A wheel starts from rest and uniformly increases angular speed to 0.19 rev/s in 32 s. (a) What is its angular acceleration in radians per second per second? rad/s2 (b) Would doubling the angular acceleration during the given period have doubled final angular speed? Yes No
A centrifuge in a biology laboratory rotates at an angular speed of 3,450 rev/min. When switched off, it rotates 48.0 times before coming to rest. Find the constant angular acceleration of the centrifuge (in rad/s2). Consider the direction of the initial angular velocity to be the positive direction, and include the appropriate sign in your result. rad/s2
The tub of a washing machine goes into its spin cycle, starting from rest and gaining angular speed steadily for 9.00 s, at which time it is turning at 5.00 rev/s. At this point, the lid of the washing machine is opened, and a safety switch turns it off. The tub then smoothly slows to rest in 14.0 s. Through how many revolutions does the tub rotate while it is in motion? rev
A racing league driver is driving her car at a constant speed of 47.0 m/s on a circular track with a radius of 280 m. (a) What is the angular speed (in rad/s) of the car? rad/s (b) What are the magnitude (in m/s2 ) and direction of the car's acceleration? magnitude m/s2 direction
A disk 7.90 cm in radius rotates at a constant rate of 1100 rev/min about its central axis. (a) Determine its angular speed. rad/s (b) Determine the tangential speed at a point 2.92 cm from its center. m/s (c) Determine the radial acceleration of a point on the rim. magnitude km/s2 direction (d) Determine the total distance a point on the rim moves in 2.10 s. m
(a) The fishing pole in the figure below makes an angle of 20.0∘ with the horizontal. What is the torque exerted by the fish about an axis perpendicular to the page and passing through the angler's hand if the fish pulls on the fishing line with a force F→ = 110 N at an angle 37.0∘ below the horizontal? The force is applied at a point L = 2.10 m from the angler's hands. (Enter the magnitude in N⋅m, accurate to at least the nearest integer. ) (i) (b) What If? To reel in the fish, the angler now pulls back on the fishing rod, increasing the angle it makes with the horizontal to 39.5∘. If the fish still applies the same force in the same direction, what is the increase in the torque exerted by the fish about an axis perpendicular to the page and passing through the angler's hand? (Enter the magnitude in N⋅m.) N⋅m
A 105-kg merry-go-round in the shape of a uniform, solid, horizontal disk of radius 1.50 m is set in motion by wrapping a rope about the rim of the disk and pulling on the rope. What constant force would have to be exerted on the rope to bring the merry go-round from rest to an angular speed of 0.800 rev/s in 2.00 s? (State the magnitude of the force.) N
Two blocks are connected to a string, and the string is hung over a pulley connected to the ceiling, as shown in the figure below. The masses of the blocks are m1 = 16.0 kg and m2 = 10.0 kg, the mass of the pulley is M = 5.00 kg, and the radius of the pulley is R = 0.300 m. Block m2 is initially on the floor, and block m1 is initially 4.90 m above the floor when it is released from rest. The pulley's axis has negligible friction. The mass of the string is small enough to be ignored, and the string does not slip on the pulley, nor does it stretch. (a) How much time (in s) does it take block m1 to hit the floor after being released? Δt1 = s (b) How would your answer to part (a) change if the mass of the pulley were neglected? (Enter the time, in seconds, it takes block m1 to hit the floor if the mass of the pulley were neglected. ) Δt2 = s
A sheet of aluminum with a rectangular shape is attached to a vertical support by a set of hinges. Assume the sheet is uniform and has height 2.00 m, width 0.840 m, and mass 25.0 kg. (a) Determine its moment of inertia (in kg⋅m2 ) for rotation on its hinges. kg⋅m2 (b) Are any pieces of data unnecessary? (Select all that apply. ) The mass of the sheet is unnecessary. The width of the sheet is unnecessary. The height of the sheet is unnecessary. No; all of the data are necessary.
