MPC Exercises (from MP)

25 July 2022
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question
Two ladybugs sit on a rotating disk, as shown in the figure (the ladybugs are at rest with respect to the surface of the disk and do not slip). (Figure 1) Ladybug 1 is halfway between ladybug 2 and the axis of rotation. 1. What is the angular speed of ladybug 1? one-half the angular speed of ladybug 2 the same as the angular speed of ladybug 2 twice the angular speed of ladybug 2 one-quarter the angular speed of ladybug 2 2. What is the direction of the vector representing the angular velocity of ladybug 2? See the figure for the directions of the coordinate axes. +x −x +y −y +z −z 3. Now assume that at the moment pictured in the figure, the disk is rotating but slowing down. Each ladybug remains "stuck" in its position on the disk. What is the direction of the tangential component of the acceleration (i.e., acceleration tangent to the trajectory) of ladybug 2? +x −x +y −y +z −z
answer
1. the same as the angular speed of ladybug 2 2. +Z 3. -Y
question
**A marching band consists of rows of musicians walking in straight, even lines. When a marching band performs in an event, such as a parade, and must round a curve in the road, the musician on the outside of the curve must walk around the curve in the same amount of time as the musician on the inside of the curve. This motion can be approximated by a disk rotating at a constant rate about an axis perpendicular to its plane. In this case, the axis of rotation is at the inside of the curve. 1.Consider two musicians, Alf and Beth. Beth is four times the distance from the inside of the curve as Alf. If Beth travels a distance s during time Δt, how far does Alf travel during the same amount of time? 4s 2s 1/2s 1/4s s 2. If Alf moves with speed v, what is Beth's speed? Speed in this case means the magnitude of the linear velocity, not the magnitude of the angular velocity. 4v v 1/4v
answer
1. The musician on the outside of the curve must travel farther than the musician on the inside 1/4s 2. The musician on the outside of the curve must travel faster than the musician on the inside 4v
question
On which of the following does the moment of inertia of an object depend? linear speed linear acceleration shape and density of the object angular speed angular acceleration total mass location of the axis of rotation
answer
shape and density of the object total mass location of the axis of rotation
question
The figure below shows two blocks suspended by a cord over a pulley. (Figure 1) The mass of block B is twice the mass of block A, while the mass of the pulley is equal to the mass of block A. The blocks are let free to move and the cord moves on the pulley without slipping or stretching. There is no friction in the pulley axle, and the cord's weight can be ignored. 1. Which of the following statements correctly describes the system shown in the figure? The acceleration of the blocks is zero. The net torque on the pulley is zero. The angular acceleration of the pulley is nonzero. 2. What happens when block B moves downward? The left cord pulls on the pulley with greater force than the right cord. The left and right cord pull with equal force on the pulley. The right cord pulls on the pulley with greater force than the left cord.
answer
1. The angular acceleration of the pulley is nonzero. 2. The right cord pulls on the pulley with greater force than the left cord. *Note that if the pulley were stationary, then the tensions in both parts of the cord would be equal. However, if the pulley rotates with a certain angular acceleration, as in the present situation, the tensions must be different.
question
The car shown in the figure has mass m (this includes the mass of the wheels). The wheels have radius r, mass mw, and moment of inertia I=kmwr2. Assume that the axles apply the same torque τ to all four wheels. For simplicity, also assume that the weight is distributed uniformly so that all the wheels experience the same normal reaction from the ground, and so the same frictional force. 1.If there is no slipping, a frictional force must exist between the wheels and the ground. In what direction does the frictional force from the ground on the wheels act? Take the positive x direction to be to the right. negative x direction positive x direction
answer
positive x direction *Friction between the wheels and the road is the only horizontal external force acting on the car.
question
** Gears are important components in many mechanical devices, from mechanical clocks to bicycles. In fact, they are present whenever a motor produces rotational motion. 1. When the cyclist encounters a steep hill, in order to maintain the same energy consumption and pedaling rate she changes her gear ratio to 0.7. Which of the following statements describes the change caused by the new gear ratio? The angular speed of the rear gear increases. The torque exerted on the rear gear wheel increases. The torque exerted on the front gear wheel increases. The power delivered to the rear gear wheel increases.
answer
The torque exerted on the rear gear wheel increases.
question
** Most systems when they are displaced from equilibrium and experience a restoring force that tends to bring them back to their equilibrium position. The resulting oscillations take the name of periodic motion. An important example of periodic motion is simple harmonic motion (SHM) and we will use the mass-spring system described here to introduce some of its properties. 1. Which of the following statements best describes the characteristic of the restoring force in the spring-mass system described in the introduction? The restoring force is constant. The restoring force is directly proportional to the displacement of the block. The restoring force is proportional to the mass of the block. The restoring force is maximum when the block is in the equilibrium position.
answer
1. The restoring force is proportional to the mass of the block. *Whenever the oscillations are caused by a restoring force that is directly proportional to displacement, the resulting periodic motion is referred to as simple harmonic motion.
