AP Physics 1 3.4: Conservation of Energy

Use energy conservation to connect kinetic and potential energy, select an effective system boundary, and account for work by nonconservative interactions.

Exam points

Apply conservation of mechanical energy when kinetic and potential energies transform within a chosen system.
Explain how system selection changes whether energy transfer appears as external work or internal change.
Determine how friction or another nonconservative interaction changes the mechanical-energy balance.

Covered syllabus points

3.4.A—Describe the energies present in a system 3.4.B—Describe the behavior of a system using conservation of mechanical energy principles 3.4.C—Describe how the selection of a system determines whether the energy of that system changes

[Maximum number: 9]

Figure for Question 3.4 Conservation of Energy question 1 — AP Physics 1: Algebra-Based

(12 points, suggested time 25 minutes) Food scientists have created a new oil. At room temperature, the oil is a liquid. As the oil gets colder, however, it stiffens (thickens) into a sticky gel. To explore the properties of the oil, the scientists fill a container with the oil to a height D, as shown in the figure above on the left. They drop a small steel ball of mass M from rest at the top of the oil. Using video to capture the ball's motion, the scientists calculate $E_{\text {lost }}$ , the mechanical energy lost by the ball-Earth system from the time the ball enters the oil to the time just before the ball strikes the bottom of the container. The scientists also define the "stiffness" S of the oil as a quantity proportional to the force required to move a rod through the oil at a standard constant speed. The scientists calculate $E_{\text {lost }}$ and S at several different temperatures, ranging from room temperature to the lowest temperature at which the ball still falls through the oil. The graph above on the right shows $E_{\text {lost }}$ as a function of S for the calculated data points and a best-fit curve.

(a)

Figure for Question (a) — AP Physics 1: Algebra-Based

(a) Give a physical reason why the curve in the graph would not reach the vertical $\left(E_{\text {lost }}\right)$ axis even if the scientists had taken data over a broader range of temperatures.

[ 2 ]

(b)

Figure for Question (b) — AP Physics 1: Algebra-Based

(c) One of the scientists, in trying to represent the relationship between the oil stiffness and the mechanical energy lost, writes down the equation $E_{\text {lost }}=C S^{2}$ , where C is a constant with appropriate units. Another scientist points out that this equation cannot be correct. Give two reasons why the equation cannot be correct.

[ 2 ]

For each of the following, up to a maximum of 2 points

1 point for each valid, independent statement that the equation does not match the graph

(maximum of 2 points).

1 point for a statement that demonstrates that $E_{\text {lost }}$ cannot increase indefinitely.

Example statements ( 1 point each, any two could be combined for 2 points):

The equation indicates that $E_{\text {lost }}$ increases indefinitely with S, but the graph has an

upper limit for $E_{\text {lost }}$ .

The equation represents a parabola (or is concave-up), which does not match the

graph.

It doesn't make physical sense that $E_{\text {lost }}$ could be more than the available

potential energy MgD.

(c)

Figure for Question (c) — AP Physics 1: Algebra-Based

(d) Further attempting to model the ball's motion, the scientists write the following equation for the time t the ball takes to fall through the oil: $t=\frac{Z}{S}$ , where Z is a constant with appropriate units. Is this equation plausible-in other words, does it make physical sense? Plausible Not plausible Briefly explain your reasoning.

[ 2 ]

(d)

Figure for Question (d) — AP Physics 1: Algebra-Based

(e) $\Delta K$ is the change in kinetic energy of the ball between the time it is released from rest and the time just before the ball strikes the bottom of the container. On the axes below, sketch $\Delta K$ as a function of S, the oil stiffness.

[ 3 ]

(e) 3 points

Figure for Answer to Question (d) — AP Physics 1: Algebra-Based

For drawing a concave-upward curve that approaches the horizontal at higher S

For having the maximum value of $\Delta K$ at the lowest graphed value of S, which can be

S=0 or a positive value

Note: This point is not earned if the graph indicates that $\Delta K$ approaches infinity as S

approaches zero.

For high values of S, the graph approaches and does not cross the horizontal axis

Note: This point is earned if the graph touches the horizontal axis without crossing it.

