AP Physics C 2nd Sem
The reel shown in the figure has a radius R and a moment of inertia I. One end of the bock of mass m is connected to a spring force constant k, and the other end is fastened to a cord wrapped around the reel. The reel axle and the incline are frictionless. The reel is wound counterclockwise so that the spring stretches a distance d from its unstretched position and the reel is then release from rest. What is the expression for the mechanical energy of the block, reel, spring system when the spring is again unstretched?
(1/2)mv^2 + (1/2)Iw^2
A newly discovered planet has twice the mass of the Earth, but the acceleration due to gravity on the new planet's surface is exactly the same as the acceleration due to gravity on the Earth's surface. The radius of the new planet in terms of the radius R of Earth is
(2)^1/2 * R
The escape speed for a rocket at Earth's surface is ve. What would be the rocket's escape speed from the surface of a planet with twice Earth's mass and the same radius as Earth?
(2)^1/2ve
A sphere of mass M, radius r and rotational inertia I is released from rest at the top of an inclined plane of height h as shown above. If the plane is frictionless, what is the speed vcm of the center of mass of the sphere at the bottom of the incline?
(2gh)^1/2
The uniform thin rod shown above has mass m and length l. The moment of inertia of the rod about an axis through its center and perpendicular to the rod is (1/12)ml2. What is the moment of inertia of the rod about an axis perpendicular to the rod and passing through point P, which is halfway between the center and the end of the rod?
(7/48)ml^2
A rod of negligible mass is pivoted at a point that is off-center, so that length l1 is different from length l2. The figures above show two cases in which masses are suspended from the ends of the rod. In each case the unknown mass m is balanced by a known mass, M1 or M2, so that the rod remains horizontal. What is the value of m in terms of the known masses?
(M1M2)^1/2
If the moment of inertia I of a disk is 0.50 kgm2. What is the angular acceleration of the disk if the radius of the disk is 15 cm and there is a tangential force of tension of 5 newtons on the outside of rim of the disk?
-1.5 rad/s^2
Consider Earth to be stationary, and the Moon as orbiting Earth in a circle of radius R. If the masses of Earth and the Moon are ME and MM , respectively, which of the following best represents the total mechanical energy of the Earth-Moon system?
-GMEMM/(2R)
A uniform 135-g meter stick rotates about an axis perpendicular to the stick passing through its center with an angular speed of 3.50 rad/s. What is the magnitude of the angular momentum of the stick?
0.0394 kg • m^2/s
8.2 Problems A uniform solid cylinder with a radius of 10 cm and a mass of 3.0 kg is rotating about its center with an angular speed of 33.4 rpm. What is its kinetic energy?
0.092 J
A uniform solid disk of mass m = 3.00 kg and radius r = 0.200 m rotates about a fixed axis perpendicular to its face with angular frequency, 𝜔, equal to 6.00 rad/s. Calculate the magnitude of the angular momentum, L in 𝑘𝑔⋅𝑚𝑠2, of the disk when the axis of rotation passes through its center of mass.
0.360 𝑘𝑔⋅𝑚/s^2
A 2 kg mass connected to a spring oscillates on a horizontal, frictionless surface with simple harmonic motion of amplitude 0.4 m. The spring constant is 50 N/m. The period of this motion is
0.4𝜋 s
A uniform solid disk of mass m = 3.00 kg and radius r = 0.200 m rotates about a fixed axis perpendicular to its face with angular frequency, 𝜔, equal to 6.00 rad/s. Calculate the magnitude of the angular momentum, L in 𝑘𝑔⋅𝑚𝑠2, of the disk when the axis of rotation passes through a point midway between the center and the rim. Hint - Use the parallel axis theorem: 𝐼=𝐼𝑐𝑜𝑚+𝑀𝐷2 where D is the distance from the center of mass to the axis of rotation.
0.540 𝑘𝑔⋅𝑚𝑠2
A simple pendulum makes 120 complete oscillations in 3.00 min at a location where g = 9.80 m/s2 . What is the length of the pendulum arm?
