AP Physics C Mechanics Unit 3 Energy

[Maximum number: 5]

Mech.2.
A block of mass 2 M rests on a horizontal, frictionless table and is attached to a relaxed spring, as shown in the figure above. The spring is nonlinear and exerts a force $F(x)=-B x^{3}$ , where B is a positive constant and x is the displacement from equilibrium for the spring. A block of mass 3 M and initial speed $v_{0}$ is moving to the left as shown.

(a)

Determine an expression for the kinetic energy of the two-block system immediately after the collision.

[ 1 ]

1 point
Using a proper expression for kinetic energy of the two-block system

\begin{aligned} K & =\frac{1}{2} m v^{2} \\ K & =\frac{1}{2}(5 M)\left(\frac{3}{5} v_{0}\right)^{2} \end{aligned}

For an answer consistent with part (b)

K=\frac{9}{10} M v_{0}^{2}

(b)

Derive an expression for the maximum distance D that the spring is compressed.

[ 4 ]

4 points
For a correct expression of the conservation of energy

1 \text { point }

\begin{aligned} & \Delta K_{\text {system }}+\Delta U_{\text {system }}=0 \\ & K_{0}=U_{\text {final }} \end{aligned}

For attempting to integrate the spring force equation
For attempting to integrate the spring force equation

1 \text { point }

For using the correct limits of integration or an appropriate constant of integration

1 \text { point }

For an answer consistent with the speed from (b) or the kinetic energy from part (c)

1 \text { point }

D=\sqrt[4]{\frac{18 M v_{0}^{2}}{5 B}}

2 points
For selecting the correct answer "Right" with a reasonable attempt at a justification
If the incorrect selection is made, no points are earned for the justification.
For an indication that at maximum compression the block of mass 2 M has an acceleration to the right due to the forces acting on the block of mass 2 M or an acceleration to the right due to the external spring force acting on the system of blocks
Example: At maximum compression the two-block system is instantaneously at rest. The only horizontal external force acting on the system is due to the spring. This force is directed to the right. The system and therefore the block of mass 2 M is accelerated to the right, which implies that the net force acting on the block of mass 2 M is also to the right.

Question 2

2 points
The magnitude of the net force is greater on the block of mass 3 M.
If the incorrect selection is made, no points are earned for the justification.
For an indication that both blocks will have the same acceleration 1 point

For a correct justification for why the net force is greater on the block of mass 3 M1 point

Example:
Because the blocks stick together, both blocks must have the same acceleration.
Because the block of mass 3 M has more mass, the net force on it must be greater than the net force on the block of mass 2 M.

[Maximum number: 10]

A box is connected to one end of a rigid rod. Both the box and the rod have negligible mass. The

other end of the rod is connected to a pivot. The box is open on one side, and a block is placed

inside the box.

The center of mass of the block is displaced a vertical distance h, as shown in Figure 1. The

block-box system is then released from rest and swings downward. There is negligible friction

about the pivot. When the system is at the lowest point of its swing, the rod collides with a rigid

stopper, as shown in Figure 2. The box comes to rest, and the block is launched horizontally out

of the box. The block moves across a horizontal surface toward a motion sensor that measures

the speed of the block. All frictional forces are negligible.

Figure 1

Figure 2

(a)

Students are asked to experimentally determine the acceleration due to gravity g using a

linear graph. To determine g, the students are permitted to use measurements from only a

meterstick and the motion sensor.

Describe an experimental procedure using the described setup to collect data that would

allow the students to determine an experimental value of g using a linear graph. Include any

steps necessary to reduce experimental uncertainty.

[ 2 ]

For a procedure that indicates a reasonable method of reducing experimental uncertainty
Examples of acceptable responses may include the following: - Repeating the experiment multiple times for the same value of h - Repeating the experiment for multiple values of h

(b)

Describe how the data collected in part A could be graphed and how that graph would be

analyzed to determine the value of g.

[ 2 ]

B For describing a graph that has linear trend that can be used to find g

Point B1

Examples of acceptable responses include the following:

- $v^{2}$ vs. 2 h

- $\frac{1}{2} v^{2}$ vs. h

- $v^{2}$ vs. h

- v vs. $\sqrt{h}$

Scoring Notes:

- Reponses that include the reciprocals of the preceding examples, in addition to

other equivalent graphs, also earn this point.

- This point may be earned independently of the response in part A.

For correctly relating the slope of the best-fit line to g

Point B2

Examples of acceptable responses include the following:

Graph	Analysis
$v^{2}$ vs. 2 h	slope =g
$\frac{1}{2} v^{2}$ vs. h	slope =g
$v^{2}$ vs. h	slope =2 g
v vs. $\sqrt{h}$	slope $=\sqrt{2 g}$

Example Response

Plot $v^{2}$ on the vertical axis and 2 h on the horizontal axis. The slope of the best-fit line

is equal to g.

