Rotation Test Review
In the figure, a given force F is applied to a rod in several different ways. In which case is the torque due to F about the pivot P greatest? (1 shows F applied perpendicularly at the end of the rod, 2 shows F applied perpendicularly at the middle of the rod, 3 shows F applied at an angle at the end of the rod, 4 shows F applied at a smaller angle than 3 at the end of the rod, 5 shows F applied parallel to the rod at the end of the rod) a. 1 b. 4 c. 5 d. 2 e. 3
a. 1 In this case, T = F^r^sin(90) = F*r. In the other cases, either r is too small or the angle is not 90. (The sin of these angles would yield a fraction)
A heavy bank-vault door is opened by the application of a force of 3.0 * 10^2 N directed perpendicular to the plane of the door at a distance of 0.80 m from the hinges. What is the torque? a. 240 N·m b. 120 N·m c. 360 N·m d. 300 N·m
a. 240 N·m T = F*r*sin(theta) = 3.0 * 10^2 * 0.80 * sin(90) = 240 N·m
If you cannot exert enough force to loosen a bolt with a wrench, which of the following should you do? a. Use a wrench with a longer handle. b. Tie a rope to the end of the wrench and pull on the rope. c. You should exert a force on the wrench closer to the bolt. d. Use a wrench with a shorter handle.
a. Use a wrench with a longer handle. T = F*r*sin(theta). Using a wrench with a longer handle will increase r (the distance from the axis of rotation) and therefore increase the applied torque.
If you want to open a swinging door with the least amount of force, where should you push on the door? a. as far from the hinges as possible b. in the middle c. close to the hinges d. It does not matter where you push.
a. as far from the hinges as possible T = F*r*sin(theta). Pushing the door as far as possible from the hinges will increase the radiu and reduce the amount of force that needs to be applied.
Where should a force be applied on a lever arm to produce the most torque? a. farthest from the axis of rotation b. It doesn't matter where the force is applied. c. closest to the axis of rotation d. in the middle of the lever arm
a. farthest from the axis of rotation The farther the force is applied from the AOR, the greater the r and greater the torque.
A spinning ice skater is able to control the rate at which she rotates by pulling in her arms. We can best understand this effect by observing that in this process: a. her angular momentum remains constant. b. her moment of inertia remains constant. c. her total velocity remains constant. d. her kinetic energy remains constant. e. she is subject to a constant non-zero torque.
a. her angular momentum remains constant. By pulling in her arms or extending them, the ice skater decreases/increases her moment of inertia, causing the rate at which she rotates (angular velocity) to change accordingly. The rate changes in order to keep the angular momentum constant.
If the planet Jupiter underwent gravitational collapse, its rate of rotation about its axis would a. increase. b. more information needed c. decrease. d. stay the same.
a. increase. L = Iω Since the radius decreases, the moment of inertia also decreases. Therefore, the angular speed (rate of rotation) will increase.
Two equal forces are applied to a door. 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 exerts the greater torque? a. the second at the doorknob b. both exert equal non-zero torques c. both exert zero torques d. additional information is needed e. the first at the midpoint
a. the second at the doorknob Torque exerted by force at the midpoint = F*r*sin(theta) Torque exerted by force at the doorknob = F*2r*sin(theta) Torque 2 > Torque 1
A disk and a (solid) sphere are released simultaneously at the top of an inclined plane. They roll down without slipping. Which will reach the bottom first? a. the sphere b. the one of smallest diameter c. the hoop d. They will reach the bottom at the same time. e. the one of greatest mass
a. the sphere Inertia of the disk (Id) = (1/2)(MR^2) Inertia of the sphere (Is) = (2/5)(MR^2) Is < Id, so the sphere will reach the bottom quicker.
Suppose a doorknob is placed at the center of a door. Compared with a door whose knob is located at the edge, what amount of force must be applied to this door to produce the torque exerted on the other door? a. two times as much b. one-half as much c. one-fourth as much d. four times as much
a. two times as much Torque when the doorknob is at the end = F*r*sin(theta) Torque when the doorknob is in the middle = F*(r/2)*sin(theta) Since the torque produced when the doorknob is in the middle is 1/2 of the torque produced when the doorknob is at the end, the F applied must be two times as much to compensate.
Two balls, one of radius R and mass M, the other of radius 2R and mass 8M, roll down an incline. They start together from rest at the top of the incline. Which one will reach the bottom of the incline first? a. It depends on the height of the incline. b. Both reach the bottom together. c. The small sphere d. The large sphere e. It depends on the length of the inclined surface.
b. Both reach the bottom together.