0.559 m
A vertical spring stretches 3.9 cm when a 10-g object M is hung from it. The object is replaced with a block of mass 25 g that oscillates up and down in simple harmonic. Calculate the period of motion in seconds. Hint: First find the force constant or spring constant k
0.62 seconds
The position of a particle is given by the expression x = 4.00cos(3.00𝜋𝑡+𝜋), where x is in meters and t is in seconds. What is period of the motion? Hint: Use 𝑥(𝑡)=𝐴cos(𝜔𝑡+𝜙), 𝜔=2𝜋𝑇, and 𝑇=1𝑓
0.667 s
The free-fall acceleration on the surface of the Moon is about one-sixth that on the surface of the Earth. The radius of the Moon is about0.250𝑅𝐸(𝑅𝐸=𝐸𝑎𝑟𝑡ℎ′𝑠𝑅𝑎𝑑𝑖𝑢𝑠=6.37𝑥106𝑚). Find the ratio of their average densities, 𝜌𝑀𝑜𝑜𝑛𝜌𝐸𝑎𝑟𝑡ℎ . Hint: Density is equal to mass divided by volume and use the volume of a sphere since the Moon and the Earth are both in the shape of a sphere. The volume of a sphere is 𝑉𝑠𝑝ℎ𝑒𝑟𝑒=4𝜋𝑅33
0.788:1
13.2 Problems What is the length of a simple pendulum with a period of 2.0 s?
0.99 m
A simple pendulum has a period of 2 s for small amplitude oscillations. The length of the pendulum is most nearly
1 m
A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as shown. A horizontal force F causes the spring to stretch a distance of 5.46 cm from its equilibrium position. What is the speed of the block when it first reaches the equilibrium position? Hint: Use 12𝑘𝐴2=12𝑚𝑣2max
1.00 m/s
A 50.0-g object connected to a spring with a force constant k of 35.0 N/m oscillates with an amplitude of 4.00 cm on a frictionless, horizontal surface. What is the speed of the object when its position is 1.00 cm? Hint: use 12𝑘𝐴2=12𝑚𝑣2+12𝑘𝑥2
1.02 m/s
Big Ben, the Parliament tower clock in London, pictured above, 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, Σ𝐿, in 𝑘𝑔⋅𝑚2𝑠 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. Use 𝐼𝑟𝑜𝑑,𝑎𝑡𝑒𝑛𝑑=13𝑀𝐿2
1.20 𝑘𝑔⋅𝑚^2/s
Plaskett's binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal. Assume the orbital speed of each star is 220 km/s and the orbital period of each is 14.4 days. Find the mass M of each star. (For comparison, the mass of our Sun is 1.99 x 1030 kg.) Hint: Use 𝐹𝑐=𝐹𝑔, 𝑣=2𝜋𝑅𝑇 and G = 6.673𝑥10−11𝑁⋅𝑚2𝑘𝑔2
1.26𝑥10^32𝑘𝑔
13.2 Problems A 0.150-kg air track cart is attached to an ideal spring with a force constant (spring constant) of 3.58 N/m and undergoes simple harmonic oscillations. What is the period of the oscillations?
1.29 s
A simple pendulum makes 120 complete oscillations in 3.00 min at a location where g = 9.80 m/s2 . What is the period of the pendulum?
1.500 s
One end of a uniform 4.00 m long rod of weight Fg =100 N is supported by a cable with a tension of 150 N at an angle 𝜃 of 40.0o with the rod. The other end rests against the wall, where it is held by friction as shown above. The coefficient of static friction between the wall and the rod is 𝜇s = 0.500. The minimum distance x from point A at which an additional object also with the same weight Fg, can be hung without causing the rod to slip at point A.
