 is started with an initial downward speed of 2.13 m/s. What is the speed of mass m2 after it has moved h = 2.47 meters?
A rectangular wire frame with 4 m×10 m size carries a current of I = 5 A and is placed as shown in the figure where the angle is θ = 37∘. There is a uniform external magnetic field B→ = 4 i^ T. (a) What is the net force acting on the wire frame? Coding note: i^:ihat, j^:jhat, k^:khat, F→ = N (b) What is the net torque acting on the wire frame? τ→ = Nm (c) Suppose that the wire frame is initially at rest. It then starts to rotate only under the effect of the external magnetic field. At the time when the frame is passing through its minimum potential energy configuration . . . what is the final kinetic energy of the frame? Kf = J
A box (mass = 100 g) is initially connected to a compressed (x = 80 cm) spring (k = 100 N/m) at point A. It was released and started moving along the horizontal surface ( μk = 0.20 ) until it moves up along the inclined surface ( μk = 0.30). The box then stops at point D alone the incline. Consider that from B to C the horizontal distance is 30 m and θ = 50∘. (a) What is the velocity of the box at point B? (b) What is the change in kinetic energy from point B to point C ? (c) What is the velocity of the box at point C? (d) What is the length of the incline, f ? (e) What is the change in gravitational potential energy from point C to point D?
Problem 4 Description: The m-kg cylinder rolls without slipping on the horizontal plane. (a) Determine the acceleration of its mass center. (b) Determine its angular acceleration. The 144−kg cylinder rolls without slipping on the horizontal plane. Part A Determine the acceleration of its mass center. Express your answer to three significant figures and include the appropriate units. Part B Determine its angular acceleration. Express your answer to three significant figures and include the appropriate units.
In the figure a small, nonconducting ball of mass m = 1.4 mg and charge q = 2.7×10−8 C (distributed uniformly through its volume) hangs from an insulating thread that makes an angle θ = 41∘ with a vertical, uniformly charged nonconducting sheet (shown in cross section). Considering the gravitational force on the ball and assuming the sheet extends far vertically and into and out of the page, calculate the surface charge density σ of the sheet. Number Units
Two metal disks, one with radius R1 = 2.50 cm and mass M1 = 0.80 kg and the other with radius R2 = 5.00 cm and mass M2 = 1.60 kg, are welded together and mounted on a frictionless axis through their common center as shown below. (a) A light string is wrapped around the edge of the smaller disk, and a 1.50−kg block is suspended from the free end of the string. What is the magnitude of the downward acceleration of the block after it is released? (b) Repeat the calculation of part (a), this time with the string wrapped around the edge of the larger disk. In which case is the acceleration of the block greater? Does your answer make sense?
(d) The potential energy (U) of two atoms when they are a distance (r) apart, is given by: U = −A rm + B rn, m = 2, n = 10 Given that the atoms form a stable molecule at a separation of 0.3 nm, with an energy of 6.4×10−19 J, calculate parameters A and B. [6]
Block pulled with a changing angle A block of mass 1.89 kg is accelerated across a rough surface by a rope passing over a pulley, as shown in the figure below. The tension in the rope is 12.4 N and the pulley is 11.5 cm above the block. The coefficient of kinetic friction is 0.370. 8 a Determine the acceleration in m/s2 of the block when x = 0.375 m. 8 b Calculate the value of x (in m) at which the acceleration becomes zero.