Rigid rods of negligible mass lying along the y axis connect three particles. The system rotates about the x axis with an angular speed of 1.50 rad/s. (a) Find the moment of inertia about the x axis. kg⋅m2 (b) Find the total rotational kinetic energy evaluated from 12 Iω2. J (c) Find the tangential speed of each particle. 4.00 kg particle m/s 2.00 kg particle m/s 3.00 kg particle m/s (d) Find the total kinetic energy evaluated from ∑12 mivi2. J (e) Compare the answers for kinetic energy in parts (b) and (d).
The four particles shown below are connected by rigid rods of negligible mass where y1 = 5.90 m. The origin is at the center of the rectangle. The system rotates in the xy plane about the z axis with an angular speed of 5.90 rad/s. (i) (a) Calculate the moment of inertia of the system about the z axis. kg⋅m2 (b) Calculate the rotational kinetic energy of the system. J
A cylinder of mass 12.0 kg rolls without slipping on a horizontal surface. At a certain instant its center of mass has a speed of 17.0 m/s. (a) Determine the translational kinetic energy of its center of mass. J (b) Determine the rotational kinetic energy about its center of mass. J (c) Determine its total energy. J
A merry-go-round on a playground consists of a horizontal solid disk with a weight of 795 N and a radius of 1.56 m. A child applies a force 49.0 N tangentially to the edge of the disk to start it from rest. What is the kinetic energy of the merry-go-round disk (in J) after 3.05 s ? J
Several species of mollusks and clams produce a crystalline style, a cylindrical protrusion 5.00 mm in length and 0.800 mm in diameter, in their stomach to act as a capstan and aid in the digestion process. The style is rotated about its long axis at an average speed of 12.0 rpm, and typically dissolves after 2.00 h. (a) What is the angular acceleration (in rad/s s2 ) of the style if, starting from rest, it reaches a rotation speed of 12.0 rpm in 12.5 s? rad/s2 (b) What is the angle (in rad) through which the style rotates to reach its final rotational speed after 12.5 seconds? rad (c) What is the rotational kinetic energy (in J) of the 3.00 g style once it reaches its final rotational speed after 12.5 seconds? J
A bar on a hinge starts from rest and rotates with an angular acceleration α = 15 + 5t, where α is in rad/s2 and t is in seconds. Determine the angle in radians through which the bar turns in the first 4.78 s. rad
In the figure below, three thin rods are all connected at their centers. The rods each have the same length L and mass m, and they are all perpendicular to one another. Each rod lies on one of the x, y, or z-axes as shown. The entire structure formed by the rods is rotated about an axis that is parallel to the y-axis and passes through the end of the rod that lies on the x-axis. What is the moment of inertia of this structure about the axis of rotation? (Use any variable or symbol stated above as necessary. ) I =
Two position vectors each start at the origin. The first has a magnitude of 37.5 cm, and is at an angle of 15.0∘ as measured counterclockwise from the +x-axis. The second has a magnitude of 23.0 cm and an angle of 58.0∘, also counterclockwise from the +x-axis. (a) What is the area of the parallelogram (in cm2 ) that has the two vectors as two of its sides? cm2 (b) What is the length of the longer diagonal of this parallelogram (in cm )? cm
The figure below shows forces acting at various points on a metal rod. The angles α = 48∘, β = 25∘, γ = 19∘. The length ℓ = 3.8 m. Find the net torque (in N⋅m ) on the rod about the following axes. (a) an axis through O perpendicular to the page magnitude N⋅m direction (b) an axis through C perpendicular to the page magnitude N⋅m direction
(a) A light, rigid rod of length ℓ = 1.00 m joins two particles, with masses m1 = 4.00 kg and m2 = 3.00 kg, at its ends. The combination rotates in the xy-plane about a pivot through the center of the rod (see figure below). Determine the angular momentum of the system about the origin when the speed of each particle is 5.60 m/s. (Enter the magnitude to at least two decimal places in kg⋅m2/s. ) magnitude kg⋅m2/s direction (b) What If? What would be the new angular momentum of the system (in kg⋅m2 /s ) if each of the masses were instead a solid sphere 14.0 cm in diameter? (Round your answer to at least two decimal places. ) kg⋅m2/s
