A size-5 soccer ball of diameter 22.6 cm and mass 426 g rolls up a hill without slipping, reaching a maximum height of 6.50 m above the base of the hill. We can model this ball as a thin-walled hollow sphere.

a) At what rate was it rotating at the base of the hill?

b) How much rotational kinetic energy did it then have?

a) At what rate was it rotating at the base of the hill?

b) How much rotational kinetic energy did it then have?

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A thin, rectangular sheet of metal has mass M and sides of length a and b. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

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If we multiply all design dimensions of an object by a scaling factor f, its volume and mass will be multiplied by f^{3}.

a) By what factor will its moment of inertia be multiplied?

b) If a (1/48)-scale model has a rotational kinetic energy of 2.5 J, what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

a) By what factor will its moment of inertia be multiplied?

b) If a (1/48)-scale model has a rotational kinetic energy of 2.5 J, what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

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A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to θ(t)= γt+ βt^{3}, where γ = 0.406 rad/s and β = 1.30×10^{−2} rad/s^{3}.

a) Calculate the angular velocity of the merry-go-round as a function of time.

b) What is the initial value of the angular velocity?

c) Calculate the instantaneous value of the angular velocity ω_{z} at t = 5.05 s .

d) Calculate the average angular velocity ω_{av - z} for the time interval t = 0 to t = 5.05 s .

a) Calculate the angular velocity of the merry-go-round as a function of time.

b) What is the initial value of the angular velocity?

c) Calculate the instantaneous value of the angular velocity ω

d) Calculate the average angular velocity ω

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The flywheel of a gasoline engine is required to give up 600 J of kinetic energy while its angular velocity decreases from 780 rev/min to 510 rev/min. What moment of inertia is required?

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A wagon wheel is constructed as shown in the figure. The radius of the wheel is 0.300 m, and the rim has mass 1.41 kg. Each of the eight spokes, that lie along a diameter and are 0.300 m long, has mass 0.260 kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel?

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Calculate the moment of inertia of each of the following uniform objects about the axes indicated.

a) A thin 3.10-kg rod of length 80.0 cm, about an axis perpendicular to it and passing through its center.

b) A thin 3.10-kg rod of length 80.0 cm, about an axis perpendicular to it and passing through one end.

c) A thin 3.10-kg rod of length 80.0 cm, about an axis parallel to the rod and passing through it.

d) A 4.00-kg sphere 31.0 cm in diameter, about an axis through its center, if the sphere is solid.

e) A 4.00-kg sphere 31.0 cm in diameter, about an axis through its center, if the sphere is a thin-walled hollow shell.

f) An 6.00-kg cylinder, of length 21.0 cm and diameter 22.0 cm , about the central axis of the cylinder, if the cylinder is solid.

g) An 6.00-kg cylinder, of length 21.0 cm and diameter 22.0 cm, about the central axis of the cylinder, if the cylinder is thin-walled and hollow.

a) A thin 3.10-kg rod of length 80.0 cm, about an axis perpendicular to it and passing through its center.

b) A thin 3.10-kg rod of length 80.0 cm, about an axis perpendicular to it and passing through one end.

c) A thin 3.10-kg rod of length 80.0 cm, about an axis parallel to the rod and passing through it.

d) A 4.00-kg sphere 31.0 cm in diameter, about an axis through its center, if the sphere is solid.

e) A 4.00-kg sphere 31.0 cm in diameter, about an axis through its center, if the sphere is a thin-walled hollow shell.

f) An 6.00-kg cylinder, of length 21.0 cm and diameter 22.0 cm , about the central axis of the cylinder, if the cylinder is solid.

g) An 6.00-kg cylinder, of length 21.0 cm and diameter 22.0 cm, about the central axis of the cylinder, if the cylinder is thin-walled and hollow.

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A basketball (which can be closely modeled as a hollow spherical shell) rolls down a mountainside into a valley and then up the opposite side, starting from rest at a height H_{0} above the bottom. In the figure, the rough part of the terrain prevents slipping while the smooth part has no friction.

a) How high, in terms of H_{0}, will it go up the other side?

b) Why doesn't the ball return to height H_{0}? Has it lost any of its original potential energy?

