Deck 10: Interactions and Potential Energy

Question 1

A potential energy function for system 1 is given by U₁(x) = Cx² + Bx³. The potential energy function for system 2 is given by U₂(x) = A + Cx² + Bx³, where A is a positive quantity. How does the force on system 1 relate to the force on system 2 at a given position?

A) The force on the two systems will be in opposite directions.
B) The force is identical on the two systems.
C) The force on the second system will be with less than the force on the first system.
D) There is no relationship between the forces on the two systems.
E) The force on the second system will be with greater than the force on the first system.

Answer 1

The force is identical on the two systems.

Question 2

A ball drops some distance and loses 30 J of gravitational potential energy. Do NOT ignore air resistance. How much kinetic energy did the ball gain?

A) more than 30 J
B) exactly 30 J
C) less than 30 J

Answer 2

less than 30 J

Question 3

Two identical balls are thrown directly upward, ball A at speed v and ball B at speed 2v, and they feel no air resistance. Which statement about these balls is correct?

A) Ball B will go twice as high as ball A because it had twice the initial speed.
B) Ball B will go four times as high as ball A because it had four times the initial kinetic energy.
C) The balls will reach the same height because they have the same mass and the same acceleration.
D) At its highest point, ball B will have twice as much gravitational potential energy as ball A because it started out moving twice as fast.
E) At their highest point, the acceleration of each ball is instantaneously equal to zero because they stop for an instant.

Answer 3

Ball B will go four times as high as ball A because it had four times the initial kinetic energy.

Question 4

A ball drops some distance and gains 30 J of kinetic energy. Do NOT ignore air resistance. How much gravitational potential energy did the ball lose?

A) more than 30 J
B) exactly 30 J
C) less than 30 J

Question 5

You do 174 J of work while pulling your sister back on a swing, whose chain is 5.10 m long. You start with the swing hanging vertically and pull it until the chain makes an angle of 32.0° with the vertical with your sister is at rest. What is your sister's mass, assuming negligible friction?

A) 22.9 kg
B) 19.5 kg
C) 26.3 kg
D) 28.4 kg

Question 6

The plot in the figure shows the potential energy of a particle, due to the force exerted on it by another particle, as a function of distance. At which of the three points labeled in the figure is the magnitude of the force on the particle greatest?

A) point X
B) point Y
C) point Z

Question 7

A tennis ball bounces on the floor three times. If each time it loses 22.0% of its energy due to heating, how high does it rise after the third bounce, provided we released it from the floor?

A) 110 cm
B) 11 cm
C) 110 mm
D) 140 cm

Question 8

Swimmers at a water park have a choice of two frictionless water slides as shown in the figure. Although both slides drop over the same height, h, slide 1 is straight while slide 2 is curved, dropping quickly at first and then leveling out. How does the speed v₁ of a swimmer reaching the end of slide 1 compares with v₂, the speed of a swimmer reaching the end of slide 2?

A) v₁ > v₂
B) v₁ < v₂
C) v₁ = v₂
D) No simple relationship exists between v₁ and v₂ because we do not know the curvature of slide 2.

Question 9

A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in the figure and eventually stops. If we repeat this experiment but instead compress the spring a distance of 2x

A) the box will go up the incline twice as high as before.
B) just as it moves free of the spring, the box will be traveling twice as fast as before.
C) just as it moves free of the spring, the box will be traveling four times as fast as before.
D) just as it moves free of the spring, the box will have twice as much kinetic energy as before.
E) just before it is released, the box has twice as much elastic potential energy as before.

Question 10

An 8.0-kg block is released from rest, with v₁ = 0.00 m/s, on a rough incline, as shown in the figure. The block moves a distance of 1.6-m down the incline, in a time interval of 0.80 s, and acquires a velocity of v₂ = 4.0 m/s. How much work does gravity do on the block during this process?

A) +81 J
B) +100 J
C) +120 J
D) -81 J
E) -100 J

Question 11

Two stones, one of mass m and the other of mass 2m, are thrown directly upward with the same velocity at the same time from ground level and feel no air resistance. Which statement about these stones is true?