A jetliner of mass 20,000 kg is flying over level ground at a constant altitude of 4.30 km with a constant velocity of 155 m/s west. Its direction is currently straight toward the top of a mountain. At a particular instant, a house is directly below the plane at ground level. (a) At this instant, what is the magnitude of the plane's angular momentum relative to the house (in kg⋅m2/s )? kg⋅m2 /s (b) As the plane continues its constant velocity motion, how does the angular momentum vector relative to the house change? The magnitude of the angular momentum vector stays the same, but the direction changes. Both the magnitude and direction of the angular momentum vector change. The direction of the angular momentum vector stays the same, but the magnitude changes. Both the magnitude and direction of the angular momentum vector stay the same. (c) At this same instant, what is the magnitude of the plane's angular momentum relative to the mountaintop (in kg⋅m2 /s )? kg⋅m2/s
A counterweight of mass m = 4.80 kg is attached to a light cord that is wound around a pulley as shown in the figure below. The pulley is a thin hoop of radius R = 6.00 cm and mass M = 1.10 kg. The spokes have negligible mass. (a) What is the net torque on the system about the axle of the pulley? magnitude N⋅m direction (b) When the counterweight has a speed v, the pulley has an angular speed ω = v/R. Determine the magnitude of the total angular momentum of the system about the axle of the pulley. (kg⋅m)v (c) Using your result from (b) and τ→ = dL→/dt, calculate the acceleration of the counterweight. (Enter the magnitude of the acceleration. ) m/s2
A particle of mass m moves in the xy plane with a velocity of v→ = vxı^ + vyj^. Determine the angular momentum of the particle about the origin when its position vector is r→ = xi^ + yj^. (Use the following as necessary: x, y, vx, vy, and m. ) L→ =
A thin, hollow sphere of radius r = 0.600 m and mass m = 14.5 kg turns counterclockwise about a vertical axis through its center (when viewed from above), at an angular speed of 2.80 rad/s. What is its vector angular momentum about this axis? (Enter the magnitude in kg⋅m2/s. ) magnitude kg⋅m2/s direction
A uniform solid sphere of radius r = 0.430 m and mass m = 14.5 kg turns counterclockwise about a vertical axis through its center (when viewed from above). Find its vector angular momentum about this axis when its angular speed is 2.9 rad/s. magnitude kg⋅m2/s direction
Big Ben, the Parliament tower clock in London, has hour and minute hands with lengths of 2.70 m and 4.50 m and masses of 60.0 kg and 100 kg, respectively. Calculate the total angular momentum of these hands about the center point. (You may model the hands as long, thin rods rotating about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively. ) magnitude kg⋅m2 /s direction
A playground merry-go-round of radius R = 1.80 m has a moment of inertia I = 230 kg⋅m2 and is rotating at 12.0 rev/min about a frictionless vertical axle. Facing the axle, a 23.0−kg child hops onto the merry-go-round and manages to sit down or edge. What is the new angular speed of the merry-go-round? rev/min
A disk of mass m1 = 70.0 g and radius r1 = 3.50 cm slides on a frictionless sheet of ice with velocity v→, where v = 14.00 m/s, as shown in a top-down view in figure (a) below. The edge of this disk just grazes the edge of a second disk in a glancing blow. The second disk has a mass m2 = 140 g, a radius r2 = 5.00 cm, and is initially at rest. As the disks make contact, they stick together due to highly adhesive glue on the edge of each, and then rotate after the collision as shown in figure (b). (i) (a) What is the magnitude of the angular momentum (in kg⋅m2 /s ) of the two-disk system relative to its center of mass? kg⋅m2/s (b) What is the angular speed (in rad/s) about the center of mass? rad/s
You are working in an observatory, taking data on electromagnetic radiation from neutron stars. You happen to be analyzing results from a neutron star similar to the one in the Formation of a Neutron Star example, verifying that the period of a 17.0 km radius neutron star is indeed 2.7 s. You go through weeks of data showing the same period. Suddenly, as you analyze the most recent data, you notice that the period has decreased to 2.0 s and remained at that level since that time. You ask your supervisor about this, who becomes excited and says that the neutron star must have undergone a glitch, which is a sudden shrinking of the radius of the star, resulting in a higher angular speed. As she runs to her computer to start writing a paper on the glitch, she calls back to you to calculate the new radius of the star (in km ), assuming it has remained spherical. She is also talking about vortices and a superfluid core, but you don't understand those words. km