The kid from last time is pulling the sled with mass m kg by a massless, taut string with constant tension T N at the angle θ shown. The pull is not large enough to lift the sled off the snow, and the block has initial velocity v0 m/s. There is friction with the coefficient μk between the block and the surface. After the block is pulled through a horizontal distance Lm, what is the change in velocity Δv = vf − v0? m = 1.7 T = 11 θ = 20∘ v0 = 1.1 μk = 0.26 L = 7 Find change in velocity Δv = Type your numeric answer and submit
Three different objects, all with different masses, are initially at rest at the bottom of a set of steps. Each step is of uniform height d. The mass of each object is a multiple of the base mass m : object 1 has mass 4.90 m, object 2 has mass 1.46 m, and object 3 has mass m. When the objects are at the bottom of the steps, define the total gravitational potential energy of the three-object system to be zero. Each answer requires the numerical coefficient to an algebraic expression that uses some combination of the variables m, g, and d, where g is the acceleration due to gravity. Enter only the numerical coefficient. (Example: If the answer is 1.23mgd, just enter 1.23) If the objects are positioned on the steps as shown, what is gravitational potential energy Ug, system of the system? If you redefine the reference height such that the total potential energy of the system is zero, how high h0 above the bottom of the stairs is the new reference height? Now, find a new reference height h0′ (measured from the base of the stairs) such that the highest two objects have the exact same gravitational potential energy. Ug, system = mgd h0 = d h0′ = d
A particle moves in a one-dimensional potential energy U(x) specified by: U(x) = ∞x < 0 (region 1) U(x) = −30 eV 0 < x < L (region 2) U(x) = −15 eV L < x < 3L (region 3) U(x) = 0 3L < x < 4 L (region 4) U(x) = ∞ x > 4 L (region 5) The energy of the particle is −5 eV. (a) Draw the potential well (b) Write down the general form of the wave function for each region and why you are selecting that as your wavefunction. Be sure to indicate how you are defining k for each region. Set up the equations for each boundary conditions that are required. (c) If we measure the position of the particle at a random time, in which region (1, 2, 3, 4, or 5) are we most likely to find the particle? EXPLAIN YOUR ANSWER.
Calculate the potential energy, kinetic energy, mechanical energy, velocity, and height of the skater at the various locations. Use g = 10 m/s2. KE = 1/2mv2 GPE = mgh ME = KE + GPE
Refer to figure 1. The 10.0 kg green cart travels down a ramp - from point A to B. Calculate its starting potential energy, kinetic energy at the bottom of the ramp, and its velocity coming off the bottom of the ramp. If the cart coasts from points B to C, did it do any work? How far up the 2nd ramp does the cart go as measured from the ground? Figure 1
Block A has a mass of 40 kg, and block B has a mass of 8 kg. The coefficients of friction between all surfaces of contact are μs = 0.20 and μk = 0.15. If P = 0, determine (a) the acceleration of block B, and (b) the tension in the cord.
Determine the elastic potential energy stored in the spring shown in the diagram below: 1.56 J 3.13 J 156 J 313 J
Find the mass and center of mass of the triangular lamina shown below, if the density function is ρ(x, y) = 12 x kilograms per square meter. mass = kilograms x¯ = meters y¯ = meters
The figure here shows a plot of potential energy U versus position x of a 0.869 kg particle that can travel only along an x axis. (non-conservative forces are negligible) The three values of potential energy at the regions defined in the graph are UA = 15 J, UBB = 35 J. and UCC = 45 J. The particle is released at x = 4.5 m with an initial speed of 7.3 m/s. As the particle heads leftward along the x-axis. a) Determine if the particle can reach x = 1.0 m. If it can reach x = 1.0, determine the speed of the particle at x = 1.0 m. If it cannot reach x = 1.0 m, at what position, x, is its turning point? Determine the b) magnitude and direction of the force on the particle as it begins to move to the left of x = 4.0 m.
A ball is thrown so that the motion is defined by the equations x = 5.8t and y = 2 + 6t − 4.9t2, where x and y are expressed in meters and t is expressed in seconds. Determine the horizontal distance the ball travels before hitting the ground. The horizontal distance the ball travels before hitting the ground is m.
A yo-yo is constructed of three disks: two outer disks of mass M, radius R, and thickness d, and an inner disk (around which the string is wrapped) of mass m, radius r, and thickness d. The yo-yo is suspended from the ceiling and then released with the string vertical (see figure below). Calculate the tension in the string as the yo-yo falls. Note that when the center of the yo-yo moves down a distance y, the yo-yo turns through an angle y/r, which in turn means that the angular speed ω is equal to vCM/r. The moment of inertia of a uniform disk is 12 MR2. (Use the following as necessary: M, R, d, m, r, and g for the acceleration due to gravity.) T =
An automobile is modeled as a lumped mass m with input from the road via a suspension system that consists of a linear spring and a linear viscous dashpot, as shown in the figure on the right. Assume that the automobile travels at a constant speed V over a road whose roughness is a known function ug(x). Determine the equation of motion for the vertical motion of the mass.