A) The heavier stone will go twice as high as the lighter one because it initially had twice as much kinetic energy.
B) Both stones will reach the same height because they initially had the same amount of kinetic energy.
C) At their highest point, both stones will have the same gravitational potential energy because they reach the same height.
D) At its highest point, the heavier stone will have twice as much gravitational potential energy as the lighter one because it is twice as heavy.
E) The lighter stone will reach its maximum height sooner than the heavier one.

Question 12

A girl throws a stone from a bridge. Consider the following ways she might throw the stone. The speed of the stone as it leaves her hand is the same in each case, and air resistance is negligible. Case A: Thrown straight up.
Case B: Thrown straight down.
Case C: Thrown out at an angle of 45° above horizontal.
Case D: Thrown straight out horizontally.
In which case will the speed of the stone be greatest when it hits the water below?

A) Case A
B) Case B
C) Case C
D) Case D
E) The speed will be the same in all cases.

Question 13

Block 1 and block 2 have the same mass, m, and are released from the top of two inclined planes of the same height making 30° and 60° angles with the horizontal direction, respectively. If the coefficient of friction is the same in both cases, which of the blocks is going faster when it reaches the bottom of its respective incline?

A) We must know the actual masses of the blocks to answer.
B) Both blocks have the same speed at the bottom.
C) Block 1 is faster.
D) Block 2 is faster.
E) There is not enough information to answer the question because we do not know the value of the coefficient of kinetic friction.

Question 14

An athlete stretches a spring an extra 40.0 cm beyond its initial length. How much energy has he transferred to the spring, if the spring constant is 52.9 N/cm?

A) 423 J
B) 4230 kJ
C) 423 kJ
D) 4230 J

Question 15

When an object is solely under the influence of conservative forces, the sum of its kinetic and potential energies does not change.

A) True
B) False

Question 16

Is it possible for a system to have negative potential energy?

A) Yes, as long as the kinetic energy is positive.
B) Yes, as long as the total energy is positive.
C) Yes, since the choice of the zero of potential energy is arbitrary.
D) No, because the kinetic energy of a system must equal its potential energy.
E) No, because this would have no physical meaning.

Question 17

A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in the figure and eventually stops. If we repeat this experiment but instead use a spring having force constant 2k

A) the box will go up the incline twice as high as before.
B) just as it moves free of the spring, the kinetic energy of the box will be twice as great as before.
C) just as it moves free of the spring, the speed of the box will be times as great as before.
D) All of the above choices are correct.
E) None of the above choices is correct.

Question 18

A box of mass m is pressed against (but is not attached to) an ideal spring of force constant k and negligible mass, compressing the spring a distance x. After it is released, the box slides up a frictionless incline as shown in the figure and eventually stops. If we repeat this experiment with a box of mass 2m

A) the lighter box will go twice as high up the incline as the heavier box.
B) just as it moves free of the spring, the lighter box will be moving twice as fast as the heavier box.
C) both boxes will have the same speed just as they move free of the spring.
D) both boxes will reach the same maximum height on the incline.
E) just as it moves free of the spring, the heavier box will have twice as much kinetic energy as the lighter box.

Question 19

It requires 6.0 J of work is needed to push a 2.0-kg object from point A to point B of the frictionless ramp as shown in the figure. What is the length s of the ramp from A to B?

Question 20

Which, if any, of the following statements concerning the work done by a conservative force is NOT true?

A) It can always be expressed as the difference between the initial and final values of a potential energy function.
B) It is independent of the path of the body and depends only on the starting and ending points.
C) When the starting and ending points are the same, the total work is zero.
D) All of the above statements are true.
E) None of the above statements are true.

Question 21

A 2.0 g bead slides along a frictionless wire, as shown in the figure. At point A, the bead is moving to the right but with negligible speed. (a) What is the potential energy of the bead at point A?
(b) What is the kinetic energy of the bead at point B?
(c) What is the speed of the bead at point B?
(d) What is the speed of the bead at point C?

Question 22

A block slides down a frictionless inclined ramp. If the ramp angle is 17.0° and its length is find the speed of the block as it reaches the bottom of the ramp, assuming it started sliding from rest at the top.