1.86 m
o, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22 𝑥105 km. From these data, determine the mass of Jupiter in kg. Note: Convert days to seconds, km to meters, and use: 𝑅3𝑇2=𝐺𝑀4𝜋2, G = 6.673𝑥10−11𝑁⋅𝑚2𝑘𝑔2
1.90𝑥10^27𝑘𝑔
A student sits on a freely rotating stool holding two dumbbells, each of mass 3.00 kg as shown. When his arms are extended horizontally, the dumbbells are 1.00 m from the axis of rotation and the student rotates with an angular speed of 0.750 rad/s. The moment of inertia (rotational inertia) of the student plus stool is 3.00 kg · m2 and is assumed to be constant. The student pulls the dumbbells inward horizontally to a position 0.300 m from the rotation axis. What is the new angular speed of the student?
1.91 rad/s
Three objects of equal mass are located at three corners of a square of edge length 𝑙 , as shown. Find the magnitude of the gravitational field at the fourth corner (at the origin) due to these objects. Hint: Find the gravitational field in the x-direction, then in the y-direction and then use pythagorean's theorem to find the magnitude of the resultant.
1.91𝐺𝑚/𝑙^2
What is the ratio of v1/v2 of the speed of the comet at position 1 to the speed at position 2? The distance between the sun and position 1 is ten times longer than the distance between position 2 and the sun.
1/10
Three solid, uniform, cylindrical flywheels, each of mass 65.0 kg and radius 1.47 m, rotate independently around a common axis through their centers. Two of the flywheels rotate in one direction at 8.94 rad/s, but the other one rotates in the opposite direction at 3.42 rad/s. Calculate the magnitude of the net angular momentum of the system.
1020 kg • m2/s
A 50.0-g object connected to a spring with a force constant k of 35.0 N/m oscillates with an amplitude of 4.00 cm on a frictionless, horizontal surface. What is the kinetic energy when its position is at 3.00 cm?
12.2 mJ
13.2 Problems The period of a simple pendulum that is 1.00 m long on another planet is What is the acceleration due to gravity on this planet if the mass of the pendulum bob is 1.5 kg?
14.3 m/s2
Two moons orbit a planet in circular orbits. Moon A has orbital radius R, and moon B has orbital radius 4R. Moon A takes 20 days to complete one orbit. How long does it take moon B to complete an orbit?
160 days
A light rigid rod of length 𝑙=1.00𝑚 joins two particles, with masses 𝑚1=4.00𝑘𝑔𝑎𝑛𝑑𝑚2=3.00𝑘𝑔, at its ends. The combination rotates in the xy plane about a pivot through the center of the rod, as shown above. What is the angular momentum of the system, L, in 𝑘𝑔⋅𝑚𝑠2 about the origin when the speed of each particle is 5.00 m/s.
17.5 𝑘𝑔⋅𝑚𝑠2, upward
A mobile is shown in the figure. The horizontal supports have negligible mass. Assume that all the numbers given in the figure are accurate to two significant figures. What mass M is required to balance the mobile? Assume the two smallest masses each have a mass of 2 grams.
18 g
A rotating platform with a radius of 2.0 m makes one complete turn every 3.0 s. The angular velocity of the platform is most nearly
2.1 rad/s
In a certain cyclotron, a proton of mass 1.67x10-27 kg moves in a circle of diameter 1.6 m with an angular speed of 2.0 x 106 rad/s. What is the angular momentum of the proton?
2.1 x 10^-21 kg • m^2/s
To test the resiliency of its bumper during low-speed collisions, a 1,000-kg automobile is driven into a brick wall. The car's bumper behaves like a spring with a force constant 5.00 x106 N/m and compresses 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming no mechanical energy is transformed or transferred away during impact with the wall? Hint: Use the law of conservation of mechanical energy.
2.23 m/s
A block with a mass of 1.00 kg attached to a spring with a spring constant of 6.50 N/m and undergoes simple harmonic motion with an amplitude of 10.0 cm. When the block is halfway between its equilibrium position and the end point, its speed is measured to be 30.0 cm/s. What is the period of the motion? Hint: 𝑇=2𝜋𝜔=2𝜋𝑚𝑘‾‾√
2.46 s
A 200-kg object and a 500-kg object are separated by 4.00 m. Find the net gravitational force exerted by these objects on a 50.0-kg object placed midway between them.