A uniform rod of length L and mass M has a point mass M attached at its end and can rotate around a frictionless axle at the other end. The rod is released form rest at an initial angle of 90∘, as shown. What is the magnitude of the angular acceleration α when the rod passes through a 30∘ angle with respect to horizontal? L = 3.5 m Find α[s−2] =
P_01: Suppose that q = −2.0 nC. Q-01: What is the contribution to the electric potential (at the location of the dot) from the charge, q? V Q-02: What is the total electric potential at the location of the dot? V Q-03: What is the potential energy of a proton located at the dot? Note: This is not the total potential energy of the collection of charges. x10−16 J
Part A Determine the mass moment of inertia of the thin plate about an axis perpendicular to the page and passing through point O. The material has a mass per unit area of 13 kg/m2. Express your answer to three significant figures and include the appropriate units. ANSWER: IO =
The graph shows part of a Potential Energy Surface (PES) for a molecule for a given degree of freedom. a) Identify points A, B, C, D, and E as local maximum, local minimum, global maximum or global minimum, justifying each assignment. b) Given a start point of S, explain what the process of optimisation will do and how it will explore the PES. c) Explain with reference to the points on the PES above how molecular dynamics is able to explore more of the PES. d) Monte Carlo is able to explore the whole PES. How does it achieve this? e) Which technique is the most likely to find the global minimum starting from point S and how can you confirm that you have found the global minimum?
In Case A two charges which are equal in magnitude but opposite in charge are separated by a distance d. In Case B the same charges are separated by a distance 2 d. Which configuration has the highest potential energy U? Case A has the highest potential energy. Case B has the highest potential energy. Both cases have the same potential energy.
This problem is identical to the second problem of Test 5 , except now you will solve it using an energy method. Three masses mA, mB and mC are initially at rest. The masses mB and mC are connected to each other with a taut and massless rope which is guided over a massless and frictionless pulley. The masses mA and mB are connected to each other with a second taut and massless rope which is guided over a second massless and frictionless pulley. The masses mB and mC are on frictionless inclines of angles θB and θC, respectively, while mass mA hangs vertically, as shown in Figure 1 . The force of gravity acts vertically downward. Figure 1. Diagram for Problem 1. a. (30 pts) The masses are released from rest. Using the Principle of Conservation of Mechanical Energy, derive an equation for the common velocity v of the masses at the moment they have moved through distance L. Express your equation for v in terms of only the aforementioned variables and g. You must make a drawing showing the various configurations of the system and labeling them. You may draw directly on Figure 1 adding all the required elements for a system configuration diagram. Simplify your equation.
In the figure below, a disk (radius R = 1.0 m, mass = 2.0 kg) is suspended from a pivot a distance d = 0.25 m above its center of mass. The angular frequency (in rad/s) for small oscillations is approximately 3.8 1.5 4.2 1.0 2.1
A crane is made from a long, horizontal arm with mass m = 2.41×104 kg and length L = 52.0 m. The crane arm is supported by a vertical tower positioned halfway along the arm as shown in the figure below. The arm is free to rotate horizontally about the axis of the tower. The moment of inertia of a uniform rod with mass m and length L pivoted about its centre is given by Irod = 112 mL2. L Part 1) The crane lifts a mass M = 1070 kg at the end of the arm on one side of the tower. To keep the crane balanced, a counterweight of mass Mc = 5420 kg is placed on the arm on the opposite side of the tower to mass M. How far from the tower should the counterweight be placed such that the centre of mass for the arm and both masses is directly over the tower? x = m Part 2) What is the moment of inertia of the crane about the axis of the tower? Include the contributions of the horizontal arm, the mass and the counterweight. I = kg m2 Part 3) A net torque of magnitude τ = 6660 Nm is applied to the crane. If the crane starts at rest, what is its angular velocity after t = 30.0 s ? ω = rad s−1
Problem 7: Crates A and B have equal mass. Crate A is at rest on an incline that makes and angle of θ = 31.1 degrees to horizontal, while crate B is at rest on a horizontal surface. Part (a) Write an expression for the ratio of the normal forces, A to B, in terms of θ. Part (b) What is the ratio of the normal forces, A to B? NA/NB =
For the configuration of two masses and two springs shown, calculate the normal frequencies and normal coordinates, assuming that the motion is restricted to the vertical plane.