A satellite in outer space has a gyroscope within it used for rotation and stabilization. The moment of inertia of the gyroscope is Ig = 21.5 kg⋅m2 about the axis of the gyroscope, and the moment of inertia of the rest of the satellite is Is = 5.00×105 kg⋅m2 about the same axis. Initially both the satellite and gyroscope are not rotating. The gyroscope is then switched on and it nearly instantly starts rotating at an angular speed of 105 rad/s. How long (in s) should the gyroscope operate at this speed in order to change the satellite's orientation by 34.0∘ ? s
The angular momentum vector of a precessing gyroscope sweeps out a cone as shown in the figure below. The angular speed of the tip of the angular momentum vector, called its precessional frequency, is given by ωp = τ/L, where τ is the magnitude of the torque on the gyroscope and L is the magnitude of its angular momentum. In the motion called precession of the equinoxes, the Earth's axis of rotation precesses about the perpendicular to its orbital plane with a period of 2.58×104 yr. Model the Earth as a uniform sphere and calculate the torque on the Earth that is causing this precession. N⋅m
Given M→ = 6 i^ + j^ − 6 k^ and N→ = 5 i^ − 4 j^ − 5 k^, calculate the vector product M→×N→. i^ + j^ + k^
A uniform solid disk of mass m = 2.99 kg and radius r = 0.200 m rotates about a fixed axis perpendicular to its face with angular frequency 5.95 rad/s. (a) Calculate the magnitude of the angular momentum of the disk when the axis of rotation passes through its center of mass. kg⋅m2 /s (b) What is the magnitude of the angular momentum when the axis of rotation passes through a point midway between the center and the rim? kg⋅m2/s
An electron at point A in (Figure 1) has a speed v0 of 1.50×106 m/s. Figure 1 of 1 Part A Find the magnitude of the magnetic field that will cause the electron to follow the semicircular path from A to B. Express your answer in teslas. Submit Request Answer Part B Find the direction of the magnetic field that will cause the electron to follow the semicircular path from A to B. into the page out of the page Part C Find the time required for the electron to move from A to B. Express your answer in seconds. t = S
A beam of protons traveling at 1.20 km/s enters a uniform magnetic field, traveling perpendicular to the field. The beam exits the magnetic field, leaving the field in a direction perpendicular to its original direction (the figure ( Figure 1)). The beam travels a distance of 1.20 cm while in the field. Figure 1 of 1 Part A What is the magnitude of the magnetic field? Express your answer in teslas. B = Submit Request Answer
A 150 V battery is connected across two parallel metal plates of area 28.5 cm2 and separation 9.20 mm. A beam of alpha particles (charge +2e, mass 6.64×10−27 kg ) is accelerated from rest through a potential difference of 1.75 kV and enters the region between the plates perpendicular to the electric field, as shown in (Figure 1). Figure 1 of 1 Part A What magnitude of magnetic field is needed so that the alpha particles emerge undeflected from between the plates? Express your answer with the appropriate units. Submit Request Answer Part B What is the direction of this magnetic field? The magnetic field is directed into the page. The magnetic field is directed out of the page. The magnetic field is directed upward. The magnetic field is directed downward. Submit Request Answer Request Answer
A long wire carrying 4.50 A of current makes two 90∘ bends, as shown in (Figure 1). The bent part of the wire passes through a uniform 0.234 T magnetic field directed as shown in the figure and confined to a limited region of space. For related problem-solving tips and strategies, you may want to view a Video Tutor Solution of Magnetic force on a straight conductor. Part A Find the magnitude of the force that the magnetic field exerts on the wire. Express your answer in newtons. Submit Request Answer Part B Find the direction of the force that the magnetic field exerts on the wire. Express your answer in degrees. Submit Request Answer