[Maximum number: 10]

A block of mass M is released from rest at position x=0 near the top of a ramp. The ramp

makes an angle of $\theta$ with the horizontal. The block slides down the ramp with negligible

friction. At x=8 D the block makes contact with an uncompressed spring with spring

constant k. The spring is then compressed and the block momentarily comes to rest at x=12 D.

Figure 1 shows the instants when the block is at x=0, x=6 D, and x=10 D, respectively.

Figure for Question 3.4 Conservation of Energy question 2 — AP Physics 1: Algebra-Based

(a)

Figure 4 shows an energy bar chart that represents the kinetic energy K of the block, the

gravitational potential energy $U_{g}$ of the block-spring-Earth system, and the spring potential

energy $U_{s}$ of the block-spring-Earth system at the instant that the block is at x=10 D. The

gravitational potential energy $U_{g}$ of the block-spring-Earth system is defined to be zero when

the block momentarily comes to rest at x=12 D.

Draw shaded bars that represent $K, U_{g}$ , and $U_{s}$ to complete the energy bar charts in

Figure 2 and Figure 3 for when the block is released from rest at x=0 and for when the

block is at x=6 D, respectively.

- Shaded bars should start at the dashed line that represents zero energy.

- Represent any energy that is equal to zero with a distinct line on the zero-energy line.

- The relative heights of each shaded bar should reflect the magnitude of the respective

energy consistent with the scale used in Figure 4.

Figure 2

Figure 3

Figure 4

[ 3 ]

A	For drawing one bar in Figure 2 that shows only gravitational potential energy $\left(U_{g}\right)$ <br>Scoring Note: The correct height of the bar is not required to earn this point.	Point A1
	For including only K and $U_{g}$ in Figure 3<br>Scoring Note: The correct heights of the bars are not required to earn this point.	Point A2
	For drawing bars in figures 2 and 3 whose total height, respectively, equals $12 E_{0}$	Point A3
Example Response
	Energy x=6 D<br> $14 E_{0}$ <br> $12 E_{0}$ <br> $10 E_{0}$ <br> $8 E_{0}$ <br> $6 E_{0}$ <br> $4 E_{0}$ <br> $2 E_{0} \pm \cdots$ <br> ${ }_{0} \pm=\cdots$ <br>\hdashline $U_{g}$
	Figure 2

(b)

Figure 5 shows the block at x=0 when the block is released from rest and the block at

x=12 D when the block momentarily comes to rest against the compressed spring.

Figure 5

Starting with conservation of energy, derive an equation for the spring constant k. Express

your answer in terms of $M, \theta, D$ , and physical constants, as appropriate. Begin your

derivation by writing a fundamental physics principle or an equation from the reference

information.

[ 4 ]

B	For a multistep derivation that includes conservation of energy	Point B1
		For equating the gravitational potential energy to the spring potential energy	Point B2
		For one of the following: - Substituting $12 D \sin \theta$ for $\Delta y$ in a gravitational potential energy expression - Substituting 4 D in for $\Delta x$ in a spring potential energy expression	Point B3
		For a correct expression for k<br>Accept any algebraic equivalent of one of the following: - $k=\frac{3}{2} \frac{M g \sin \theta}{D}$ - $\frac{3}{2} \frac{M g \sin \theta}{D}$	Point B4
		Example Response $\begin{aligned} & E_{0}=E_{f} & U_{g}=U_{s} & m g \Delta y=\frac{1}{2} k(\Delta x)^{2} & M g(12 D \sin \theta)=\frac{1}{2} k(4 D)^{2} & k=\frac{3}{2} \frac{M g \sin \theta}{D} \end{aligned}$

(c)

Figure 6 shows a graph of the energy of the system as a function of the position of the block

from x=8 D to x=12 D. The spring potential energy $U_{s}$ of the block-spring-Earth system is

shown on the graph.

On the axes shown in Figure 6, do the following.

[ 1 ]

(i)

Sketch and label a line or curve that represents the total mechanical energy E for the

block-spring-Earth system as a function of the position of the block from x=8 D

to x=12 D.