A 23.0 N force pulls horizontally on a three-mass system connected by two horizontal strings that slides on a rough horizontal surface. Starting from the left, the masses of the blocks are 3.9 kg, 1.5 kg and 2.6 kg. The coefficient of kinetic friction μk is 0.19 for all the blocks and the surface. a. Draw free body diagrams for both all three blocks. (3) b. Write Newtons 2 nd law for the blocks. (6) c. What is acceleration of the blocks? (1) d. (Bonus) Find tension T1? (1) e. (Bonus) Find tension T2? (1)
Sliding Along A hollow sphere of mass M and radius R is placed under a plank of mass 3M and length 2R. The plank is hinged to the floor, and it initially makes an angle θ = π 3 rad to the horizontal. Under the weight of the plank, the cylinder starts rolling without slipping across the floor. What is the cylinder's initial translational acceleration? Assume the plank is frictionless. A not-to-scale picture of the sphere-plank setup.
Consider the spring-mass system shown below, where the mass is subjected to the gravitational force mg. (a) Represent the free-body diagram needed to derive the equation of motion. Identify the magnitude and direction of all forces. (b) Using Newton's second law, derive the equation of motion as a linear, time-invariant ODE. (c) Using the differentiation theorem, find the Laplace transform of the equation of motion, and express the function Y(s) = L[y(t)]. Assume initial conditions y(0) = 1 and y˙(0) = 0.
(b) What is the potential energy of this oscillator? Give the expression in terms of the mass m, angle θ, length L, and the gravitational constant g. Obtain total energy, i. e. , Kinetic Energy plus Potential Energy of this oscillator Define that the potential energy is zero when the angle θ = 0. (5 points) Fig. 1(a)
An annular cylinder (mass 1.6 kg) with inner radius R1 = 21 cm and outer radius R2 = 35 cm is mounted on a disk (mass 3.4 kg) of radius R3 = 51 cm (see below). This system spins around the central axis with rotational kinetic energy of 407 Joules. Find the angular speed of the system in radians per second.
Two men, one of mass m1 = 50 kg and another of mass m2 = 50 kg are sitting in the boat of mass m3 = 150 kg. Initially, the boat is at rest. Define the displacement of the boat if the man of mass m2 moves a distance a = 2 m in the direction of the boat bow as shown in Figure 6. Neglect any resistance to motion afforded by the water. Figure 6: Illustration to Problem 6.
In the figure, a small block of mass m = 0.023 kg can slide along the frictionless loop-the-loop, with loop radius R = 13 cm. The block is released from rest at point P, at height h = 6.1R above the bottom of the loop. (a) How much work does the gravitational force do on the block as the block travels from point P to point Q? J (b) How much work does the gravitational force do on the block as the block travels from point P to the top of the loop? J If the gravitational potential energy of the block-Earth system is taken to be zero at the bottom of the loop, find the following. (c) the potential energy when the block is at point P J (d) the potential energy when the block is at point Q (e) the potential energy when the block is at the top of the loop J (f) If, instead of being released, the block is given some initial speed downward along the track, do the answers to (a) through (e) increase, decrease, or remain the same? increase decrease remain the same
The equation below shows the mathematical representation of spring potential energy (also called the elastic potential energy) of an object that is being acted upon by a linear ideal spring with a spring constant, k (also called the stiffness of the spring). A linear ideal spring either pushes or pulls (that is, it responds with a restoring force -of equal strength in either case) in direct proportion (k) to the distance it is either compressed or stretched away from its equilibrium position (also called its "relaxed length"). Match each term in the equation with the correct description from this list: (1) Spring Constant; (2) Distance from Equilibrium; (3) Spring Potential Energy Distance from Equilibrium Spring Constant Spring Potential Energy
Problem 14: The diagram shows a crate with mass m = 20.1 kg being pushed up an incline that makes an angle φ = 15 degrees with horizontal. The pushing force is horizontal, with magnitude P = 450 N. Starting from rest, the crate achieves a speed of 4.01 m/s as it moves a distance d = 4.01 m. What is the work done by friction, in joules? W = J
The man has a mass of 70 kg and the crate has a mass of 90 kg. If the coefficient of static friction between his shoes and the ground is μs = 0.3 and between the crate and the ground is μs = 0.4, determine if the man is able to move the crate using the rope-and-pulley system shown.