An ice skater doing a spin pulls in her arms, decreasing her moment of inertia by a factor of two. How does her angular speed change? a. It is reduced by a factor of two. b. It increases by a factor of two. c. It increases by a factor of four. d. It is reduced by a factor of four. e. It doesn't change.
b. It increases by a factor of two. L = Iω Initial Inertia = I Final Inertia = I/2 Therefore, the angular speed has to increase by a factor of 2 to keep L constant.
A disk and a hoop of the same mass and radius are released at the same time at the top of an inclined plane. Which object reaches the bottom of the incline first? a. It depends on the length of the inclined surface. b. The disk c. Both reach the bottom at the same time. d. The hoop e. It depends on the angle of inclination.
b. The disk Inertia of the hoop (Ih) = MR^2 Inertia of the disk (Id) = Inertia of a solid cylinder = (1/2)MR^2 Id < Ih, so the disk will reach the bottom quicker.
Consider a solid sphere of radius R and mass M rolling without slipping. Which form of kinetic energy is larger, translational or rotational? a. You need to know the speed of the sphere to tell. b. Translational kinetic energy is larger. c. Rotational kinetic energy is larger. d. Both are equal. e. You need to know the acceleration of the sphere to tell.
b. Translational kinetic energy is larger. Translational kinetic energy = (1/2)mv^2 Rotational kinetic energy = (1/2)((2/5)MR^2)(v^2/r^2) = (1/5)mv^2
The point around which an object naturally spins is called its: a. orbit. b. center of mass. c. satellite. d. trajectory.
b. center of mass.
A uniform solid sphere has mass M and radius R. If these are increased to 2M and 3R, what happens to the sphere's moment of inertia about a central axis? a. increases by a factor of 5 b. increases by a factor of 18 c. increases by a factor of 12 d. increases by a factor of 6 e. increases by a factor of 54
b. increases by a factor of 18 Initial Inertia = (2/5)MR^2 Changed Inertia = (2/5)(2M)(3R^2) = (36/5)MR^2 Factor of change = (36/5)/(2/5) = 18
A solid sphere (I = (2/5)MR^2) and a solid cylinder (I = (1/2)MR^2) of the same mass and radius roll without slipping at the same speed. It is correct to say that the total kinetic energy of the solid sphere is a. more than the total kinetic energy of the cylinder. b. less than the total kinetic energy of the cylinder. c. impossible to compare to the total kinetic energy of the cylinder. d. equal to the total kinetic energy of the cylinder.
b. less than the total kinetic energy of the cylinder. Total KE of the sphere = Rotational KE + Translational KE = (1/2)Iω^2 + (1/2)mv^2 = (1/2)((2/5)MR^2)(v^2/R^2) + (1/2)mv^2 = (1/5)(mv^2) + (1/2)(mv^2) = (7/10)mv^2 Total KE of the cylinder = Rotational KE + Translational KE = (1/2)Iω^2 + (1/2)mv^2 = (1/2)((1/2)MR^2)(v^2/R^2) + (1/2)mv^2 = (1/4)(mv^2) + (1/2)(mv^2) = (3/4)mv^2 KE of the cylinder > KE of the sphere
Suppose a solid sphere of mass M and radius R rolls without slipping down an inclined plane starting from rest. The linear velocity of the sphere at the bottom of the incline depends on a. both the mass and the radius of the sphere. b. neither the mass nor the radius of the sphere. c. the mass of the sphere. d. the radius of the sphere.
b. neither the mass nor the radius of the sphere.
A huge rotating cloud of particles in space gravitate together to form an increasingly dense ball. As it shrinks in size, the cloud a. cannot rotate. b. rotates faster. c. rotates at the same speed. d. rotates slower.
b. rotates faster. L = Iω = (MR^2)ω Since the size (radius) of the cloud decreased, the inertia also decreased. Therefore, the cloud's angular speed must increase (it must rotate faster) to conserve angular momentum.