A) 13.1 m/s
B) 172 m/s
C) 9.26 m/s
D) 24.0 m/s

Question 23

A car on a roller coaster starts at zero speed at an elevation above the ground of 26 m. It coasts down a slope, and then climbs a hill. The top of the hill is at an elevation of 16 m. What is the speed of the car at the top of the hill? Neglect any frictional effects.

A) 14 m/s
B) 18 m/s
C) 10 m/s
D) 9.0 m/s
E) 6.0 m/s

Question 24

A 50.0-kg skier starting from rest travels 200 m down a hill that has a 20.0° slope and a uniform surface. When the skier reaches the bottom of the hill, her speed is 30.0 m/s.
(a) How much work is done by friction as the skier comes down the hill?
(b) What is the magnitude of the friction force if the skier travels directly down the hill?

Question 25

A 60.0-kg person drops from rest a distance of 1.20 m to a platform of negligible mass supported by an ideal stiff spring of negligible mass. The platform drops 6.00 cm before the person comes to rest. What is the spring constant of the spring?

A) 2.56 × 10⁵ N/m
B) 3.92 × 10⁵ N/m
C) 5.45 × 10⁴ N/m
D) 4.12 × 10⁵ N/m
E) 8.83 × 10⁴ N/m

Question 26

A very small 100-g object is attached to one end of a massless 10-cm rod that is pivoted without friction about the opposite end. The rod is held vertical, with the object at the top, and released, allowing the rod to swing. What is the speed of the object at the instant that the rod is horizontal?

A) 0.71 m/s
B) 4.0 m/s
C) 1.4 m/s
D) 2.8 m/s
E) 1.8 m/s

Question 27

An object is attached to a hanging unstretched ideal and massless spring and slowly lowered to its equilibrium position, a distance of 6.4 cm below the starting point. If instead of having been lowered slowly the object was dropped from rest, how far then would it then stretch the spring at maximum elongation?

A) 13 cm
B) 9.1 cm
C) 6.4 cm
D) 18 cm
E) 26 cm

Question 28

A small hockey puck slides without friction over the icy hill shown in the figure and lands 6.20 m from the foot of the cliff with no air resistance. What was its speed v₀ at the bottom of the hill?

A) 20.8 m/s
B) 17.4 m/s
C) 14.4 m/s
D) 13.7 m/s
E) 4.71 m/s

Question 29

In the figure, a 5.00-kg block is moving at 5.00 m/s along a horizontal frictionless surface toward an ideal massless spring that is attached to a wall. After the block collides with the spring, the spring is compressed a maximum distance of 0.68 m. What is the speed of the block when it has moved so that the spring is compressed to only one-half of the maximum distance?

Question 30

A projectile is fired from ground level at an angle of 40.0° above horizontal at a speed of 30.0 m/s. What is the speed of the projectile when it has reached a height equal to 50.0% of its maximum height?

A) 26.0 m/s
B) 27.4 m/s
C) 28.7 m/s
D) 26.7 m/s
E) 28.1 m/s

Question 31

An 8.0-m massless rod is loosely pinned to a frictionless pivot at 0, as shown in the figure. A very small 4.0-kg ball is attached to the other end of the rod. The ball is held at A, where the rod makes a 30° angle above the horizontal, and is released. The ball-rod assembly then swings freely with negligible friction in a vertical circle between A and B. The tension in the rod when the ball passes through the lowest point at D is closest to

A) 160 N.
B) 200 N.
C) 120 N.
D) 80 N.
E) 40 N.

Question 32

A mass is pressed against (but is not attached to) an ideal horizontal spring on a frictionless horizontal surface. After being released from rest, the mass acquires a maximum speed v and a maximum kinetic energy K. If instead the mass initially compresses the spring twice as far:

A) Its maximum speed will be 2v and its maximum kinetic energy will be 2K.
B) Its maximum speed will be 2v and its maximum kinetic energy will be K.
C) Its maximum speed will be v and its maximum kinetic energy will be 2K.
D) Its maximum speed will be 2v and its maximum kinetic energy will be 4K.
E) Its maximum speed will be 4v and its maximum kinetic energy will be 2K.

Question 33

A 2.0-kg object is moving without friction along the x-axis. The potential energy curve as a function of position is shown in the figure, and the system is conservative. If the speed of the object at the origin is 4.0 m/s, what will be its speed at 7.0 m along the +x-axis?

