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.

Accepted Answer

The force on each system is related to the negative gradient of its respective potential energy function. Since both potential energy functions have the same form except for the constant term A in the second, the only difference in the forces will be this constant term, which does not depend on position. Therefore, the force on system 1 will be identical to the force on system 2 at a given position.

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

Accepted Answer

Due to air resistance, some of the gravitational potential energy is converted into other forms of energy, such as thermal energy and sound energy, rather than into kinetic energy. Therefore, the amount of kinetic energy gained by the ball will be less than the amount of gravitational potential energy lost.

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.

Accepted Answer

The height reached by an object thrown upward is proportional to the square of its initial velocity. Since kinetic energy is proportional to the square of the velocity ($KE = \frac{1}{2}mv^2$), doubling the initial speed ($2v$) results in four times the initial kinetic energy ($4 \times \frac{1}{2}mv^2 = 2mv^2$). This increased energy allows the object to reach a height four times greater than if it were thrown at speed $v$, assuming no air resistance and constant acceleration due to gravity.

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

Accepted Answer

When the ball drops, it loses some gravitational potential energy and gains an equal amount of kinetic energy. However, since air resistance is not being ignored, some of the energy is lost to air friction (thermal energy), which means that the ball must have lost more than 30 J of gravitational potential energy for it to gain 30 J of kinetic energy.

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

Accepted Answer

The answer of You do 174 J of work while...

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

Accepted Answer

The answer of The plot in the figure shows the...

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

Accepted Answer

The answer of A tennis ball bounces on the floor...

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.

Accepted Answer

The answer of Swimmers at a water park have a...

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.

Accepted Answer

The answer of A box of mass m is pressed...

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

Accepted Answer

The answer of An 8.0-kg block is released from rest,...

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.

Accepted Answer

The answer of Two stones, one of mass m and...

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.

Accepted Answer

In the absence of air resistance, all objects near the Earth's surface accelerate downwards at the same rate regardless of their initial motion. This means that the final speed of the stone when it hits the water depends only on the height from which it is dropped and not on the direction in which it is thrown. Therefore, the speed will be the same in all cases when the stone hits the water.

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.

Accepted Answer

The answer of Block 1 and block 2 have the...

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

Accepted Answer

The energy transferred to the spring can be calculated using the formula for the potential energy stored in a spring: $E = \frac{1}{2}kx^2$, where $E$ is the energy in joules, $k$ is the spring constant in N/m, and $x$ is the displacement in meters. Given $k = 52.9 \, \text{N/cm} = 5290 \, \text{N/m}$ (since $1 \, \text{N/cm} = 100 \, \text{N/m}$) and $x = 40.0 \, \text{cm} = 0.4 \, \text{m}$, the energy transferred is $E = \frac{1}{2} \times 5290 \times (0.4)^2 = 423 \, \text{J}$.

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

Accepted Answer

In a system where only conservative forces are acting, the total mechanical energy (sum of kinetic and potential energies) remains constant, as energy is conserved.

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.

Accepted Answer

The choice of the zero of potential energy is arbitrary, so it is possible for a system to have negative potential energy. For example, we could choose the zero of potential energy to be higher than the actual lowest point of the gravitational potential energy of a ball, resulting in a negative potential energy value.

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.

Accepted Answer

The answer of A box of mass m is pressed...

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.

Accepted Answer

The answer of A box of mass m is pressed...

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?

Accepted Answer

The answer of It requires 6.0 J of work is...

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.

Accepted Answer

All the statements provided are true characteristics of a conservative force. Conservative forces, such as gravity or spring forces, have the property that the work done is independent of the path taken, depends only on the initial and final positions, and is zero if the starting and ending points are the same.

Quiz 10: Interactions and Potential Energy

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 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?

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

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?

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 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?

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?

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?

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?

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

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

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?

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

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?

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?