Consider two uniform solid spheres where both have the same diameter, but one has twice the mass of the other. The ratio of the larger moment of inertia to that of the smaller moment of inertia is a. 6. b. 8. c. 2. d. 12. e. 10.
c. 2. Inertia of the lighter sphere (I1) = (2/5)(MR^2) Inertia of the heavier sphere (I2) = (2/5)(2MR^2) = (4/5)(MR^2) Ratio of I2 to I1 = (4/5)/(2/5) = 2
Two forces are applied to a doorknob, perpendicular to the door. The first force is twice as large as the second force. The ratio of the torque of the first to the torque of the second is a. 8. b. 1/4. c. 2. d. 4. e. 1/2.
c. 2. Torque exerted by the first force = 2F*r*sin(theta) Torque exerted by the second force = F*r*sin(theta) Ratio of 1 to 2 = 2:1 = 2
If the torque required to loosen a nut on a wheel has a magnitude of 40.0 N·m and the force exerted by a mechanic is 133 N, how far from the nut must the mechanic apply the force? a. 1.20 m b. 60.2 cm c. 30.1 cm d. 15.0 cm
c. 30.1 cm T = F*r*sin(theta) 40.0 = 133 * r * sin(90) r = 40.0/133 = 0.301 m = 30.1 cm
In the figure are scale drawings of four objects, each of the same mass and uniform thickness. Which has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing? In each case the axis passes through point P. a. A b. C c. B d. The moment of inertia is the same for all of these objects. e. D
c. B Inertia of A: (2/5)(MR^2) Inertia of B: M*(2R)^2 = 4(MR^2) Inertia of C: (2/5)(M(2R^2)) = (8/5)(MR^2) Inertia of D: (1/12)(M(2R^2)) = (1/3)(MR^2)
An ice skater performs a pirouette (a fast spin) by pulling in his outstretched arms close to his body. What happens to his angular momentum about the axis of rotation? a. It decreases. b. It changes, but it is impossible to tell which way. c. It does not change. d. It increases.
c. It does not change. While the moment of inertia does decrease, the angular speed also increases to compensate, thus keeping L constant.
Suppose that a heavy person and a light person are balanced on a teeter-totter made of a plank of wood. Each person now moves in toward the fulcrum a distance of 25 cms. What effect will this have on the balance of the teeter-totter? a. Only if the plank has significant mass will the light person's end go down. b. The heavy person's end will go down. c. The light person's end will go down. d. One cannot tell whether either end will rise or fall without knowing the relative mass of the plank. e. The teeter-totter will remain in balance.
c. The light person's end will go down.
A croquet mallet balances when suspended from its center of mass, as shown in Figure 11-2. If you cut the mallet in two at its center of mass, as shown, how do the masses of the two pieces compare? a. The masses are equal. b. The piece with the head of the mallet has the smaller mass. c. The piece with the head of the mallet has the greater mass. d. It is impossible to tell.
c. The piece with the head of the mallet has the greater mass. The center of mass is located closer to the left. Therefore, the left piece must have the greater mass.
A figure skater is spinning slowly with arms outstretched. She brings her arms in close to her body and her angular speed increases dramatically. The speed increase is a demonstration of a. Newton's second law for rotational motion: she exerts a torque and so her angular speed increases. b. This has nothing to do with mechanics, it is simply a result of her natural ability to perform. c. conservation of angular momentum: her moment of inertia is decreased, and so her angular speed must increase to conserve angular momentum. d. conservation of energy: her moment of inertia is decreased, and so her angular speed must increase to conserve energy.
c. conservation of angular momentum: her moment of inertia is decreased, and so her angular speed must increase to conserve angular momentum.
A person sits on a freely spinning lab stool (no friction). When this person extends her arms, a. her moment of inertia decreases and her angular velocity increases. b. her moment of inertia increases and her angular velocity increases. c. her moment of inertia increases and her angular velocity decreases. d. her moment of inertia decreases and her angular velocity decreases. e. her moment of inertia increases and her angular velocity remains the same.
c. her moment of inertia increases and her angular velocity decreases. L = Iω The "radius" increases because the person extends their arms. This increases the moment of inertia and decreases the angular velocity.
An ice skater doing a spin pulls in her arms, decreasing her moment of inertia by a factor of two, and doubling her angular speed. Her final rotational kinetic energy a. is cut in fourth. b. is cut in half. c. is doubled. d. quadruples. e. remains unchanged.
c. is doubled. Initial Rotational KE = (1/2)Iω^2 Final Rotational KE = (1/2)(I/2)(2ω)^2 = Iω^2
"The total angular momentum of a system of particles changes when a net external force acts on the system." This statement is a. sometimes true. It depends on the force's magnitude. b. always true. c. sometimes true. It depends on the force's point of application. d. never true.
c. sometimes true. It depends on the force's point of application.