A) 4.0 m/s
B) 4.2 m/s
C) 4.4 m/s
D) 4.6 m/s
E) 9.8 m/s

Question 34

A roller coaster of mass 80.0 kg is moving with a speed of 20.0 m/s at position A as shown in the figure. The vertical height above ground level at position A is 200 m. Neglect friction. (a) What is the total mechanical energy of the roller coaster at point A?
(b) What is the total mechanical energy of the roller coaster at point B?
(c) What is the speed of the roller coaster at point B?
(d) What is the speed of the roller coaster at point C?

Question 35

In the figure, a stunt car driver negotiates the frictionless track shown in such a way that the car is barely in contact with the track at the top of the loop. The radius of the track is 9.9 m and the mass of the car is 1800 kg. Find the magnitude of the force of the car on the track when the car is at point A. You can treat the car as a point mass.

Question 36

Consider the motion of a 1.00-kg particle that moves with potential energy given by U(x) = (-2.00 J ∙ m)/x + (4.00 J ∙ m²)/x². Suppose the particle is moving with a speed of 3.00 m/s when it is located at x = 1.00 m. What is the speed of the object when it is located at ?

A) 2.13 m/s
B) 3.00 m/s
C) 4.68 m/s
D) 3.67 m/s

Question 37

A 2.0 kg mass is moving along the x-axis. The potential energy curve as a function of position is shown in the figure. The kinetic energy of the object at the origin is 12 J. The system is conservative, and there is no friction. (a) What will be the kinetic energy at 2.0 m along the +x-axis?
(b) What will be the speed of the object at 6.0 m along the +x-axis?

Question 38

In the figure, a 4.0-kg ball is on the end of a 1.6-m rope that is fixed at 0. The ball is held at point A, with the rope horizontal, and is given an initial downward velocity. The ball moves through three quarters of a circle with no friction and arrives at B, with the rope barely under tension. The initial velocity of the ball, at point A, is closest to

A) 4.0 m/s
B) 5.6 m/s
C) 6.3 m/s
D) 6.9 m/s
E) 7.9 m/s

Question 39

A spring-loaded dart gun is used to shoot a dart straight up into the air, and the dart reaches a maximum height of 24 meters above its point of release. The same dart is shot up a second time from the same gun, but this time the spring is compressed only half as far (compared to the first shot). How far up does the dart go this time? (Neglect friction and assume the spring is ideal and massless.)

A) 6.0 m
B) 12 m
C) 3.0 m
D) 48 m

Question 40

In the figure, a very small toy race car of mass m is released from rest on the loop-the-loop track. If it is released at a height 2R above the floor, how high is it above the floor when it leaves the track, neglecting friction?

A) 1.67 R
B) 2.00 R
C) 1.50 R
D) 1.33 R
E) 1.25 R

Question 41

A 5.00-kg object moves clockwise around a 50.0 cm radius circular path. At one location, the speed of the object is 4.00 m/s. When the object next returns to this same location, the speed is 3.00 m/s.
(a) How much work was done by nonconservative (dissipative) forces as the object moved once around the circle?
(b) If the magnitude of the above nonconservative (dissipative) forces acting on the object is constant, what is the value of this magnitude?

Question 42

A potential energy function is given by U(x) = ( 3.00 N/m)x - ( 1.00 N/m³)x³. At what position or positions is the force equal to zero?

A) m and - m
B) 0.00 m, m and - m
C) 1.00 m and -1.00 m
D) 3.00 m and -3.00 m
E) The force is not zero at any location.

Question 43

The only force acting on an object moving along the x-axis is the conservative force given by F(x) = (2.00 N/m)x + (1.00 N/m³)x³.
(a) What is the change in potential energy when the object moves from x = 1.00 m to x = 2.00 m?
(b) What is the change in kinetic energy when the object moves from x = 1.00 m to x = 2.00 m?

Question 44

A potential energy function is given by U(x) = (3.00 J)x + (1.00 J/m²)x³. What is the force function F(x) that is associated with this potential energy function?

Question 45

An object of mass 4.0 kg starts at rest from the top of a rough inclined plane of height 10 m as shown in the figure. If the speed of the object at the bottom of the inclined plane is 10 m/s, how much work does friction do on this object as it slides down the incline?