Part A The system in the Figure is in equilibrium. A mass M1 = 228.0 kg hangs from the end of a uniform strut which is held at an angle θ = 40.0∘ with respect to the horizontal. The cable supporting the strut is at angle a = 27.6∘ with respect to the horizontal. The strut has a mass of 59.5 kg. Find the magnitude of the tension T in the cable. Part B Find the magnitude of the horizontal component of the force exerted on the strut by the hinge? Part C Find the magnitude of the vertical component of the force exerted on the strut by the hinge?
QUESTION 05 (a) (i) What is meant by kinetic energy? (ii) What is the relationship between potential and kinetic energy? (b) (i) Explain Hooke's law? What is restoring force? (ii) What is conservation of energy? (c) In Figure, a single frictionless rollercoaster car of mass m = 825 kg tops the first hill with speed v0 = 17.0 m/s at height h = 42.0 m. How much work does the gravitational force do on the car from that point to (i) point A, (ii) point B, and (iii) point C ? If the gravitational potential energy of the car-Earth system is taken to be zero at C, what is its value when the car is at (iv) B and (v) A? (vi) If mass m were doubled, would the change in the gravitational potential energy of the system between points A and B increase, decrease, or remain the same?
If an object's potential energy U(x) changes depending on its position x as shown, which of the following best describes the force Fx generated by this potential energy? a. Fx < 0 everywhere b. Fx < 0 for x < 1 cm c. Fx > 0 for x > 1 cm d. Fx > 0 for x < 1 cm e. Fx > 0 everywhere
The pulley in the figure has radius r and total mass m. The rope does not slip on the pulley rim. The weights and the pulley are not moving initially. Calculate the angular speed of the pulley at the moment that the 4 kg block strikes the floor. r = 11 cm m = 2.1 kg Find ω [s−1] = Type your numeric answer and submit
Problem 12: The diagram shows a crate with mass m = 24.3 kg being pushed up an incline that makes an angle ϕ = 25.4 degrees with horizontal. The pushing force is horizontal, with magnitude P = 552 N. Starting from rest, the crate achieves a speed of 4.17 m/s as it moves a distance d = 4.62 m. What is the work done by friction, in joules? W = J
A body of mass 2.0 kg is moving along the x-axis with a speed of 3.0 m/s at the instant represented below. (a) What is the acceleration (in m/s^2) of the body?
A plot of potential energy versus position is shown for a 0.203 kg particle that can move along an x-axis as it is acted upon by a conservative force. In the graph, UA = 8.6 J, UC = 23 J and UD = 30 J (corresponding to x > 8 ). The particle is observed at the point where the potential energy U forms a curved "hill" to have kinetic energy of 4.98 J (At this point UB = 13 J). What is the difference in particle speed when it is located at x = 6.5 m compared to x = 3.5 m ? Assume mechanical energy is conserved. [Find the speed for x = 6.5 m and subtract the speed for x = 3.5 m] Δv = m/s What is the range of possible particle motion? [Find the turnaround location for the left and right side. The particle turns around when v = 0. Take the difference in these (right value minus left value) to get the range of the particle's motion] |Δx| = m
(a) The graph shows part of a Potential Energy Surface (PES) for a molecule for a given degree of freedom. (i) Identify points A, B, C, D, and E as local maximum, local minimum, global maximum or global minimum, justifying each assignment. (ii) Given a start point of S, explain what the process of optimisation will do and how it will explore the PES. (iii) Explain briefly with reference to the points on the PES above how molecular dynamics is able to explore more of the PES.