(c)

The experiment is repeated, but the horizontal surface on which the block slides is replaced

with a new rough surface, as shown in Figure 3. The coefficient of kinetic friction between

the block and the new surface is $\mu$ .

Figure 3

The block-box system is pulled aside so that the center of mass of the block is displaced

various vertical distances h and then released from rest. For each vertical distance, students

measure the position $x=x_{\text {max }}$ at which the block comes to rest.

The students' measurements of h and $x_{\text {max }}$ are shown in Table 1.

Table 1

[ 4 ]

(i)

Indicate two quantities, either measured quantities from Table 1 or additional calculated

quantities, that could be graphed to produce a straight line that could be used to

determine $\mu$ .

Vertical axis:

Horizontal axis:

[ 1 ]

C (i) For indicating appropriate quantities that could be plotted to produce a linear graph that

can be used to determine $\mu$ , such as h vs. $x_{\text {max }}$

Scoring Note: Reponses that include the reciprocal of the preceding example, in

addition to other equivalent graphs, also earn this point.

(ii)

On the grid provided, create a graph of the quantities indicated in part C (i).

- Use Table 2 to record the measured or calculated quantities that you will plot.

- Clearly label the axes, including units as appropriate.

- Plot the points you recorded in Table 2.

Figure for Question (ii) — AP Physics C: Mechanics

[ 2 ]

(iii)

Draw a best-fit line to the data graphed in part C (ii).

[ 1 ]

(d)

Using the best-fit line that you drew in part C (iii), calculate an experimental value for $\mu$ .

[ 2 ]

D For correctly relating the slope of the best-fit line to the value of $\mu$

Examples of acceptable responses includes the following:

Graph	Analysis
h vs. $x_{\text {max }}$	slope $=\mu$

For a value of $\mu$ within a range of 0.30 to 0.50

Example Response

$\begin{aligned} & E_{0}+W=E_{f} \\ & m g h-F_{f} x_{\max }=0 \\ & m g h=\mu m g x_{\max } \\ & h=\mu x_{\max } \end{aligned}$ $\begin{aligned} & y=m x+b \\ & h=\mu x_{\max } \\ & \text { slope }=\mu \\ & \text { slope }=\frac{0.95 \mathrm{~m}-0.30 \mathrm{~m}}{2.4 \mathrm{~m}-0.70 \mathrm{~m}} \\ & \text { slope }=0.38 \\ & \mu=\text { slope }=0.38 \end{aligned}$

[Maximum number: 10]

In Scenario 1, a system composed of two springs, A and B, and a block of mass m is at rest on

a horizontal surface. Friction between the block and the surface is negligible. Each spring is

attached to a fixed wall and the block, as shown in Figure 1. Spring A has a spring constant k

and Spring B has a spring constant 2 k. Each spring is at its relaxed length when the block is at

position x=0, as shown.

Figure 1

The block is moved to $x=x_{1}$ and held at rest, as shown in Figure 2.

Figure 2

(a)

An energy bar chart can be used to represent the elastic potential energy $U_{\mathrm{A}}$ of Spring A,

the elastic potential energy $U_{\mathrm{B}}$ of Spring B, and the kinetic energy $K_{\text {block }}$ of the block. On the

energy bar chart in Figure 3, draw shaded bars to represent the energy of the system for

when the block is at $x=x_{1}$ .

- The height of the shaded bars should be proportional to the relative values of $U_{\mathrm{A}}, U_{\mathrm{B}}$ ,

and $K_{\text {block }}$ .

- Any energy that is equal to zero should be represented by a distinct line on the zero-energy line.

Figure 3

[ 3 ]

A	For indicating that $K_{\text {block }}$ is zero	Point A1
		For drawing bars with positive heights for $U_{\mathrm{A}}$ and $U_{\mathrm{B}}$	Point A2
		For drawing a bar for U_B with a height that is twice the height of the bar drawn for U_A. Scoring Note: This point may be earned regardless of the signs of either bar.	Point A3

Example Response
Figure 3

(b)

The block is released from rest at $x=x_{1}$ and begins to oscillate. Derive an expression for the

speed v of the block as the block passes through $x=\frac{1}{2} x_{1}$ . Express your answer in terms of m,

$k, x_{1}$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental

physics principle or an equation from the reference information.