2.50𝑥10^−7𝑁𝑡𝑜𝑤𝑎𝑟𝑑𝑡ℎ𝑒500𝑘𝑔𝑜𝑏𝑗𝑒𝑐𝑡.
If the earth were twice as far from the sun as it presently is, how long (in terms of the present year) would it take it to make one orbit around the sun?
2.8 years
The position of a particle is given by the expression x = 4.00cos(3.00𝜋𝑡+𝜋), where x is in meters and t is in seconds. What is the position of the particle at t = 0.250 s.
2.83 m
A disk is free to rotate about an axis perpendicular to the disk through its center. If the disk starts from rest and accelerates uniformly at the rate of 3 radians/s2 for 4 s, its angular displacement during this time is
24 radians
13.2 Problems A 0.250-kg stone is attached to an ideal spring and undergoes simple harmonic oscillations with a period of 0.640 s. What is the force constant (spring constant) of the spring?
24.1 N/m
A 0.60-kg block attached to a spring with force constant of 130 N/m is free to move on a frictionless, horizontal surface as in the diagram above. The block is released from rest when the spring is stretched 0.13 m. At the instant the block is released, What is its acceleration? Hint: 𝐹𝑠=−𝑘𝑥𝑎𝑛𝑑𝐹𝑠=𝑚𝑎
28 𝑚/𝑠^2 to the left
A triangular rod, shown above, has a length L, mass M, and a nonuniform linear mass density given by the equation λ = 2 M L 2 x, where x is the distance from one end of the rod. What is the rotational inertia of the rod about its left end. (Hint: λ = d m d x as well as the equation given above)
2ML^2/3
The angular displacement θ of a rotating wheel is described by the equation θ = θ 0 + at2 - bt3, where t is time and θ , a, and b are positive constants. The angular acceleration of the wheel as a function of time t is
2a - 6bt
A solid cylinder of mass m and radius R has a string wound around it. A person holding the string pulls it vertically upward, as shown, such that the cylinder is suspended in midair for a brief time interval of Δt and its center of mass does not move. The tension in the string is T, and the rotational inertia of the cylinder about its axis is 0.5mR2. The linear acceleration of the person's hand during the time interval Δt is
2g
Two blocks, one of mass M and one of mass 3M, are connected by a massless string over a pulley that is a uniform disk of mass 2M and moment of inertia MR2. The two masses are released from rest, and the masses accelerate as the pulley rotates. Assume there is negligible friction between the pulley and the axle. What is the linear acceleration, a, of the masses?
2g/5
The equation of motion of a simple harmonic oscillator is d2x/dt2 = -9x where x is displacement and t is time. The period of oscillation is
2𝜋/3
13.2 Problems The quartz crystal in a digital watch has a frequency of 32.8 kHz. What is its period of oscillation?
30.5 µs
An object weighing 300 N is suspended by means of two cords, as shown above. The tension in the horizontal cord is
300 N
A disk rotating at 5 rad/s experiences a constant torque for 4 seconds that results in an angular acceleration of 2 rad/s2. The angular displacement of the disk during those 4 seconds is most nearly
36 radians
A 1.0 kg mass is attached to the end of a vertical ideal spring with a force constant of 400 N/m. The mass is setin simple harmonic motion with an amplitude of 10 cm. The speed of the 1.0 kg mass at the equilibriumposition is(A) 2 m/s (B) 4 m/s (C) 20 m/s (D) 40 m/s (E) 200 m/s39. An object of mass m hanging from a spring of spring constant k oscillates with a certain frequency. What is thelength of a simple pendulum that has the same frequency of oscillation?(A) mk / g (B) mg / k (C) kg / m (D) k / mg (E) g / mk40. A platform of mass 2 kg is supported by a spring of negligible mass as shown.
