Ch 8 - Rotational Motion and Torque

All of the following objects rotate around the same axis. Do any of them have the same rotational kinetic energy K=12I2? If so, which ones? Explain your reasoning. (i) sphere with moment of inertia 4.0 kg • m2 rotating clockwise at 2.0 rad/s (ii) cylinder with moment of inertia 2.0 kg • m2 rotating clockwise at 4.0 rad/s (iii) disk with moment of inertia 4.0 kg • m2 rotating counterclockwise at 4.0 rad/s (iv) hoop with moment of inertia 8.0 kg • m2 rotating counterclockwise at 2.0 rad/s

(ii) and (iv) have the same angular kinetic energy because and both equal 16 J [1/2 times 2 times 42 for (ii) and 1/2 times 8 times 22 for (iv)].

Three point objects with mass m1 = 1.4 kg, m2 = 2.8 kg, and m3 = 1.8 kg are arranged in the configuration shown in the figure. The distance to mass m1 is d1 = 28 cm and the distance mass m3 is d3 = 41 cm. The distances are measured from the axis O. What is the combined moment of inertia I for the three point objects about the axis O?

I = 0.41 kg x m^2

A uniform rod of mass M = 3.61 kg pivots about an axis through its center and perpendicular to its length. Two small bodies, each of mass m = 0.299 kg, are attached to the ends of the rod. What must the length of L of the rod be so that the moment of inertia of the three body system with respect to the described axis is I = 0.865 kg x m^2.

L = 1.39 m

A child's top is held in place upright on a frictionless surface. The axle has a radius of r = 2.96 mm. Two strings are wrapped around the axle, and the top is set spinning by applying T = 3.15 N of constant tension to each string. If it takes 0.470 s for the string to unwind, how much angular momentum L does the top acquire? Assume the strings do not slip as tension is applied. Point P is located on the outer surface of the top, a distance h=27.0 mm above the ground. The angle that the outer surface of the top makes with the rotation axis of the top is theta = 24 degrees. If the final tangential speed vt of point P is 1.75v m/s, what is the top's moment of inertia I?

L = 8.76 x 10^-3 kgxm^2/s I = 6.02 x 10 ^-5 kg x m^2

The following objects all rotate around the same axis, and all have the same rotational kinetic energy. Rank them in order of their angular speed, from fastest speed to slowest speed. If any two objects have the same angular speed, state this. Explain how you made your ranking. (i) ball with moment of inertia 4.0 kg • m2 (ii) cylinder with moment of inertia 2.0 kg • m2 (iii) disk with moment of inertia 3.0 kg • m2 (iv) hoop with moment of inertia 5.0 kg • m2

Since all the objects have the same angular kinetic energy, which is , the objects with the smallest moment of inertia have the largest angular speed. Thus the order of angular speeds is (fastest) (ii) > (iii) > (i) > (iv) (slowest). No speeds are equal because no moments of inertia are equal.

After the object in part (a) has descended a vertical distance h, it has kinetic energy due to translation as well as kinetic energy due to rotation around its center of mass. On which of the following does the ratio of rotational kinetic energy to translational kinetic energy depend? (There may be more than one correct answer.) Explain your answers. (i) mass of the object (ii) radius of the object (iii) angle of the ramp (iv) how the mass of the object is distributed within its volume (v) value of h

The kinetic energy will have two parts, and . Let us assume that , where R is the radius and is a factor that depends on the distribution of the mass. Since for rolling without slipping , the ratio of the rotational kinetic energy to translational kinetic energy depends only on , which is (iv), and nothing else.

A solid object starts at rest at the top of a ramp. It rolls without slipping down the ramp. After it has descended a vertical distance h, the object has a certain speed. On which of the following does this speed depend? (There may be more than one correct answer.) Explain your answers. (i) mass of the object (ii) radius of the object (iii) angle of the ramp (iv) how the mass of the object is distributed within its volume (v) value of h

The kinetic energy will have two parts, and . Let us assume that , where R is the radius and is a factor that depends on the distribution of the mass. Since for rolling without slipping , the total energy of the rolling object is . This will equal the change in potential energy = mgh. So the speed, V, will depend only on (iv) and (v).