Calculate the Electric Potential Energy for the next configuration of particles. q1 = 5 C q2 = −4 C q3 = 6 C q4 = −7 C r12 = 0.3 cm r13 = 0.4 cm r14 = 0.8 cm r23 = 0.7 cm r24 = 0.5 cm r34 = 1.2 cm
A spaceship (with mass m and moving at speed v relative to an observer) separates into two pieces due to internal forces, one of them 3.8 times as massive as the other. After the separation the less massive piece is at rest as seen by the observer (in the observer's frame of reference). How does the kinetic energy of the system change as a result of the separation, as seen in the observer's frame of reference? Express your answer as a ratio of the change in kinetic energy to the initial kinetic energy: ΔKKi, where Ki is the initial kinetic energy: ΔKi Ki =
Consider a potential energy barrier like that of the figure but whose height U0 is 8.8 eV and whose thickness L is 0.53 nm. What is the energy of an incident electron whose transmission coefficient is 0.0018? Energy Number Units
The image below shows a collection of point masses laid out along the x- and y-axes, connected by massless rods. The mass M in the lower-left is located at the origin. What are the x- and y-coordinates of the center of mass for this system of objects? Give your answers in terms of the parameter L.
(a) Which particle above is positively charged (the magnetic field is into the page)? (b) Which particle has the largest kinetic energy?
Consider the mass spectrometer shown schematically in the figure below. The magnitude of the electric field between the plates of the velocity selector is 2.70×10 3 V/m, and the magnetic field in both the velocity selector and the deflection chamber has a magnitude of 0.0400 T. Calculate the radius of the path for a singly charged ion having a mass m = 2.32×10−26 kg.
The mass of the homogeneous thin metal plate is 65.0 kg. b) Find the mass moment of inertia about the z-axis. c) Find the mass moment of inertia about the axis parallel to the z-axis, but going through the center of mass. Don't forget to include the units for everything.
Consider the three charges in Figure 2. Each charge has magnitude of 1.0 nC. Their electric potential energy (using the usual reference potential energy) is a. 3.0×10−7 J b. 5.0×10−7 J c. 7.0×10−7 J d. 9.0×10−7 J e. None of the aboveConsider the three charges in Figure 2. Each charge has magnitude of 1.0 nC. The electric potential at the unoccupied corner of the rectangle, point P indicated with a dot (using the usual reference potential), is a. 3.0×102 V b. 5.0×102 V c. 7.0×102 V d. 9.0×102 V e. None of the above
An object of mass m starts from the left with velocity v0 and travels along a track moving right from position 1 to position 5 as shown. The track is completely frictionless, except for a horizontal region of length Δxf, where the coefficient of kinetic friction between the track and the object is μk. The object finally collides with a spring with a spring constant k and comes to rest in position 5 , compressing the formerly relaxed spring by a maximum amount, Δx. Which statements about the mechanical energy of the object-earth-spring system as the object moves from position 1 to position 5 is FALSE? Assume that the center of mass of the block is at the exact same vertical heights as positions 1, 3, and 5 . Note that while the object is moving at positions 1 through 4, the magnitudes of the velocity vectors shown may not be to scale. Note that more than more than one statement might be false (unlike what you will see on the final exam, where you will only be able to select a single response). The potential energy is the same at positions 1, 3 , and 5 . The gravitational potential energy is the same at positions 1, 3, and 5 . The total mechanical energy is the same at positions 1, 2, 3, 4, and 5. The kinetic energy is the same at positions 1 and 3 . The total mechanical energy is the same at positions 4 and 5 . The total mechanical energy is the same at positions 1, 2 , and 3.