38. correct answer is (A) From conservation of energy it shows: (1/2)kx^2 = (1/2)mv^2 => v=\sqrtkx2m=2m/s 39. correct answer is (B)
The position of a particle is given by the expression x = 4.00cos(3.00𝜋𝑡+𝜋), where x is in meters and t is in seconds. What is the amplitude of the motion?
4.00 m
How much energy (External Work) is required to move a 1,000-kg object from the Earth's surface to an altitude twice the Earth's radius? Hint: External Work = Δ𝑈,𝑈=−𝐺𝑀𝑚𝑟,𝑎𝑛𝑑𝑔=𝐺𝑀𝐸𝑅2𝐸⟹𝐺=𝑔𝑅2𝐸𝑀𝐸 𝑈𝑠𝑒𝑔=9.8𝑚𝑠2 and 𝑅𝐸=6.37𝑥106𝑚
4.17 x 10^10 J
A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as shown. A horizontal force F causes the spring to stretch a distance of 5.46 cm from its equilibrium position. What is the magnitude of the horizontal force, F?
4.58 N
A 7.00-kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object is set into vertical oscillations having a period of 2.60 s. What is the force constant k of the spring. Hint: Use 𝑇=2𝜋𝑚𝑘‾‾√
40.9 m/s
Satellite X moves around Earth in a circular orbit of radius R. Satellite Y is also in a circular orbit around Earth, and it completes one orbit for every eight orbits completed by satellite X. What is the orbital radius of satellite Y?
4R
A satellite is in an elliptical orbit about a planet, as shown in the figure above. At apogee, a distance R1 from the planet, the satellite's angular speed is 𝜔 . What is the angular speed of the satellite at perigee, a distance R1 = 2R2 from the planet?
4w
A frictionless pendulum of length of 3 m swings with an amplitude of 10o. At its maximum displacement, the potential energy of the pendulum is 10 J. What is the kinetic energy of the pendulum when its potential energy is 5 J?
5 J
An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 8.50 m/s2. The period of the satellite is
5621 seconds
A newly discovered planet is found to have twice the radius and five times the mass of Earth. If the acceleration of gravity at the surface of Earth is g, the acceleration of gravity at the surface of the new planet is
5g/4
A figure skater rotating at 5.00 rad/s with arms extended has a moment of inertia of 2.25 kg • m2. If he pulls in his arms so his moment of inertia decreases to 1.80 kg • m2, what will be his new angular speed?
6.25 rad/s
An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 9.00 𝑚𝑠2. What is the distance from the center of the satellite to the center of the Earth, R, in meters? Use G = 6.673𝑥10−11𝑁⋅𝑚2𝑘𝑔2 and 𝑀𝐸𝑎𝑟𝑡ℎ=5.97𝑥1024𝑘𝑔. Hint: use 𝑔=𝐺⋅𝑀𝐸𝑅2 and solve for R.
6.653𝑥10^6𝑚
In a supermarket, you place a 22.3-N (around 5 lb) bag of oranges on a scale, and the scale starts to oscillate at 2.7 Hz. What is the force constant (spring constant) of the spring of the scale?
650 N/m
A playground merry-go-round of radius R = 2.00 m has a moment of inertia (AKA rotational inertia) 𝐼=250𝑘𝑔⋅𝑚2 and is rotating at 10.0 rev/min about a frictionless, vertical axle. Facing the axle, a 25.0 kg child hops onto the merry-go-round and manages to sit down on the edge. What is the new angular speed of the merry-go-round?
7.14 rpm
In introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant G uses lead spheres with masses of 1.50 kg and 15.0 g whose centers are separated by about 4.50 cm. Calculate the gravitational force between these spheres, treating each as a particle located at the sphere's center. Hint: Make sure all the units agree.
