The right side of a truck has a constant speed of 6 m/s as it turns to the left on a path with of curvature of 16 m. The direction of the angular velocity vector of the rear right wheel is changing as the truck turns. What is the magnitude and direction (relative to the truck) of the angular acceleration vector, α→, of the rear right wheel?
In the figure, a force P→ acts on a block weighing 49.0 N. The block is initially at rest on a plane inclined at angle θ = 17.0∘ to the horizontal. The positive direction of the x-axis is up the plane. The coefficients of friction between block and plane are μs = 0.540 and μk = 0.340. In unit-vector notation, what is the frictional force on the block from the plane when P→ is (a) (-5.00 N) i^, (b) (-8.40 N) i^, and (c) (−15.2 N)i^?
A particle moving along a straight line has an acceleration which varies according to position as shown. If the velocity of the particle at the position x = −4 ft is v = 5 ft/sec, determine the velocity v when x = 11 ft. Assume x1 = −4 ft, x2 = 5 ft, x3 = 8 ft, x4 = 11 ft, a1 = −3 ft/sec2, a2 = 5 ft/sec2. Answer: v = ft/sec
A block of mass m rests on a rough horizontal surface and is attached to a spring of stiffness k. The coefficients of both static and kinetic friction are μ. The block is displaced a distance x0 to the right of the unstretched position for the spring and released from rest. If the value for x0 is large enough, the spring force will overcome the maximum available static friction force and the block will slide toward the unstretched position of the spring with an acceleration a = μg − k/mx, where x represents the amount of stretch (or compression) in the spring at any given location in the motion. Use the values m = 10 kg, k = 115 N/m, μ = 0.44, and x0 = 530 mm and determine the final spring stretch (or compression) xf when the block comes to a complete stop. The distance xf will be positive if to the right (spring stretched) and negative if to the left (spring compressed). Answer: xf = mm
A small block is loaded onto the storage platform of a lorry, at a distance d away from its open edge, as shown in the figure. The static and kinetic coefficients of friction are μs and μk respectively. At time t = 0, the lorry accelerates from rest towards the right with a magnitude of a = Bt, where B is a constant. The block grips onto the platform until it starts slipping at t = t1, and eventually falls off the lorry at t = t2. The value of d = .
A 61.0 kg circus performer slides 3.40 m down a pole to the circus floor, starting from rest. What is the kinetic energy of the performer as she reaches the floor if the frictional force on her from the pole (a) is negligible (she will be hurt) and (b) has a magnitude of 380 N? (a) Number Units (b) Number Units
In the figure here, a small block is sent through point A with a speed of 6.4 m/s. Its path is without friction until it reaches the section of length L = 14 m, where the coefficient of kinetic friction is 0.75. The indicated heights are h1 = 5.4 m and h2 = 2.2 m. What are the speeds of the block at (a) point B and (b) point C? (c) Does the block reach point D? (d) If so, what is its speed there if not, how far through the section of friction does it travel? (a) Number Unit (b) Number Unit (c)
In the figure, a ball is shot directly upward from the ground with an initial speed of v0 = 7.10 m/s. Simultaneously, a construction elevator cab begins to move upward from the ground with a constant speed of vC = 3.50 m/s. What maximum height does the ball reach relative to (a) the ground and (b) the cab floor? At what rate does the speed of the ball change relative to (c) the ground and (d) the cab floor? (Give the magnitude of the rate of change)
In the figure an electron is shot at an initial speed of v0 = 2.71×106 m/s, at angle θ0 = 43.2∘ from an x axis. It moves through a uniform electric field E→ = (4.92 N/C)j^. A screen for detecting electrons is positioned parallel to the y axis, at distance x = 2.73 m. What is the y component of the electron's velocity (sign included) when the electron hits the screen? Number Units
A small 0.40−kg box is launched from rest by a horizontal spring as shown in the figure below. The block slides on a track down a hill and comes to rest at a distance d from the base of the hill. The coefficient of kinetic friction between the box and the track is 0.40 along the entire track. The spring has a spring constant of 32.5 N/m and is compressed 30.0 cm with the box attached. The block remains on the track at all times. (a) What would you include in the system? Explain your choice. This answer has not been graded yet. (b) Calculate d. m (c) Compare your answer with a similar situation where the inclined part of the track is frictionless.
A small spherical insulator of mass 3.26×10−2 kg and charge +0.600 μC is hung by a thin wire of negligible mass. A charge of −0.900 μC is held 0.150 m away from the sphere and directly to the right of it, so the wire makes an angle θ with the vertical (see the drawing). Find (a) the angle θ and (b) the tension in the wire. (a) Number Units (b) Number Units
An electron is released from rest at the negative plate of a parallel plate capacitor and accelerates to the positive plate (see the drawing). The plates are separated by a distance of 1.6 cm, and the electric field within the capacitor has a magnitude of 2.5×106 V/m. What is the kinetic energy of the electron just as it reaches the positive plate? KEpositive
(a) A particle which is moving in a straight line with constant acceleration 2 m/s2 is initially at rest. Find the distance covered by the particle in the third second of its motion. [4 marks] (b) The time variation of the position of a particle in rectilinear motion is given by x = 2t3 + t2 + 2t. If v is the velocity and a is the acceleration of the particle. Find the velocity and acceleration of the particle when t = 3 s. [4 marks] (c) A bicycle moves along a straight road such that its position is described by the graph shown. Construct the v−t graph and a−t graph for 0 ≤ t ≤ 30 s.
