AP Physics 1 5.5 Rotational Equilibrium

[Maximum number: 10]

Students are investigating balancing systems using the following setup. The students have a spring scale of negligible mass that is fixed to one end of a uniform meterstick. The center of the meterstick is attached to a stand on which the meterstick can pivot. There is a hook of negligible mass fixed to the top of a block of mass $m_{0}$ . The hook can be attached to the meterstick through one of the small holes in the meterstick, as shown in Figure 1. The students do not have a direct way to measure the mass of the block. The block cannot be attached to the spring scale.

Figure 1

The students are asked to take measurements that will allow the students to create a linear graph whose slope could be used to determine the mass $m_{0}$ of the block.

(a)

Describe an experimental procedure to collect data that would allow the students to determine $m_{0}$ . Include any steps necessary to reduce experimental uncertainty.

[ 2 ]

A For describing a procedure that includes both of the following:

Point A1

- Attaching the block to the meterstick

- Measuring the force exerted on the left end of the meterstick while the block is

attached

For a procedure that indicates a reasonable method of reducing experimental

Point A2

uncertainty

Examples of acceptable responses may include the following:

- Repeating the force measurement for one location of the block

- Collecting force measurements for multiple locations of the block

Example Response

Attach the block of unknown mass $m_{0}$ through one of the small holes in the meterstick.

Measure the distance from the stand to the block. Record the force reading from the

spring scale when the meterstick is horizontal. Repeat data collection by attaching the

block to the meterstick through a different small hole in the meterstick. Repeat the

measurements by attaching the block at each of the small hole locations numerous

times.

(b)

Describe how the data collected in part A could be graphed and how that graph would be analyzed to determine $m_{0}$ .

[ 2 ]

B For indicating appropriate quantities that can be plotted so that $m_{0}$ can be determined

Scoring Notes:

- Any response that correctly identifies the functional dependence between force

and distance earns this point, regardless of any coefficients that contain

numbers or physical/fundamental constants, or which axis is chosen to graph

each of those quantities.

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

For describing how the slope is related to the unknown mass $m_{0}$

Examples of acceptable responses may include the following:

Graph	Analysis
Force as a function of the distance from the stand to the block	The slope is equal to $m_{0}$ times g divided by the distance from the stand to the spring scale.<br>OR<br> $m_{0}$ is equal to the slope times the distance from the stand to the spring scale divided by g.
Force as a function of the ratio of the distance from the stand to the block to the distance from the stand to the spring scale	The slope is equal to $m_{0}$ times g.<br>OR<br> $m_{0}$ is equal to the slope divided by g.
Force as a function of g times the distance from the stand to the block	The slope is equal to $m_{0}$ divided by the distance from the stand to the spring scale.<br>OR<br> $m_{0}$ is equal to the slope times the distance from the stand to the spring scale.
Force as a function of g times the ratio of the distance from the stand to the block to the distance from the stand to the spring scale	The slope is equal to $m_{0}$ .<br>OR<br> $m_{0}$ is equal to the slope.
Force times the distance from the stand to the spring scale as a function of the distance from the stand to the block	The slope is equal to $m_{0}$ times g.<br>OR<br> $m_{0}$ is equal to the slope divided by g.
Force times the distance from the stand to the spring scale as a function of g times the distance from the stand to the block	The slope is equal to $m_{0}$ .<br>OR<br> $m_{0}$ is equal to the slope.

Example Response

A graph of the force as a function of the distance from the stand to the block could be

used to determine the unknown mass. The slope of the graph times the distance from the

stand to the spring scale divided by g will result in the value of the unknown mass $m_{0}$ .

(c)

The students have an identical meterstick of mass M that is now attached to an axle that is fixed to a wall. The meterstick is free to rotate with negligible friction about the axle. The meterstick is suspended horizontally by a string that is connected to a spring scale of negligible mass, as shown in Figure 2.

Figure 2

The angle $\theta$ that the string makes with the meterstick can be varied by attaching the string to one of the pegs located along the wall. The students use the spring scale to measure the tension $F_{T}$ required to hold the meterstick horizontal. Table 1 shows the measured values of $\theta$ and $F_{T}$ .

Table 1

The students correctly determine that the relationship between $F_{T}$ and $\theta$ is given by

F_{T}=\frac{5 M g}{6 \sin \theta} .

The students create a graph with $\frac{1}{\sin \theta}$ plotted on the horizontal axis.

[ 4 ]

(i)

Indicate what measured or calculated quantity could be plotted on the vertical axis to yield a linear graph whose slope can be used to calculate an experimental value for the mass M of the meterstick.

Vertical axis:

Horizontal axis: $\frac{1}{\sin \theta}$

[ 1 ]

C (i) For listing a quantity that could be plotted on the vertical axis to produce a linear graph

Point C1

whose slope can be used to determine the mass of the meterstick ${ }^{1}$

Examples of acceptable responses may include the following:

- $F_{T}$

- $\frac{F_{T}}{g}$

- $\frac{6}{5} \frac{F_{T}}{g}$

- $\frac{6}{5} F_{T}$

Scoring Note: Any response that correctly identifies the functional dependence

between tension and angle earns this point, regardless of any coefficients that contain

numbers or physical/fundamental constants.

Example Response

Vertical axis: $F_{T}$

(ii)

On the blank grid provided, create a graph of the quantities indicated in part C (i) that can be used to determine M.

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

- Clearly label the vertical axis, including units as appropriate.

- Plot the points you recorded in Table 2.

Figure 3

[ 2 ]

(iii)

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

[ 1 ]

(iii) For drawing an appropriate best-fit line that approximates the trend of the plotted data

Point C4

Scoring Note: If the graph produced is nonlinear, then a line or curve that approximates

the trend of that data can earn this point.

${ }^{1}$ Update made post-scoring to correct typo that did not impact scoring.

Example Response

Figure 3

(d)

Using the best-fit line that you drew in part C (iii), calculate an experimental value for the mass M of the meterstick.

[ 2 ]

D For correctly relating the slope of the best-fit line to the value of M

Examples of acceptable responses may include the following:

Graph	Analysis
$F_{T}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=\frac{5}{6} M g \text { OR } M=\frac{6}{5} \frac{\text { slope }}{g} \end{gathered}$
$\frac{F_{T}}{g}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=\frac{5}{6} M \text { OR } M=\frac{6}{5} \text { slope } \end{gathered}$
$\frac{6}{5} \frac{F_{T}}{g}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=M \text { OR } M=\text { slope } \end{gathered}$
$\frac{6}{5} F_{T}$ vs. $\frac{1}{\sin \theta}$	$\begin{gathered} \text { slope }=M g \text { OR } M=\frac{\text { slope }}{g} \end{gathered}$

For a value of M within a range of 0.90 kg to 1.15 kg

AP Physics 1: Algebra-Based 5.5 Rotational Equilibrium and Newtons First Law in Rotational Form Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (iii)

Question (d)