7.41𝑥10^−10𝑁
An ice skater has a moment of inertia of 5.0 kg • m2 when her arms are outstretched, and at this time she is spinning at 3.0 rev/s. If she pulls in her arms and decreases her moment of inertia to 2.0 kg • m2, how fast will she be spinning?
7.5 rev/s
A 15-kg child is sitting on a playground teeter-totter, 1.5 m from the pivot. What is the magnitude of the minimum force, applied 0.30 m on the other side of the pivot, that is needed to make the child lift off the ground? Assume the teeter-totter has no mass and use g = 9.8 m/s2.
740 N
An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 9.00 𝑚𝑠2. Use G = 6.673𝑥10−11𝑁⋅𝑚2𝑘𝑔2 and 𝑀𝐸𝑎𝑟𝑡ℎ=5.97𝑥1024𝑘𝑔. Determine the orbital period, T, of the satellite in minutes. Hint: Use 𝐹𝑐=𝐹𝑔 and 𝑣=2𝜋𝑅𝑇
90.0 minutes
A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as in Figure P15.23. A horizontal force F S causes the spring to stretch a distance of 5.46 cm from its equilibrium position. (a) Find the magnitude of F S. (b) What is the total energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block just after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, would your answer to part (d) be larger or smaller? (f) What other information would you need to know to find the actual answer to part (d) in this case? (g) What is the largest value of the coefficient of friction that would allow the block to reach the equilibrium position?
A. 4.58 N B. 0.125 J C. 18.3 m/s2 D. 1 m/s E. smaller F. coefficient of kinetic friction G. 0.934
8.1 Conceptual Questions The figure shows scale drawings of four objects, each of the same mass and uniform thickness, with the mass distributed uniformly. Which one has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P?
B (wheel looking one)
A 200-kg object and a 500-kg object are separated by 4.00 m. At what position (other than an infinitely remote one) can the 50.0-kg object be placed so as to experience a net force of zero from the other two objects?
Between the objects, and 2.45 m from the 500 kg object.
Angular momentum is defined as 𝐼𝜔 in your text. It is also defined in your text as 𝑚𝑣𝑟. Which equation is correct?
Both are correct
The sphere-rod combination can be pivoted about an axis that is perpendicular to the plane of the page and that passes through one of the five lettered points. Through which point should the axis pass for the moment of inertia of the sphere-rod combination about this axis to be greatest?
E
Two small balls, A and B, attract each other gravitationally with a force of magnitude F. If we now double both masses and the separation of the balls, what will now be the magnitude of the attractive force on each one?
F
If F1 is the magnitude of the force exerted by the Earth on a satellite in orbit about the Earth and F2 is the magnitude of the force exerted by the satellite on the Earth, then which of the following is true?
F1 is equal to F2.
A solid disk of mass M, radius R, and rotational inertia I is free to rotate about a fixed frictionless axis that is perpendicular to the disk through its center, as shown above. A force of constant direction and constant magnitude F is exerted on the disk. If the disk accelerates from rest at time t = 0, through what angle θ does the disk rotate from t = 0 to t = T?
FRT2/(2I)
A motor drives a shaft of radius R/8 that is attached to the center of a wheel of radius R. The motor is turned off, and the force that the motor exerts on the shaft, Fmotor, varies with time as shown. There is also a constant frictional force of 0.4 N applied to the rim of the wheel in the opposite direction of the motion. During which time interval does the rotational kinetic energy increase and then decrease?
From t = 3 s and to t = 5 s
A disk sliding on a horizontal surface that has negligible friction collides with a rod that is free to move and rotate on the surface, as shown in the top view above. Which of the following quantities must be the same for the disk-rod system before and after the collision? I. Linear Momentum II. Angular Momentum III. Kinetic Energy
I and II only
A satellite of mass M moves in a circular orbit of radius R with constant speed v. True statements about this satellite include which of the following? I. Its angular speed is v/R II. Its tangential acceleration is zero III. The magnitude of its centripetal acceleration is constant
I, II, and III
A student is asked to determine the mass of Jupiter. Knowing which of the following about Jupiter and one of its moons will allow the determination to be made? I. The time it takes for Jupiter to orbit the Sun II. The time it takes for the moon to orbit Jupiter III. The average distance between the moon and Jupiter
II and III only
A comet moves in the Sun's gravitational field, following the path shown above. What happens to its angular momentum as it moves from point X to point Y?
