MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics IC_W05D1 ConcepTests

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1 Reading Question MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 IC_W05D1 ConcepTests Two objects are pushed on a frictionless surface from a starting line to a finish line with equal constant forces. One object is four times as massive as the other. Both objects are initially at rest. Which of the following statements is true when the objects reach the finish line? 1. The kinetic energies of the two objects are equal. 2. Object of mass 4m has the greater kinetic energy. 3. Object of mass m has the greater kinetic energy. 4. Not information is given to decide. 5. Unsure. I did not do the reading yet. Answer (1): The change in kinetic energy is equal to the work done on the objects which is the force times the distance displaced for a constant force. Because the work done is the same for both objects, the change in kinetic energy is also the same. Since they both started from rest, their final kinetic energy is the same.

2 Concept Questions: Work and Kinetic Energy 1. Compared to the amount of energy required to accelerate a car from rest to 10 miles per hour, the amount of energy required to accelerate the same car from 10 mph to 20 mph is 1. the same 2. twice as much 3. three times as much 4. four times as much 5. unsure. Answer: 3. The energy increase in going from zero speed to speed v is 1/2mv 2. To go from v to 2v is 1/2m(2v) 2-1/2mv 2 = 3/2mv A particle starts from rest at x = 0 and moves to x = L under the action of a variable force F(x), which is shown in the figure. What is the particle's kinetic energy at x=l/2 and at x=l? 1. F max L/2, F max L 2. F max L/4, 0 3. F max L, 0 4. F max L/4, F max L/2 5. F max L/2, F max L/4 Answer: 4. The work is equal the area under the graph of force vs. displacement. The work is also equal to the kinetic energy. At x =L/s, the area is F max L/4. At x = L the area is F max L/2.

3 3. A ball is given an initial horizontal velocity and allowed to fall under the influence of gravity, as shown below. The work done by the force of gravity on the ball is: 1. positive 2. zero 3. negative Answer: 1. The force of gravity causes the ball to accelerate downward, so the force displacement has a component in the same direction as the force. Hence the work done is positive. (The dot product of the ball's displacement and the downward force of gravity is positive.) 4. A comet is speeding along a hyperbolic orbit toward the Sun. While the comet is moving away from the Sun, the work done by the Sun on the comet is: 1. positive 2. zero 3. negative Answer: 3. The displacement of the comet has a component in the opposite direction as the force on the comet so the work done is negative. (The comet's acceleration is always toward the Sun; when the comet moves away from the Sun, the work is negative.)

4 5. A tetherball of mass m is attached to a post of radius by a string. Initially it is a distance r 0 from the center of the post and it is moving tangentially with a speed v 0. The string passes through a hole in the center of the post at the top. The string is gradually shortened by drawing it through the hole. Ignore gravity and any dissipative forces. Until the ball hits the post, 1. The kinetic energy of the ball is constant. 2. The kinetic energy of the ball changes. 3. Not enough information is given to determine whether the kinetic energy of the ball changes or not. Answer: 3. Since the path of the ball is not circular, a small displacement of the ball has a radial component inward so the dot product between the force by the rope on the ball with the displacement is non-zero hence the work done by the force is not zero. Therefore the kinetic energy of the ball changes..

5 6. A tetherball of mass m is attached to a post of radius R by a string. Initially it is a distance r 0 from the center of the post and it is moving tangentially with a speed v 0. The string wraps around the outside of the post. Ignore gravity and any dissipative forces. Until the ball hits the post, 1. The kinetic energy of the ball is constant. 2. The kinetic energy of the ball changes. 3. Not enough information is given to determine whether the kinetic energy of the ball changes or not. Answer 1. A small displacement of the ball is always perpendicular to string since at each instant in time the ball undergoes an instantaneous circular motion about the string contact point with pole. Therefore the dot product between the force by the rope on the ball with the displacement is zero hence the work done by the force is zero. Therefore the kinetic energy of the ball does not change.

6 7. A streetcar is freely coasting (no friction) around a large circular track. It is then switched to a small circular track. When coasting on the smaller circle its linear speed is 1. greater 2. less. 3. unchanged. Answer: 3. Unchanged. Carefully draw a free body diagram for the streetcar while it is on the link of track connecting the two circular tracks. In the vertical direction, the force of gravity at all times balances the vertical component of the normal force of the track on the car, so there is no acceleration in the vertical direction and we may ignore it for the rest of this problem. In the horizontal direction, the train is guided by the horizontal component of the normal force of the track on the wheels. This force is always normal to the direction of motion, so no work is done by this force. Therefore the streetcar's total energy does not change, and its speed remains constant.