[ 4 ]

\multirow[t]{4}{*}{B} & For a multistep derivation that includes energy conservation or simple harmonic motion & Point B1 \\ \hline & For relating the presence of both springs to the behavior of the system & Point B2 \\ \hline & For relating positions $x=x_{1}$ and $x=\frac{1}{2} x_{1}$ to the oscillation of the block & Point B3 \\ \hline & For a correct expression for v in terms of given quantities & Point B4 \\ \hline \end{tabular}

Example Responses

		$x=x_{\text {max }} \cos (\omega t+\phi)$
$E_{\text {tot }, 0}=U_{\mathrm{A}}\left(x_{1}\right)+U_{\mathrm{B}}\left(x_{1}\right)+K_{\text {block }}\left(x_{1}\right)$
$E_{\text {tot }, f}=U_{\mathrm{A}}\left(\frac{1}{2} x_{1}\right)+U_{\mathrm{B}}\left(\frac{1}{2} x_{1}\right)+K_{\text {block }}\left(\frac{1}{2} x_{1}\right)$		$\begin{aligned} & x(t)=x_{1} \cos \omega t & v(t)=-x_{1} \omega \sin \omega t \end{aligned}$
$\begin{aligned} & E_{\mathrm{tot}, 0}=\frac{1}{2}(k)\left(x_{1}\right)^{2}+\frac{1}{2}(2 k)\left(x_{1}\right)^{2}+0 & E_{\mathrm{tot}, f}=\frac{1}{2}(k)\left(\frac{1}{2} x_{1}\right)^{2}+\frac{1}{2}(2 k)\left(\frac{1}{2} x_{1}\right)^{2}+K_{\mathrm{block}, \frac{1}{2} x_{1}} \end{aligned}$	OR	$\begin{aligned} & k_{\text {eff }}=k+2 k=3 k & \omega=\sqrt{\frac{3 k}{m}} \end{aligned}$
$\begin{aligned} & E_{\text {tot }, 0}=E_{\text {tot }, f} & \frac{1}{2} k x_{1}^{2}+k x_{1}^{2}=\frac{1}{8} k x_{1}^{2}+\frac{1}{4} k x_{1}^{2}+K_{\text {block }, \frac{1}{2} x_{1}} & \frac{3}{2} k x_{1}^{2}=\frac{3}{8} k x_{1}^{2}+K_{\text {block, } \frac{1}{2} x_{1}} \end{aligned}$		$\begin{aligned} & \frac{x_{1}}{2}=x_{1} \cos \omega t_{\frac{1}{2} x_{1}} & \cos \omega t_{\frac{1}{2} x_{1}}=\frac{1}{2} & t_{\frac{1}{2} x_{1}}=\frac{\pi}{3 \omega} \end{aligned}$
$\begin{aligned} & K_{\text {block }, \frac{1}{2} x_{1}}=\frac{9}{8} k x_{1}^{2} & \frac{1}{2} m v^{2}=\frac{9}{8} k x_{1}^{2} \end{aligned}$		$\begin{aligned} v_{\frac{1}{2} x_{1}} & =-x_{1} \omega \sin \omega t_{\frac{1}{2} x_{1}} v_{\frac{1}{2} x_{1}} & =-x_{1} \sqrt{\frac{3 k}{m}} \sin \frac{\pi}{3} \end{aligned}$
$v=\frac{3}{2} x_{1} \sqrt{\frac{k}{m}}$		$v=\left\|v_{\frac{1}{2}} x_{1}\right\|=\frac{3}{2} x_{1} \sqrt{\frac{k}{m}}$

(c)

In Scenario 1, the block oscillates with period T. The position x of the block in Scenario 1 as a

function of time t is shown in Figure 4.

Scenario 1

In Scenario 2, the block-springs system is placed on a new surface. There is friction between

the block and the new surface. The block is again moved to the same position $x=x_{1}$ and

released from rest. The block completes multiple oscillations with the same period as in

Scenario 1 before coming to rest.

On the axes shown in Figure 5, sketch a graph of the kinetic energy K of the block as a

function of t for Scenario 2.

Scenario 2

[ 3 ]

\multirow[t]{4}{*}{C} & For sketching a curve that starts at zero and is always positive or zero & Point C1 \\ \hline & For sketching a periodic curve with zeros that have a period of $\frac{1}{2} T$ & Point C2 \\ \hline & For sketching a periodic curve with a decreasing amplitude & Point C3 \\ \hline &

Example Response
Scenario 2
Figure 5

& \\ \hline

[Maximum number: 1]

An electrical motor provides 0.50 W of mechanical power. How much time will it take the motor to lift a 0.1 kg mass at constant speed from the floor to a shelf 2.0 m above the floor?

A

0.25 s

B

0.40 s

C

1.0 s

D

2.0 s

E

4.0 s

Figure for Question 3 Work, Energy, and Power question 4 — AP Physics C: Mechanics

AP Physics C Mechanics Unit 3: Work, Energy, and Power

Covered syllabus points

Covered syllabus points

3 Work, Energy, and Power question 1

Question (a)

Question (b)

3 Work, Energy, and Power question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (iii)

Question (d)

3 Work, Energy, and Power question 3

Question (a)

Question (b)

Question (c)

3 Work, Energy, and Power question 4