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A string is wrapped several times around the rim of a small hoop with radius 8.00 cm and mass 0.180 kg. The free end of the string is held in place and the hoop is released from rest (the figure). After the hoop has descended 75.0 cm , calculate

a) the angular speed of the rotating hoop and

b) the speed of its center

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Biomedical measurements show that the arms and hands together typically make up 13.0 % of a persons mass, while the legs and feet together account for 37.0 % . For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. Let us consider a 76.0 kg person having arms 68.0 cm long and legs 93.0 cm long. The person is running at 12.0 km/h , with his arms and legs each swinging through 30**°** in 1/2 s. Assume that the arms and legs are kept straight.

a) What is the average angular velocity of his arms and legs?

b) Calculate the amount of rotational kinetic energy in this persons arms and legs as he walks.

c) What is the total kinetic energy due to both his forward motion and his rotation?

d) What percentage of his kinetic energy is due to the rotation of his legs and arms?

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A sphere of radius = 23.0 cm and mass m = 1.20 kg starts from rest and rolls without slipping down a 36.0 degree incline that is 13.0 m long.

a) Calculate its translational speed when it reaches the bottom.

b) Calculate its rotational speed when it reaches the bottom.

c) What is the ratio of translational to rotational kinetic energy at the bottom?

d) Does your answer in part A depend on mass or radius of the ball? Part B? Part C?

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A bowling ball of mass 7.6 kg and radius 9.0 cm rolls without slipping down a lane at 3.6 m/s. Calculate its total kinetic energy.

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Calculate the translational speed of a cylinder when it reaches the foot of an incline 7.30 m high. Assume it starts from rest and rolls without slipping.

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A roller in a printing press turns through an angle θ(t) given by θ(t) = γt^{2} - βt^{3} , where γ = 3.20 rad/s^{2} and β = 0.500 rad/s^{3}.

a) Calculate the angular velocity of the roller as a function of time.

b) Calculate the angular acceleration of the roller as a function of time.

c) What is the maximum positive angular velocity?

d) At what value of t does it occur?

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A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [*Hint*: Use conservation of energy.]

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A merry-go-round has a mass of 1550 kg and a radius of 7.60 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 9.00 s ? Assume it is a solid cylinder.

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Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass and radius about an axis perpendicular to the hoops plane at an edge.

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A hollow spherical shell has mass 8.20 kg and radius 0.225 m . It is initially at rest and then rotates about a stationary axis that lies along a diameter with a constant acceleration of 0.895 rad/s^{2} . What is the kinetic energy of the shell after it has turned through 6.25 rev ?

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A fan blade rotates with angular velocity given by ω_{z}(t) = γ - βt^{2}.

a) Calculate the angular acceleration as a function of time.

b) If γ = 5.05 rad/s and β = 0.805 rad/s^{3} , calculate the instantaneous angular acceleration α_{z} at t = 3.10 s .

c) If γ = 5.05 rad/s and β = 0.805 rad/s^{3} , calculate the average angular acceleration α_{av - z} for the time interval t = 0 to t = 3.10 s .

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Small blocks, each with mass * m *, are clamped at the ends and at the center of a rod of length * L * and negligible mass. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through:

a) the center of the rod

b) a point one-fourth of the length from one end

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An airplane propeller is rotating at 1910 rev/min.

a) Compute the propellers angular velocity in rad/s.

b) How long in seconds does it take for the propeller to turn through 36°?

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At t=0 a grinding wheel has an angular velocity of 26.0 rad/s . It has a constant angular acceleration of 31.0 rad/s^{2} until a circuit breaker trips at time t = 2.10 s. From then on, it turns through an angle 436 rad as it coasts to a stop at constant angular acceleration.

a) Through what total angle did the wheel turn between t=0 and the time it stopped?

b) At what time did it stop?

c) What was its acceleration as it slowed down?

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The angular acceleration of a wheel, as a function of time, is α = 5.0 t^{2} - 8.5 t, where α is in rad/s^{2} and t is in seconds. If the wheel starts from rest (θ = 0, ω = 0, at t = 0), determine a formula for

a) the angular velocity ω as a function of time

b) the angular position θ as a function of time

c) evaluate ω at t = 4.0 s

d) evaluate θ at t = 4.0 s

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A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 6.00 kg , while the balls each have mass 0.300 kg and can be treated as point masses. Find the moment of inertia of this combination about an axis:

a) perpendicular to the bar through its center

b) perpendicular to the bar through one of the balls

c) parallel to the bar through both balls

d) parallel to the bar and 0.500 m from it

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a) Calculate the angular velocity of the second hand of a clock. State in rad/s.

b) Calculate the angular velocity of the minute hand of a clock. State in rad/s.

c) Calculate the angular velocity of the hour hand of a clock. State in rad/s.

d) What is the angular acceleration in each case?

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