[ 1 ]

C (i) For sketching a horizontal line at $12 E_{0}$ , continuous from 8 D to 12 D, with a label that

indicates total mechanical energy

(d)

Indicate whether the speed $v_{9 D}$ of the block at x=9 D is greater than, less than, or equal to

the speed $v_{8 D}$ of the block at x=8 D.

$v_{9 D}>v_{8 D}$

$v_{9 D}<v_{8 D}$

$v_{9 D}=v_{8 D}$

Justify how your response is consistent with the energy lines or curves you drew in Figure 6

in part C.

[ 2 ]

D For one of the following:

- Indicating $v_{9 D}>v_{8 D}$ if the graph in part C shows

$U_{g}+U_{s} \text { at } x=8 D \text { is greater than } U_{g}+U_{s} \text { at } x=9 D$

- Indicating $v_{9 D}<v_{8 D}$ if the graph in part C shows

$U_{g}+U_{s} \text { at } x=8 D \text { is less than } U_{g}+U_{s} \text { at } x=9 D$

- Indicating $v_{9 D}=v_{8 D}$ if the graph in part C shows

$U_{g}+U_{s} \text { at } x=8 D \text { is equal to } U_{g}+U_{s} \text { at } x=9 D$

For a justification consistent with the graph that correctly relates the speed to the

Point D2

gravitational potential energy, spring potential energy, and either kinetic energy or total

energy

Example Response

The speed at x=9 D is greater than the speed at x=8 D. From the graph, the kinetic

energy at x=8 D must be $8 E_{0}$ . At x=9 D, the spring potential energy is $<E_{0}$ and

the gravitational potential energy is $3 E_{0}$ , leaving $>8 E_{0}$ of the energy as kinetic.

Therefore, the speed must be greater at x=9 D than at x=8 D.

[Maximum number: 4]

(7 points, suggested time 13 minutes)

A rod with a sphere attached to the end is connected to a horizontal mounted axle and carefully balanced so that it rests in a position vertically upward from the axle. The center of mass of the rod-sphere system is indicated with a ⊗, as shown in Figure 1. The sphere is lightly tapped, and the rod-sphere system rotates clockwise with negligible friction about the axle due to the gravitational force.

A student takes a video of the rod rotating from the vertically upward position to the vertically downward position. Figure 2 shows five frames (still shots) that the student selected from the video.
Note: these frames are not equally spaced apart in time.

Figure 2

(a)

The rod-sphere system has mass M and length L, and the center of mass is located a distance $\frac{3}{4} L$ from the axle, shown in Figure 3.

[ 4 ]

(i)

Derive an expression for the change in kinetic energy of the rod-sphere-Earth system from the moment shown in Frame A to the moment shown in Frame E. Express your answer in terms of M, L, and fundamental constants, as appropriate.

[ 3 ]

(i) For a multistep derivation that begins with conservation of energy 1 point

E_{i}=E_{f} \quad \mathbf{O R} \quad \Delta E=0 \quad \mathbf{O R} \quad U_{g i}+K_{i}=U_{g f}+K_{f}

For indicating the change in height is equal to $\frac{3}{2} L$ 1 point

\Delta y=\frac{3}{2} L

For an answer consistent with the height change indicated previously in the response 1 point

K_{f}=\frac{3}{2} M g L

Scoring Note: A correct answer of $K_{f}=\frac{3}{2} M g L$ with no supporting work can earn only this point.
Example Response

\begin{aligned} & E_{i}=E_{f} \\ & U_{g i}+K_{i}=U_{g f}+K_{f} \\ & \Delta K=U_{g i}-U_{g f} \\ & \Delta K=M g \Delta y \\ & \Delta y=\frac{3 L}{4}+\frac{3 L}{4}=\frac{3}{2} L \\ & \Delta K=\frac{3}{2} M g L \end{aligned}

(ii)

Briefly explain why the rod and sphere gain kinetic energy, even if Earth is not included in the system.

[ 1 ]

All question bank results loaded

Covered syllabus points

3.4 Conservation of Energy question 1

Question (a)

Question (b)

Question (c)

Question (d)

3.4 Conservation of Energy question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (d)

3.4 Conservation of Energy question 3

Question (a)

Question (i)

Question (ii)