A thin, 52.0 cm long metal bar with mass 750 g rests on, but is not attached to, two metallic supports in a uniform magnetic field with a magnitude of 0.440 T, as shown in ( Figure 1). A battery and a resistor of resistance 29.0 Ω are connected in series to the supports. For related problem-solving tips and strategies, you may want to view a Video Tutor Solution of Magnetic force on a straight conductor. Figure 1 of 1 Part A What is the largest voltage the battery can have without breaking the circuit at the supports? Express your answer in volts. Submit Request Answer Part B The battery voltage has the maximum value calculated in part (a). If the resistor suddenly gets partially short-circuited, decreasing its resistance to 2.00 Ω, find the initial acceleration of the bar. Express your answer in meters per second squared. a = m/s2
The circuit shown in the figure is used to make a magnetic balance to weigh objects. The mass m to be measured is hung from the center of the bar, that is in a uniform magnetic field of 1.50 T, directed into the plane of the figure. The battery voltage can be adjusted to vary the current in the circuit. The horizontal bar is 60.0 cm long and is made of extremely light-weight material. It is connected to the battery by thin vertical wires that can support no appreciable tension; all the weight of the suspended mass m is supported by the magnetic force on the bar. A resistor with R = 5.00 Ω is in series with the bar; the resistance of the rest of the circuit is much less than this. (Figure 1) Figure 1 of 1 Part A Which point, a or b, should be the positive terminal of the battery? a b Submit Request Answer Part B If the maximum terminal voltage of the battery is 175 V, what is the greatest mass m that this instrument can measure? Express your answer in kilograms. m = kg
A 2.60 N metal bar, 0.850 m long and having a resistance of 10.0 Ω, rests horizontally on conducting wires connecting it to the circuit shown in (Figure 1). The bar is in a uniform, vertical, 1.60 T magnetic field and is not attached to the wires in the circuit. Figure 1 of 1 Part A What is the acceleration of the bar just after the switch S is closed? Express your answer with the appropriate units. Submit Request Answer
The rectangular loop of wire shown in the figure (Figure 1) has a mass of 0.20 g per centimeter of length and is pivoted about side ab on a frictionless axis. The current in the wire is 9.0 A in the direction shown. Figure 1 of 1 Part A Find the magnitude of the magnetic field parallel to the y-axis that will cause the loop to swing up until its plane makes an angle of 30.0∘ with the yz-plane. Express your answer in teslas. Submit Request Answer Part B Find the direction of the magnetic field parallel to the y-axis that will cause the loop to swing up until its plane makes an angle of 30.0∘ with the yz-plane. −y-direction +y-direction
A plastic circular loop of radius R and a positive charge q is distributed uniformly around the circumference of the loop. The loop is then rotated around its central axis, perpendicular to the plane of the loop, with angular speed ω. Part A If the loop is in a region where there is a uniform magnetic field B→ directed parallel to the plane of the loop, calculate the magnitude of the magnetic torque on the loop. Express your answer in terms of the variables q, ω, R, and B.
The lower end of the thin uniform rod in (Figure 1) is attached to the floor by a frictionless hinge at point P. The rod has mass 0.0940 kg and length 18.0 cm and is in a uniform magnetic field 0.130 T that is directed into the page. The rod is held at an angle 58.0∘ above the horizontal by a horizontal string that connects the top of the rod to the wall. The rod carries a current 12.0 A in the direction toward P. Figure 1 of 1 Part A Calculate the tension in the string. Use the fact that τ = 12 IBL2 for a uniform bar of length L carring a current I in a magnetic field B. Express your answer with the appropriate units.
Block A in (Figure 1) has mass 1.00 kg, and block B has mass 3.00 kg. The blocks are forced together, compressing a spring S between them; then the system is released from rest on a level, frictionless surface. The spring, which has negligible mass, is not fastened to either block and drops to the surface after it has expanded. Block B acquires a speed of 1.50 m/s. Figure 1 of 1 Part A What is the final speed of block A ? Express your answer in meters per second. Submit Request Answer Part B How much potential energy was stored in the compressed spring? Express your answer in joules.