A solid sphere, solid cylinder, and a hollow pipe all have equal masses and radii. If the three are released simultaneously at the top of an inclined plane, which will reach the bottom first? a. pipe b. cylinder c. sphere d. It depends on the angle of inclination. e. they all reach bottom in the same time
c. sphere Inertia of the solid sphere = (2/5)MR^2 Inertia of the solid cylinder = (1/2)MR^2 Inertia of the hollow pipe = MR^2 The inertia of the sphere is the smallest. Therefore, it will reach first.
A disk, a hoop, and a solid sphere are released at the same time at the top of an inclined plane. They all roll without slipping. In what order do they reach the bottom? a. hoop, disk, sphere b. disk, hoop, sphere c. sphere, disk, hoop d. hoop, sphere, disk e. hoop, sphere, disk
c. sphere, disk, hoop Inertia of the disk (Id) = (1/2)(MR^2) Inertia of the sphere (Is) = (2/5)(MR^2) Inertia of the hoop (Ih) = MR^2 Order in which the objects reach the bottom = Order of inertia from least to greatest = Is < Id < Ih
A merry-go-round spins freely when Janice moves quickly to the center along a radius of the merry-go-round. It is true to say that a. the moment of inertia of the system decreases and the angular speed decreases. b. the moment of inertia of the system increases and the angular speed decreases. c. the moment of inertia of the system decreases and the angular speed increases. d. the moment of inertia of the system decreases and the angular speed remains the same. e. the moment of inertia of the system increases and the angular speed increases.
c. the moment of inertia of the system decreases and the angular speed increases. L = Iω The radius decreases because Janice moves to the center. Therefore, the moment of inertia decreases and the angular speed increases to compensate.
Which of the following quantities measures the ability of a force to rotate or accelerate an object around an axis? a. lever arm b. axis of rotation c. torque d. tangential force
c. torque
What condition or conditions are necessary for static equilibrium? a. ΣFx = 0, ΣFy = 0 b. ΣΤ = 0 c. ΣFx = 0, ΣFy = 0, ΣΤ = 0 d. ΣFx = 0
c. ΣFx = 0, ΣFy = 0, ΣΤ = 0
Two uniform solid spheres have the same mass, but one has twice the radius of the other. The ratio of the larger sphere's moment of inertia to that of the smaller sphere is a. 1/2. b. 8/5. c. 4/5. d. 4. e. 2.
d. 4. Inertia of the smaller sphere (I1) = (2/5)(MR^2) Inertia of the larger sphere (I2) = (2/5)(M(2R)^2) = (8/5)(MR^2) Ratio of I2 to I1 = (8/5)/(2/5) = 4
Consider two uniform solid spheres where one has twice the mass and twice the diameter of the other. The ratio of the larger moment of inertia to that of the smaller moment of inertia is a. 2. b. 10. c. 4. d. 8. e. 1/2. f. 6.
d. 8. Inertia of the lighter sphere (I1) = (2/5)(MR^2) Inertia of the heavier sphere with the greater diameter (I2) = (2/5)(2M(2R^2)) = (16/5)(MR^2) Ratio of I2 to I1 = (16/5)/(2/5) = 8
An ice skater performs a pirouette (a fast spin) by pulling in his outstretched arms close to his body. What happens to his moment of inertia about the axis of rotation? a. It changes, but it is impossible to tell which way. b. It increases. c. It does not change. d. It decreases.
d. It decreases.
An ice skater performs a pirouette (a fast spin) by pulling in his outstretched arms close to his body. What happens to his rotational kinetic energy about the axis of rotation? a. It does not change. b. It changes, but it is impossible to tell which way. c. It decreases. d. It increases.
d. It increases. Rotational KE = (1/2)Iω^2 The moment of inertia decreases and the angular speed increases. Because the angular speed is squared, it contributes more to the overall change in Rotational KE. Therefore, the KE increases.
In what circumstances can the angular velocity of system of particles change without any change in the system's angular momentum? a. This can happen if a net external force acts on the system's center of mass. b. This cannot happen under any circumstances. c. This can happen if an external net torque is applied properly to the system. d. This can happen if the only forces acting are internal to the system.
d. This can happen if the only forces acting are internal to the system.
A ball is released from rest on a no-slip surface, as shown. After reaching its lowest point, the ball begins to rise again, this time on a frictionless surface as shown in Figure 10-1. When the ball reaches its maximum height on the frictionless surface, it is a. at a greater height as when it was released. b. impossible to tell without knowing the radius of the ball. c. at the same height as when it was released. d. at a lesser height as when it was released. e. impossible to tell without knowing the mass of the ball.
d. at a lesser height as when it was released. When going down the no-slip surface, the ball's rotational kinetic energy > translational kinetic energy. Therefore, while going up the frictionless surface, the ball is rotating more than translating, so it will cover less distance going up than the distance it covered going down.