The dragster starts from rest and travels along a straight track with an acceleration-deceleration described by the graph below. Construct the v-s graph for 0 ≤ s ≤ s1 and determine the distance s1 traveled before the dragster again comes to rest.
A particle moving in the x-y plane has a velocity at time t = 5.28 s given by (6.9i + 6.9j) m/s, and at t = 5.38 s its velocity has become (7.07i + 7.03j) m/s. Calculate the magnitude a of its average acceleration during the 0.10-s interval and the angle it makes with the x axis. Answers: a = m/s2 θ =
A police car is traveling at a velocity of 20.0 m/s due north, when a car zooms by at a constant velocity of 41.0 m/s due north. After a reaction time 0.500 s the policeman begins to pursue the speeder with an acceleration of 6.00 m/s^2. Including the reaction time, how long does it take for the police car to catch up with the speeder?
Calculate the minimum possible magnitude u of the muzzle velocity which a projectile must have when fired from point A to reach a target B on the same horizontal plane 7.2 km away. Answer: u = m/s
Electrostatics - Coulomb's Law Consider the three-point charges at the corners of a triangle as shown in figure, where q1 = 6×10−9 C, q2 = −2×10−9 C, and q3 = 5×10−9 C. Find the magnitude and direction of the resultant force on q3.
Object of m2 = 7.90 kg and m1 (unknown) are attached with a string Static friction coefficient (μs = 0.30) and kinetic friction coefficient (μk = 0.25). System is moving with acceleration of a = 1.25 m/s2 (m1 towards +x and m2 towards −y). a) Consider system is moving ( m1 to right and m2 to down) and draw two free-body-diagrams (one free-body-diagram for each object)? (3 pts) b) Apply Newton 2nd law to m2 and find the algebraic equation for string tension force and then find numerical value? (3 pts) c) Apply Newton 2 nd law to m1 and find algebraic equation for mass m1 and numerical value? (4 pts)
The velocity v→ of a particle moving in the xy plane is given by v→ = (6.0t − 4.0t2)i^ + 7.1 j^, with v→ in meters per second and t( > 0) is in seconds. (a) What is the acceleration when t = 1.1 s? (Express your answer in vector form.) a→ = m/s2 (b) When (if ever) is the acceleration zero? (Enter 'never' if appropriate.) s (c) When (if ever) is the velocity zero? (Enter 'never' if appropriate.) s (d) When (if ever) does the speed equal 10 m/s ? (Enter 'never' if appropriate.) s
A projectile of mass 0.799 kg is shot straight up with an initial speed of 16.2 m/s. (a) How high would it go if there were no air resistance? (b) If the projectile rises to a maximum height of only 7.24 m, determine the magnitude of the average force due to air resistance.
A motorcycle starts from rest with an initial acceleration of 3.6 m/s2, and the acceleration then changes with distance s as shown. Determine the velocity v of the motorcycle when s = 340 m. At this point also determine the value of the derivative dv ds.
Block 1 of mass 2 m is connected to Block 2 of mass m using an inextensible rope and pulley of negligible mass as shown in the diagram below. The coefficient of kinetic friction between Block 1 and the inclined plane is μ1 whereas that between Block 2 and the inclined plane is μ2. It is observed that Block 1 accelerates down the incline pulling Block 2 up the incline. (a) Derive an expression for the acceleration of the system, a in terms of the given variables only m, μ1, μ2 and any constants, such as g. (b) Calculate the tension in the string given m = 3.0 kg, θ1 = 60∘, θ2 = 30∘, μ1 = 0.2, μ2 = 0.4 (c) If the inclined planes are frictionless, what is the acceleration of the system? Hint: Modify the expression you derived in part (a) to obtain an expression for the acceleration of the system in this case.
A block of mass m sits on top of a slab of mass 4 m. The slab rests on a frictionless floor. The coefficient of static friction between the slab and the block is 0.60 whereas the coefficient of kinetic friction is 0.40 . The block is now pulled by a horizontal force of magnitude F to the left. (a) It is observed that the blocks move together as a system. In this case, is the friction between the contact surfaces of the block and the slab static friction or kinetic friction? Justify. (b) If m = 10 kg, what is the maximum value of the applied horizontal force, F for which the slab and the block move together as a system? (c) If the applied horizontal force is increased beyond the value you determined in part (b), describe what would be the change, if any, in the motion of the block and slab. Explain. (d) For F = 100 N, what would be the accelerations of the slab and the block individually if you expect them to be different?