Two skaters collide and grab on to each other on frictionless ice. One of them, of mass 72.0 kg, is moving to the right at 4.00 m/s, while the other, of mass 63.0 kg, is moving to the left at 2.50 m/s. Part A What is the magnitude of the velocity of these skaters just after they collide? Express your answer with the appropriate units. v = Value Units Submit Request Answer Part B What is the direction of this velocity? to the left to the right
A 12.0 g rifle bullet is fired with a speed of 360 m/s into a ballistic pendulum with mass 10.0 kg, suspended from a cord 70.0 cm long. Part A Compute the initial kinetic energy of the bullet. Express your answer in joules. Part B Compute the kinetic energy of the bullet and pendulum immediately after the bullet becomes embedded in the pendulum. Express your answer in joules. Submit Request Answer Part C Compute the vertical height through which the pendulum rises. Express your answer in centimeters. h = cm
Part A Find the position of the center of mass of the system of the sun (1.99×1030 kg) and Jupiter (1.90×1027 kg). The orbit radius of Jupiter is 7.78×1011 m. (Since Jupiter is more massive than the rest of the planets combined, this is essentially the position of the center of mass of the solar system. ) Express your answer in meters to three significant figures. x = m from the center of the sun Submit Request Answer Part B Does the center of mass lie inside or outside the sun? The sun's radius is 6.96×108 m. inside outside
A 70 kg astronaut floating in space in a 110 kg MMU (manned maneuvering unit) experiences an acceleration of 0.029 m/s2 when he fires one of the MMU's thrusters. For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Acceleration of a rocket. Part A If the speed of the escaping N2 gas relative to the astronaut is 490 m/s, how much gas is used by the thruster in 5.0 s ? Express your answer with the appropriate units. m = Submit Request Answer Part B What is the thrust of the thruster? F =
You throw a 3.00 N rock vertically into the air from ground level. You observe that when it is 16.0 m above the ground, it is traveling at 23.0 m/s upward. Part A Use the work-energy theorem to find the rock's speed just as it left the ground. Express your answer with the appropriate units. v2 = Submit Request Answer Part B Use the work-energy theorem to find its maximum height. Express your answer with the appropriate units. hmax =
A block of ice with mass 2.00 kg slides 1.53 m down an inclined plane that slopes downward at an angle of 36.9∘ below the horizontal. Part A If the block of ice starts from rest, what is its final speed? Ignore friction. Express your answer with the appropriate units. v =
To stretch an ideal spring 9.00 cm from its unstretched length, 18.0 J of work must be done. What is the force constant of this spring? Express your answer with the appropriate units. k = Value Units Submit Request Answer Part B What magnitude force is needed to stretch the spring 9.00 cm from its unstretched length? Express your answer with the appropriate units. F = Part C How much work must be done to compress this spring 4.00 cm from its unstretched length? Express your answer with the appropriate units. W = Submit Request Answer Part D What force is needed to compress the spring this distance? Express your answer with the appropriate units. F =
A 5.0 kg box moving at 3.0 m/s on a horizontal, frictionless surface runs into a light horizontal spring of force constant 95 N/cm. For related problemsolving tips and strategies, you may want to view a Video Tutor Solution of Motion with a varying force. Part A Use the work-energy theorem to find the maximum compression of the spring. Express your answer with the appropriate units. x =
A small glider is placed against a compressed spring at the bottom of an air track that slopes upward at an angle of 49.0∘ above the horizontal. The glider has mass 9.00×10−2 kg. The spring has 580 N/m and negligible mass. When the spring is released, the glider travels a maximum distance of 1.80 m along the air track before sliding back down. Before reaching this maximum distance, the glider loses contact with the spring. Part A What distance was the spring originally compressed? x = m Part B When the glider has traveled along the air track 0.800 m from its initial position against the compressed spring, is it still in contact with the spring? Yes No Submit Request Answer Part C What is the kinetic energy of the glider at this point? Express your answer in joules.
You are applying a constant horizontal force F→ = (−7.60 N)i^ + (3.00 N)j^ to a crate that is sliding on a factory floor. Part A At the instant that the velocity of the crate is v→ = (3.20 m/s)i^ + (2.20 m/s)j^, what is the instantaneous power supplied by this force? Express your answer with the appropriate units.
An elevator has mass 600 kg, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 s, and it is driven by a motor that can provide up to 40 hp to the elevator. Part A What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 kg. n =