How would a spinning disk's kinetic energy change if its moment of inertia was five times larger but its angular speed was five times smaller? a) 0.2 times as large as before b) 5 times as large as before c) 10 times as large as before d) 0.1 times as large as before e) the same as before

a)

(a) Rank the four forces in order of the magnitude of the torque they produce around axis of rotation 1, from greatest to smallest. If two or more forces produce torques of the same magnitude, state which ones. Explain how you made your ranking. b) Rank the four forces in order of the magnitude of the torque they produce around axis of rotation 2, from greatest to smallest. If two or more forces produce torques of the same magnitude, state which ones. Explain how you made your ranking.

a) (Greatest torque) (smallest torque) F3 has the greatest moment arm around axis 1 and the force is perpendicular to the moment arm. F2 is at an angle less than 90˚ for the moment arm and so produces less torque than F1, and F4 produces zero torque. b) (Greatest torque) F1 > F1 > F3 = F4 (smallest torque) F1 has the greatest moment arm around axis 2 and the force is perpendicular to the moment arm. F2 is at an angle less than 90˚ for the same moment arm and so produces less torque than F1, and F3 and F4 produce zero torque since they are at the rotation axis.

In both cases shown, a hula hoop with mass M and radius R is spun with the same angular velocity w about a vertical axis through its center. In case 1, the plane of the hoop is perpendicular to the rotation axis, and in Case 2, it is parallel. In which case doe the spinning hoop have the most kinetic energy? a) Case 1 b) Case 2 c) same in both cases

a) All points of mass are all radius R away from the rotational axis, whereas in case 2, most of the hoop's mass is less than radius R away from the rotational axis. Thus, case 1 will have the greatest moment of inertia, and thus have the most kinetic energy.

A student sitting at rest on a chair that is free to rotate holds a spinning bicycle wheel that rotates in the horizontal plane, shown in Figure A. When the student flips the bicycle wheel over, he will a) spin in the opposite direction as the flipped bicycle wheel b) spin in the same direction as the flipped bicycle wheel c) not spin at all

a) Angular momentum should be conserved (momentum of the system is always zero, thus if the bicycle wheel now has a negative angular momentum, the student must rotate in the opposite direction to have a positive momentum to cancel out that of the bicycle wheel)

A disk rotates freely around a vertical axis. Initially it is rotating at a constant angular speed. As the disk rotates you drop a ball from rest so that it falls a short distance and lands on a point near the rim of the disk, as shown. The ball is covered with glue, so it sticks to the surface of the rotating disk. The rotating disk is on top of a frictionless table (not shown), so it remains at the same vertical position after the ball lands on it. (a) After the ball lands on the disk, has the angular speed of the disk increased, decreased, or remained the same? Explain your answer. (b) After the ball lands on the disk, has the angular momentum around the axis of rotation of the system of disk and ball increased, decreased, or remained the same? Explain your answer. c) After the ball lands on the disk, has the total kinetic energy of the system of disk and ball increased, decreased, or remained the same? Explain your answer.

a) The angular speed of the disk decreased because the moment of inertia of the ball-disk system is greater than that of the disk, but the angular momentum of the ball-disk system after the collision must equal the angular momentum of the disk before the collision. b) The angular momentum of the ball-disk system after the collision is the same as the angular momentum of the disk before the collision because there was no external torque on the system. c) The total kinetic energy decreased. Conservation of momentum gives , so the final kinetic energy is , which is less than the initial kinetic energy by the factor .

A hoop, a solid disk, and a solid sphere, all with the same mass and same radius, are set rolling without slipping up an incline, all with the same initial kinetic energy. Which goes furthest up the incline? a) the hoop b) the disk c) the sphere d) they all roll to the same height The same three objects are set rolling without slipping up an incline, all with the same initial linear speed. Which goes furthest up the incline. a) the hoop b) the solid disk c) the solid sphere d) they all roll to the same height

d) Conservation of energy (has the same kinetic energy, all kinetic energy will be expended to potential energy, which will translate to the same height) a) the hoop (apply the conservation of energy, use inertia equations for each object)

Suppose that there are two solid steel spheres. The second sphere has a radius twice as large as the radius of the first sphere. What is the ratio of the moments of inertia of the spheres, I2 : I1? a) 4 b) 16 c) 8 d) 2 e) 32

e)

The bones of the forearm (radius and ulna) are hinged to the humerus at the elbow. The biceps muscle connects to the bones of the forearm about 2.15 cm beyond the joint. Assume the forearm has a mass of 2.45kg and a length of 0.465 m. When the humerus and the biceps are nearly vertical and the forearm is horizontal, if a person wishes to hold an object of mass 7.75 kg so that her forearm remains motionless, what is the force exerted by the biceps muscle?

force = 1904 N

A student holds a bike wheel and starts it spinning with an initial angular speed of 9.0 rotations per second. The wheel is subject to some friction, so it gradually slows down. In the 10.0s period following the initial spin, the bike wheel undergoes 80.0 complete rotations. Assuming the frictional torque remains constant, how much more time will it take the bike wheel to come to a complete stop? The bike wheel has a mass of 0.625 kg and a radius of 0.385 m. If all the mass of the wheel is assumed to be located on the rim, find the magnitude of the frictional torque Tf that was acting on the spinning wheel.

t = 35 s Tf = 0.116 Nm