Engineering a highway curve. If a car goes through a curve too fast, the car tends to slide out of the curve. For a banked curve with friction, a frictional force acts on a fast car to oppose the tendency to slide out of the curve; the force is directed down the bank (in the direction water would drain). Consider a circular curve of radius R = 220 m and bank angle θ, where the coefficient of static friction between tires and pavement is μs. A car (without negative lift) is driven around the curve as shown in the figure below. Find an expression for the car speed vmax that puts the car on the verge of sliding out, in terms of R, θ, and μs. Evaluate vmax for a bank angle of θ = 11∘ and for (a) μs = 0.77 (dry pavement) and (b) μs = 0.047 (wet or icy pavement). (Now you can see why accidents occur in highway curves when icy conditions are not obvious to drivers, who tend to drive at normal speeds. ) (a) Number Units (b) Number Units
A loaded penguin sled weighing 66.0 N rests on a plane inclined at angle θ = 24.0∘ to the horizontal (see the figure). Between the sled and the plane, the coefficient of static friction is 0.220, and the coefficient of kinetic friction is 0.100. (a) What is the minimum magnitude of the force F→, parallel to the plane, that will prevent the sled from slipping down the plane? (b) What is the minimum magnitude F that will start the sled moving up the plane? (c) What value of F is required to move the sled up the plane at constant velocity? (a) Number Units (b) Number Units (c) Number Units
A person wants to slide the 100−lb box across the floor by pushing on it with the force P located as shown. The coefficients of static and kinetic friction between the box and the floor are 0.20. a) Is it possible to slide the box with an acceleration of 8.00 ft/s2 to the right without tipping the box? Support your answer with a calculation. b) If so, calculate the value of P that gives this acceleration.
Engineering a highway curve. If a car goes through a curve too fast, the car tends to slide out of the curve. For a banked curve with friction, a frictional force acts on a fast car to oppose the tendency to slide out of the curve; the force is directed down the bank (in the direction water would drain). Consider a circular curve of radius R = 220 m and bank angle θ, where the coefficient of static friction between tires and pavement is μs. A car (without negative lift) is driven around the curve as shown in the figure below. Find an expression for the car speed vmax that puts the car on the verge of sliding out, in terms of R, θ, and μs. Evaluate vmax for a bank angle of θ = 11∘ and for (a) μs = 0.77 (dry pavement) and (b) μs = 0.047 (wet or icy pavement). (Now you can see why accidents occur in highway curves when icy conditions are not obvious to drivers, who tend to drive at normal speeds.)
A person lowers a bucket into a well by turning the hand crank, as the drawing illustrates. The crank handle moves with a constant tangential speed of 1.13 m/s on its circular path. The rope holding the bucket unwinds without slipping on the barrel of the crank. Find the linear speed with which the bucket moves down the well.
A circular loop has radius R and carries current I2 in a clockwise direction. The center of the loop is a distance D above a long, straight wire. a) What is the direction of the current I1 in the wire if the net magnetic field at the center of the loop is zero? [1 mark] b) Derive an equation for the magnitude of the current I1 [5 marks]
An aerialist on a high platform holds onto a trapeze attached to a support by an 8.9−m cord. (See the drawing.) Just before he jumps off the platform, the cord makes an angle of 45∘ with the vertical. He jumps, swings down, then back up, releasing the trapeze at the instant it is 0.79 m below its initial height. Calculate the angle θ that the trapeze cord makes with the vertical at this instant.
The cart impacts the safety barrier with speed v0 = 4.00 m/s and is brought to a stop by the nest of nonlinear springs which provide a deceleration a = −k1x − k2x3, where x is the amount of spring deflection from the undeformed position and k1 and k2 are positive constants. If the maximum spring deflection is 490 mm and the velocity at half-maximum deflection is 3.56 m/s, determine the values for the constants k1 and k2. Answers: k1 = s−2 k2 = m−2 s−2
The string in the figure is L = 107 cm long, has a ball attached to one end, and is fixed at its other end. The distance d from the fixed end to a fixed peg at point P is 81 cm. When the initially stationary ball is released with the string horizontal as shown, it will swing along the dashed arc. What is its speed when it reaches (a) its lowest point and (b) its highest point after the string catches on the peg?
A radar tower sends out a signal at an angle of 41.0∘ above the horizontal. The signal reflects off of a plane and returns to the transmitter in a total time of 7.50 μs. Determine the height of the plane.
The velocity of a particle is given by v = 16t2 − 118t + 40, where v is in meters per second and t is in seconds. Plot the velocity v and acceleration a versus time for the first 8.1 seconds of motion and evaluate the velocity when a is zero. Make the plots and then answer the questions. Questions: When t = 1.5 s, v = m/s, a = m/s2 When t = 4.7 s, v = m/s, a = m/s2 When t = 7.1 s, v = m/s, a = m/s2 When a = 0, v = m/s
You throw a ball toward a wall at speed 25.0 m/s and at angle above the horizontal (See the figure). The wall is distance from the release point of the ball. (a) How far above the release point does the ball hit the wall? What are the (b) horizontal and (c) vertical components of its velocity as it hits the wall? (d) When it hits, has it passed the highest point on its trajectory?