AP Physics 1 3.4 Mechanical Energy Conservation

[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.B—Describe the behavior of a system using conservation of mechanical energy principles question 1 — 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.

3.4.B—Describe the behavior of a system using conservation of mechanical energy principles question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (d)