It remains constant
A wheel with rotational inertia I is mounted on a fixed, frictionless axle. The angular speed 𝜔f in a time interval T. What is the average power input to the wheel during this time interval? What is the average power input to the wheel during this time interval?
I𝜔f^2/(2T)
A long board is free to slide on a sheet of frictionless ice. As shown in the top view above, a skater skates to the board and hops onto one end, causing the board to slide and rotate. In this situation, which of the following occurs?
Linear momentum and angular momentum are both conserved.
A wheel of mass M and radius R rolls on a level surface without slipping. If the angular velocity of the wheel is ω , what is its linear momentum?
MwR
When is the angular momentum of a system constant?
Only when no net external torque acts on the system.
What is required to have pure rolling motion?
Rolling friction
Given the graph above of angular acceleration versus time for a rigid body rotating about a fixed axis, which segment of the graph represents the time interval during which the largest magnitude of net torque is exerted on the rigid body?
Segment A
Four round objects of equal mass and radius roll without slipping along a horizontal surface that then bends upward and backward into an arc of a half circle. The objects all have the same linear speed initially. The objects are a hollow sphere, a solid cylinder, a solid sphere, and a hollow sphere. The objects go up the arc and exit the arc going in the opposite direction they entered without falling off the arc. Now, several trials are run for each object. For each trial, the initial speed of the object is reduced until the object does not make it through the full arc. The speed needed for each object to just make it through the arc is recorded. Which of the following correctly lists the objects in order from fastest to slowest speed needed to make it through the arc?
Solid sphere, solid cylinder, hollow sphere, hollow cylinder
In an experiment, students roll several hoops down the same incline plane. Each hoop has the same mass but a different radius. Each hoop rolls down the incline without slipping. Which of the following graphs best shows the linear speed v of the hoops at the bottom of the incline as a function of the radius r of the hoop?
Straight horizontal line
13.1 Conceptual Questions Two simple pendulums, A and B, are each 3.0 m long, and the period of pendulum A is T. Pendulum A is twice as heavy as pendulum B. What is the period of pendulum B?
T
A mass M suspended by a spring with force constant k has a period T when set into oscillation on Earth. Its period on Mars, whose mass is about 1/9 and radius 1/2 that of Earth, is most nearly
T
13.2 Problems On the Moon, the acceleration of gravity is g/6. If a pendulum has a period T on Earth, what will its period be on the Moon?
T(6)^1/2
If 𝜃1 < 𝜃2 and the mass of the stoplight is 15 kg, what can you determine about the tension in the string T2? (Stoplight)
T1 < T2
Two blocks of masses m1 and m2 are connected by a massless string that passess over a wheel of mass m, as shown above. The string does not slip on the wheel and exerts forces T1 and T2 on the blocks. When the wheel is released from rest in the position shown, it undergoes an angular acceleration and rotates clockwise. Which of the following statements about T1 and T2 is correct?
T2 is greater than T1 because an unbalanced clockwise torque is needed to accelerate the wheel clockwise.
A figure skater goes into a spin with arms fully extended. Which of the following describes the changes in the rotational kinetic energy and angular momentum of the skater as the skater's arms are brought toward the body?
The rotational kinetic energy increases and the angular momentum remains the same.
A sphere of mass M, radius r, and rotational inertia I is released from rest at the top of an inclined plane of height h as shown above. If the plane has friction so that the sphere rolls without slipping, what is the speed vcm of the center of mass at the bottom of the incline?