When the displacement in SHM is equal to 1 /4 of the amplitude xm, what fraction of the total energy is (a) kinetic energy and (b) potential energy? (c) At what displacement, in terms of the amplitude, is the energy of the system half kinetic energy and half potential energy? (Give the ratio of the answer to the amplitude) (a) Number Units (b) Number Units (c) Number Units *amplitude
Two constant forces act on an object of mass m = 5.80 kg moving in the xy plane as shown in Figure below. Force F1 is 24.5 N at 35.0°, and force F2 is 42.0 N at 150°. At time t = 0, the object is at the origin and has velocity (3.40ı^ + 2.15 j^) m/s. (a) Express the two forces in unit-vector notation. F→1 = N F→2 = N (b) Find the total force exerted on the object. N (c) Find the object's acceleration. m/s2 Now, consider the instant t = 3.00 s. (d) Find the object's velocity. m/s (e) Find its position. m (f) Find its kinetic energy from 1/2 mvf2. kJ (g) Find its kinetic energy from 1/2 mvi2 +ΣF→⋅Δr→. kJ
Consider a frictionless track as shown in the figure below. A block of mass m1 = 2.00 kg is released from point (A) from a height of h = 1.15 m. It makes a perfectly inelastic collision at point (B) with a block of mass m2 = 2.00 kg that is initially at rest. After the collision, the block of mass m2 moves to the right and collides with a spring at point ( C . The spring is attached to a wall and has a spring constant of 8.59 kN/m. (a) Calculate the energy loss during the collision. Enter the magnitude. J (b) Calculate the maximum compression of the spring. cm (c) Instead of an inelastic collision, consider a head-on elastic collision at point B. During this specific elastic collision when the masses are equal, m1 transfers its kinetic energy to m2 and as a result, m1 comes to rest. Calculate the maximum compression of the spring. cm
A child's pogo stick (figure below) stores energy in a spring (k = 2.95×104 N/m). At position (A) (x1 = −0.100 m), the spring compression is a maximum and the child is momentarily at rest. At position (B) (x = 0), the spring is relaxed and the child is moving upward. At position C, the child is again momentarily at rest at the top of the jump. Assume that the combined mass of child and pogo stick is 28.0 kg. (a) Calculate the total energy of the system if both potential energies are zero at x = 0. J (b) Determine x2. m (c) Calculate the speed of the child at x = 0. m/s (d) Determine the value of x for which the kinetic energy of the system is a maximum. mm (e) Obtain the child's maximum upward speed. m/s
In the figure, an electron with an initial kinetic energy of 3.80 keV enters region 1 at time t = 0. That region contains a uniform magnetic field directed into the page, with magnitude 0.00760 T. The electron goes through a half-circle and then exits region 1, headed toward region 2 across a gap of 25.0 cm. There is an electric potential difference ΔV = 1900 V across the gap, with a polarity such that the electron's speed increases uniformly as it traverses the gap. Region 2 contains a uniform magnetic field directed out of the page, with magnitude 0.0220 T. The electron goes through a half-circle and then leaves region 2. At what time t does it leave?
A block of mass 0.60 kg is dropped onto a vertical weightless spring. The spring is initially unstretched. The speed of the block just before it makes contact with the spring is 2.0 ms−1. The block instantaneously stops when the spring is compressed through a distance of 5.1 cm. a. Calculate: i. the initial kinetic energy of the block ii. the work done on the block by the gravitational force, since the first contact with the spring until it stops iii. the elastic potential energy stored in the spring at the instant when the block is at rest. b. Hence, calculate the spring constant. c. Determine the acceleration of the block at the instant when the block is at rest.
A photon having energy E0 = 0.880 MeV is scattered by a free electron initially at rest such that the scattering angle θ of the scattered electron is equal to that of the scattered photon as shown in the following figure. (a) Determine the scattering angle of the photon and the electron. (b) Determine the energy and momentum of the scattered photon. (c) Determine the kinetic energy and momentum of the scattered electron.
In the figure below, a charged particle moves perpendicular to a uniform B-field (in a vacuum) and the magnetic field is the dominant factor determining the motion. The particle's kinetic energy and speed remain constant, and the direction of motion and speed are unaffected. Because the magnetic force is perpendicular to the direction of travel, the charged particle follows a curved path in the magnetic field. And the particle continues to follow this curved path until it forms a complete circle. The magnetic force is perpendicular to the velocity, where the velocity changes in direction and magnitude. The magnetic force is perpendicular to the velocity and velocity changes in direction not magnitude. In this situation, where a charged particle moves perpendicular to a uniform B-field (in a vacuum), the magnetic force supplies the centripetal force Fc = mv2 r. The magnetic force is always perpendicular to velocity, and it does no work on the charged particle. A negatively charged particle moves in the plane of the paper in a region where the magnetic field is perpendicular out of the paper.