question
** An object of mass m attached to a spring of force constant k oscillates with simple harmonic motion. The maximum displacement from equilibrium is A and the total mechanical energy of the system is E. 1. What is the system's potential energy when its kinetic energy is equal to 34E? kA^2 kA^2/2 kA^2/4 kA^2/8
answer
1. kA^2/8
question
** The shaker cart, shown in the figure,(Figure 1) is the latest extreme sport craze. You stand inside of a small cart attached to a heavy-duty spring, the spring is compressed and released, and you shake back and forth, attempting to maintain your balance. Note that there is also a sandbag in the cart with you. At the instant you pass through the equilibrium position of the spring, you drop the sandbag out of the cart onto the ground. 1. What effect does dropping the sandbag out of the cart at the equilibrium position have on the amplitude of your oscillation? It increases the amplitude. It decreases the amplitude. It has no effect on the amplitude. 2. What effect does dropping the sandbag out of the cart at the equilibrium position have on the maximum speed of the cart? It increases the maximum speed. It decreases the maximum speed. It has no effect on the maximum speed. 3. What effect does dropping the sandbag at the cart's maximum distance from equilibrium have on the amplitude of your oscillation? It increases the amplitude. It decreases the amplitude. It has no effect on the amplitude. 4. What effect does dropping the sandbag at the cart's maximum distance from equilibrium have on the maximum speed of the cart? It increases the maximum speed. It decreases the maximum speed. It has no effect on the maximum speed.
answer
1. It decreases the amplitude 2. It has no effect on the maximum speed 3. It has no effect on the amplitude 4. It increases the maximum speed
question
** A simple pendulum consisting of a bob of mass m attached to a string of length L swings with a period T. 1. If the bob's mass is doubled, approximately what will the pendulum's new period be? T/2 T 2√T 2T 2. If the pendulum is brought on the moon where the gravitational acceleration is about g/6, approximately what will its period now be? T/6 T/6√ 6√T 6T 3. If the pendulum is taken into the orbiting space station what will happen to the bob? It will continue to oscillate in a vertical plane with the same period. It will no longer oscillate because there is no gravity in space. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall. It will oscillate much faster with a period that approaches zero.
answer
1. T 2. 6√T 3. It will no longer oscillate because both the pendulum and the point to which it is attached are in free fall.
question
Two identical pulses are moving in opposite directions along a stretched string that has one fixed end and the other movable, as shown in (Figure 1). The two pulses reflect off the boundaries of the string, and at some later time, they pass through the middle of the string and interfere. 1. Consider the point where the two pulses start to overlap, point O in (Figure 2) . What is the displacement of point O as these pulses interfere? It varies with time. It remains zero. It depends on the (identical) amplitude of the pulses. It is zero only when the pulses begin to overlap. 2. Why does destructive interference occur when the two pulses overlap instead of constructive interference? because the pulses are traveling in opposite directions because a pulse is inverted upon reflection because the pulses are identical and cancel each other out because constructive interference occurs only when the pulses have the same amplitude 3. As the pulses interfere destructively there is a point in time when the string is perfectly straight. Which of the following statements is true at this moment? The energy of the string is zero. The string is not moving either up or down. The string has only kinetic energy.
answer
1. It remains zero 2. because a pulse is inverted upon reflection 3. The string has only kinetic energy
question
Which of the following explains why the bottle breaks? The pressure that the water exerts on the walls of the glass bottle is greater than the pressure required to break the glass. The force that the water exerts on the bottom of the bottle is greater than the failure pressure of the glass. The force that the water exerts on the bottom of the bottle is greater than the force required to break the glass. When the hand strikes the bottle, it sends a shock wave through the bottle, causing the water molecules to get very close together. When the mass of extra water molecules strike the bottom of the bottle, it breaks.
answer
The pressure that the water exerts on the walls of the glass bottle is greater than the pressure required to break the glass.
question
Two massive spheres are mounted on a light rod that can be rotated by a string wrapped around a central cylinder, forming a winch as shown in the figure. (Figure 1) A force of magnitude F is applied to the string to turn the system. With respect to the variables given in the figure, the equation for the magnitude of the angular acceleration α is α=rF/3mR^2. 1. If the sphere on the left is moved closer to the central cylinder and placed at a distance R/2 from the axis of rotation, what is the magnitude of the angular acceleration α of the modified system? Assume that the rest of the system doesn't change. 2.Consider again the original system. Instead of applying the force to the string, a force with the same magnitude F is applied to the rod at a point R/2 from the sphere of mass 2m and in a direction perpendicular to the rod, as shown in the figure.What is the magnitude of the angular acceleration α of the system?
answer
1. α=2rF/3mR^2 2. α=F/6mR
question
A U-tube is filled with water, and the two arms are capped.(Figure 1) The tube is cylindrical, and the right arm has twice the radius of the left arm. The caps have negligible mass, are watertight, and can freely slide up and down the tube. 1.A one-inch depth of sand is poured onto the cap on each arm. (Figure 2) After the caps have moved (if necessary) to reestablish equilibrium, is the right cap higher, lower, or the same height as the left cap? higher lower the same height 2.The sand is removed and a 1.0-kg-mass block is placed on each cap. (Figure 3) After the caps have moved (if necessary) to reestablish equilibrium, is the right cap higher, lower, or the same height as the left cap? higher lower the same height
answer
1. The same height 2. Higher