[(2Mghr^2)/(I +Mr^2)]^1/2
Planet X has a mass M, radius R, and no atmosphere. An object of mass m is located a distance 2R above the surface of planet X, as shown in the figure above. The object is released from rest and falls to the surface of the planet. What is the speed of the object just before it reaches the surface of planet X ?
[(4GM)/(3R)]^1/2
For the wheel-and-axle system shown, which of the following expresses the condition required for the system to be in static equilibrium?
am1 = bm2
The elliptical orbit of a comet is shown above. Positions 1 and 2 are, respectively, the farthest and nearest positions to the Sun, and at position 1 the distance from the comet to the Sun is 10 times that at position 2. At position 2, the comet's kinetic energy is
at its maximum value for the orbit.
A 2 kg mass connected to a spring oscillates on a horizontal, frictionless surface with simple harmonic motion of amplitude 0.4 m. The spring constant is 50 N/m. The maximum velocity occurs where the
displacement from equilibrium is equal to zero
A ball is tossed straight up from the surface of a small, spherical asteroid with no atmosphere. The ball rises to a height equal to the asteroid's radius and then falls straight down toward the surface of the asteroid. The acceleration of the ball at the top of its path is
equal to one-fourth the acceleration at the surface of the asteroid.
A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is g, is taken to another planet, where its period is 2 s. The acceleration due to gravity on the other planet is most nearly
g/4
A person sits on a freely spinning lab stool that has no friction in its axle. When this person extends her arms,
her moment of inertia increases and her angular speed decreases.
What is a proper unit for rotational inertia?
kgm^2
A satellite S is in an elliptical orbit around a planet P, as shown above, with r1 and r2 being its closest and farthest distances, respectively, from the center of the planet. If the satellite has a speed v1 at its closest distance, what is its speed at its farthest distance?
r1v1/r2
A simple pendulum consists of a l.0 kilogram brass bob on a string about 1.0 meter long. It has a period of 2.0 seconds. The pendulum would have a period of 1.0 second if the
string were replaced by one about 0.25 meter long.
Two forces produce equal torques on a door about the door hinge. The first force is applied at the midpoint of the door; the second force is applied at the doorknob. Both forces are applied perpendicular to the door. Which force has a greater magnitude?
the first force (at the midpoint)
A merry-go-round spins freely when Diego moves quickly to the center along a radius of the merry-go-round. As he does this, it is true to say that
the moment of inertia of the system decreases and the angular speed increases.
A small uniform disk and a small uniform sphere are released simultaneously at the top of a high inclined plane, and they roll down without slipping. Which one will reach the bottom first?
the sphere
A can of condensed mushroom soup is placed on its side at the top of an inclined plane and allowed to roll down the plane. What is the direction of rolling friction acting on the can as it rolls down the incline? What would be the direction of rolling friction if it rolled up the incline?
up the incline; up the incline
Two identical starts, a fixed distance D apart, revolve in a circle about their mutual center of mass, as shown. Each star has a mass M and speed v. G is the universal gravitational constant. Which of the following is a correct relationship among these quantities?
v2 = GM/2D
A solid cylinder of mass m and radius R has a string wound around it. A person holding the string pulls it vertically upward, as shown, such that the cylinder is suspended in midair for a brief time interval of Δt and its center of mass does not move. The tension in the string is T, and the rotational inertia of the cylinder about its axis is 0.5mR2. The net force on the cylinder during the time interval Δt is
zero
A student sits on a freely rotating stool holding two dumbbells, each of mass 3.00 kg as shown. When his arms are extended horizontally, the dumbbells are 1.00 m from the axis of rotation and the student rotates with an angular speed of 0.750 rad/s. The moment of inertia (rotational inertia) of the student plus stool is 3.00 kg · m2 and is assumed to be constant. The student pulls the dumbbells inward horizontally to a position 0.300 m from the rotation axis. What is the kinetic energy of the rotating system before and after he pulls the dumbbells inward?
𝐾𝑖=2.53𝐽,𝐾𝑓=6.44𝐽