The center of mass of an object: a. does not exist if the object has an irregular shape. b. is always somewhere inside the object. c. is always the same as the center of gravity. d. is sometimes outside of the object.
d. is sometimes outside of the object.
The rotating systems shown in Figure 11-1 differ only in that the two identical movable masses are positioned a distance r from the axis of rotation (left), or a distance r/2 from the axis of rotation (right). a. the block at left lands first. b. both blocks land at the same time. c. it is impossible to tell which block reaches the bottom first. d. the block at right lands first.
d. the block at right lands first. The rod to which the block on the left is attached has the movable masses closer to the center, producing less torque than the right. It is therefore "easier" for the block on the left to fall than the block on the right.
Two equal forces are applied to a door at the doorknob. The first force is applied perpendicular to the door; the second force is applied at 30° to the plane of the door. Which force exerts the greater torque? a. additional information is needed b. both exert zero torques c. the second applied at an angle d. the first applied perpendicular to the door e. both exert equal non-zero torques
d. the first applied perpendicular to the door Torque exerted by the force applied perpendicular: F*r*sin(90) = F*r Torque exerted by the force applied at 30 degrees: F*r*sin(30) = (F*r)/2 Torque 1 > Torque 2
Suppose a solid sphere of mass M and radius R rolls without slipping down an inclined plane starting from rest. The angular velocity of the sphere at the bottom of the incline depends on a. the radius of the sphere. b. the mass of the sphere. c. neither the mass nor the radius of the sphere. d. both the mass and the radius of the sphere.
d. the radius of the sphere.
What condition or conditions are necessary for rotational equilibrium? a. ΣFx = 0, ΣΤ = 0 b. ΣFx = 0 c. ΣFx = 0, ΣFy = 0 d. ΣΤ = 0
d. ΣΤ = 0
A solid cylinder is rolling without slipping. What fraction of its kinetic energy is rotational? a. 2/3 b. 1/2 c. 3/4 d. 1/4 e. 1/3
e. 1/3 Total KE = Rotational KE + Translational KE = (1/2)Iω^2 + (1/2)mv^2 = (1/2)(1/2MR^2)(v^2/R^2) + (1/2)(mv^2) = (1/4)mv^2 + (1/2)mv^2 = (3/4)mv^2 Fraction of KE that is rotational = (1/4)/(3/4) = 1/3
A figure skater is spinning slowly with arms outstretched. She brings her arms in close to her body and her moment of inertia decreases by 1/2. Her angular speed increases by a factor of a. 1 b. sqrt(2) c. 1/2. d. 4. e. 2.
e. 2. Initial L = Iω Final L = (I/2)*2ω Therefore, the angular speed increased by a factor of 2 to conserve angular momentum.
Consider a hoop of radius R and mass M rolling without slipping. Which form of kinetic energy is larger, translational or rotational? a. Translational kinetic energy is larger. b. Rotational kinetic energy is larger. c. You need to know the speed of the hoop to tell. d. You need to know the acceleration of the hoop to tell. e. Both are equal.
e. Both are equal. Translational kinetic energy = (1/2)mv^2 Rotational kinetic energy = (1/2)(MR^2)(v^2/r^2) = (1/2)mv^2
When is the angular momentum of a system constant? a. When the moment of inertia is constant. b. When the linear momentum and the energy are constant. c. When the total kinetic energy is constant. d. When no net external force acts on the system. e. When no torque acts on the system.
e. When no torque acts on the system.
In rotational dynamics, moment of inertia plays a role analogous to one of the quantities encountered in linear motion, namely a. translational kinetic energy. b. impulse. c. momentum. d. a "couple" or "moment" of a force. e. mass.
e. mass.
Suppose you are at the center of a large freely-rotating horizontal turntable in a carnival funhouse. As you crawl toward the edge, the angular momentum of you and the turntable a. decreases in direct proportion to your decrease in RPMs. b. none of these c. decreases. d. increases. e. remains the same, but the RPMs decrease.
e. remains the same, but the RPMs decrease.
Angular momentum cannot be conserved if a. the angular velocity changes. b. there is a net force on the system. c. the moment of inertia changes. d. the angular acceleration changes. e. the net torque is not zero.
e. the net torque is not zero.