Question 46

A 1.86-kg block is held in place against the spring by a 81-N horizontal external force (see the figure). The external force is removed, and the block is projected with a velocity v₁ = 1.2 m/s upon separation from the spring. The block descends a ramp and has a velocity at the bottom. The track is frictionless between points A and B. The block enters a rough section at B, extending to E. The coefficient of kinetic friction over this section is 0.28. The velocity of the block is v₃ = 1.4 m/s at C. The block moves on to D, where it stops. The height h of the ramp is closest to

A) 11
B) 7.3
C) 15
D) 17
E) 18

Question 47

A 2.5-kg box, sliding on a rough horizontal surface, has a speed of 1.2 m/s when it makes contact with a spring (see the figure). The block comes to a momentary halt when the compression of the spring is 5.0 cm. The work done by the friction, from the instant the block makes contact with the spring until is comes to a momentary halt, is -0.50 J. (a) What is the spring constant of the spring?
(b) What is the coefficient of kinetic friction between the box and the rough surface?

Question 48

In the figure, a block of mass m is moving along the horizontal frictionless surface with a speed of 5.70 m/s. If the slope is 11.0° and the coefficient of kinetic friction between the block and the incline is 0.260, how far does the block travel up the incline?

Question 49

The potential energy for a certain mass moving in one dimension is given by U(x) = (2.0 J/m³)x³ - (15 J/m²)x² + (36 J/m)x - 23 J. Find the location(s) where the force on the mass is zero.

A) 4.0 m, 5.0 m
B) 1.0 m
C) 2.0 m, 3.0 m
D) 3.0 m, 5.0 m

Question 50

A force on an object is given by F(x) = (2.00 N/m)x - (3.00 N/m³)x³. What is a potential energy function U(x) for this conservative force?

Question 51

When a particle is a distance r from the origin, its potential energy function is given by the equation U(r) = kr, where k is a constant and r = (a) What are the SI units of k?
(b) Find a mathematical expression in terms of x, y, and z for the y component of the force on the particle.
(c) If U = 3.00 J when the particle is 2.00 m from the origin, find the numerical value of the y component of the force on this particle when it is at the point (-1.00 m, 2.00 m, 3.00 m).

Question 52

A force on an object is given by F(x) = ( -4.00 N/m)x + ( 2.00 N/m³)x³. What is the change in potential energy in moving from x = 1.00 m to x = 2.00 m?

A) 10.0 J
B) -1.50 J
C) -10.0 J
D) 1.50 J
E) 12.0 J

Question 53

A particle experiences a force given by F(x) = α - βx³. Find the potential field U(x) the particle is in. (Assume that the zero of potential energy is located at x = 0.)

A) U(x) = -αx + x⁴
B) U(x) = αx - x⁴
C) U(x) = 3βx²
D) U(x) = -3βx²

Question 54

A 1.37-kg block is held in place against the spring by a 74-N horizontal external force (see the figure). The external force is removed, and the block is projected with a velocity v₁ = 1.2 m/s upon separation from the spring. The block descends a ramp and has a velocity at the bottom. The track is frictionless between points A and B. The block enters a rough section at B, extending to E. The coefficient of kinetic friction over this section is 0.24. The velocity of the block is at C. The block moves on to D, where it stops. The initial compression of the spring is closest to

A) 2.7 cm.
B) 1.4 cm.
C) 0.96 cm.
D) 5.3 cm.
E) 3.6 cm.

Question 55

An 0.80-kg block is held in place against the spring by a 67-N horizontal external force (see the figure). The external force is removed, and the block is projected with a velocity v₁ = 1.2 m/s upon separation from the spring. The block descends a ramp and has a velocity v₂ = 1.9 m/s at the bottom. The track is frictionless between points A and B. The block enters a rough section at B, extending to E. The coefficient of kinetic friction over this section is 0.39. The velocity of the block is v₃ = 1.4 m/s at C. The block moves on to D, where it stops. The spring constant of the spring is closest to

A) 3900 N/m.
B) 2600 N/m.
C) 2000 N/m.
D) 1600 N/m.
E) 1100 N/m.