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AP Physics 1- 5.4 Rotational Inertia - Exam Style questions - FRQs- New Syllabus

Rotational Inertia AP  Physics 1 FRQ

Unit 5: Torque and Rotational Dynamics

Weightage : 10-15%

AP Physics 1 Exam Style Questions – All Topics

Question

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.
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 of the block, \( m_0 \).
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.
B. Describe how the data collected in part \( \mathrm{A} \) could be graphed and how that graph would be analyzed to determine \( m_0 \).
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 \( 2 \).
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 force \( F_T \) required to hold the meterstick horizontal. Table \( 1 \) shows the measured values of \( \theta \) and \( F_T \).
\( \theta \) \( (\mathrm{degrees}) \)\( F_T \) \( (\mathrm{N}) \)
\( 22 \)\( 21 \)
\( 31 \)\( 17 \)
\( 36 \)\( 13 \)
\( 45 \)\( 12 \)
\( 80 \)\( 8 \)
The students correctly determine that the relationship between \( F_T \) and \( \theta \) is given by
\( F_T = \dfrac{5Mg}{6\sin\theta} \)
The students create a graph with \( \dfrac{1}{\sin\theta} \) plotted on the horizontal axis.
C.
(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: \( \dfrac{1}{\sin\theta} \)
(ii) On the blank grid provided, create a graph of the quantities indicated in part \( \mathrm{C(i)} \) 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 \( 2 \).
(iii) Draw a straight best-fit line for the data graphed in part \( \mathrm{C(ii)} \).
D. Using the best-fit line that you drew in part \( \mathrm{C(iii)} \), calculate an experimental value for the mass \( M \) of the meterstick.

Most-appropriate topic codes (AP Physics 1):

• Topic \( 5.4 \) — Torque and the Second Condition of Equilibrium (Part \( \mathrm{A} \), Part \( \mathrm{B} \), Part \( \mathrm{C} \), Part \( \mathrm{D} \))
• Topic \( 5.5 \) — Rotational Equilibrium and Newton’s First Law in Rotational Form (Part \( \mathrm{A} \), Part \( \mathrm{B} \), Part \( \mathrm{C} \), Part \( \mathrm{D} \))
• Topic \( 2.2 \) — Forces and Free-Body Diagrams (Part \( \mathrm{A} \), Part \( \mathrm{C} \))
• Topic \( 1.1 \) — Scalars and Vectors in One Dimension (Part \( \mathrm{C} \), graphing and measured quantities)
▶️ Answer/Explanation

A
Attach the block of unknown mass \( m_0 \) to one of the small holes in the meterstick. Keep the meterstick horizontal and record the force reading from the spring scale.

Repeat the measurement for several different block positions \( r_b \) measured from the stand \( (\text{pivot}) \). For each position, also record the distance from the stand to the spring scale attachment point, which stays fixed.

To reduce experimental uncertainty, repeat the spring-scale reading multiple times for each block position and average the values. It also helps to use many different hole positions so the graph is based on several data points instead of only one or two.

A good experimental habit is to make sure the meterstick is level before reading the scale each time and to read the scale at eye level to reduce parallax error.

B
One valid graph is force as a function of the distance from the stand to the block.

Taking torques about the stand,

\( F_s r_s = m_0 g\, r_b \)

where \( F_s \) is the spring-scale force, \( r_s \) is the distance from the stand to the spring scale, and \( r_b \) is the distance from the stand to the block.

Rearranging,

\( F_s = \left(\dfrac{m_0 g}{r_s}\right) r_b \)

So a graph of \( F_s \) on the vertical axis versus \( r_b \) on the horizontal axis is linear. Its slope is

\( \text{slope} = \dfrac{m_0 g}{r_s} \)

Therefore,

\( m_0 = \dfrac{(\text{slope})\,r_s}{g} \)

Any equivalent linear graph that correctly relates force and distance also earns full credit.

C(i)
Vertical axis: \( F_T \)

Since \( F_T = \dfrac{5Mg}{6\sin\theta} = \left(\dfrac{5Mg}{6}\right)\left(\dfrac{1}{\sin\theta}\right) \), a plot of \( F_T \) versus \( \dfrac{1}{\sin\theta} \) is linear.

C(ii)
Use the calculated values of \( \dfrac{1}{\sin\theta} \) together with the measured values of \( F_T \).

Table \( 2 \)\( \theta \) \( (\mathrm{degrees}) \)\( \dfrac{1}{\sin\theta} \)\( F_T \) \( (\mathrm{N}) \)
Point \( 1 \)\( 22 \)\( 2.67 \)\( 21 \)
Point \( 2 \)\( 31 \)\( 1.94 \)\( 17 \)
Point \( 3 \)\( 36 \)\( 1.70 \)\( 13 \)
Point \( 4 \)\( 45 \)\( 1.41 \)\( 12 \)
Point \( 5 \)\( 80 \)\( 1.02 \)\( 8 \)

The graph should have vertical axis \( F_T\ (\mathrm{N}) \) and horizontal axis \( \dfrac{1}{\sin\theta} \). Plot the points:

\( (2.67,21),\ (1.94,17),\ (1.70,13),\ (1.41,12),\ (1.02,8) \)

These points lie approximately on a straight line.

C(iii)
Draw a straight best-fit line through the plotted data. It should have positive slope and pass close to the origin.

Because the equation has the form \( F_T = \left(\dfrac{5Mg}{6}\right)\left(\dfrac{1}{\sin\theta}\right) \), the slope of the best-fit line is the important quantity for finding \( M \).

D
From a reasonable best-fit line, the slope is about \( 8.4\ \mathrm{N} \).

Since

\( F_T = \left(\dfrac{5Mg}{6}\right)\left(\dfrac{1}{\sin\theta}\right) \),

the slope is \( \dfrac{5Mg}{6} \).

Therefore,

\( M = \dfrac{6(\text{slope})}{5g} \)

\( M = \dfrac{6(8.4\ \mathrm{N})}{5(9.8\ \mathrm{N/kg})} \)

\( M \approx 1.03\ \mathrm{kg} \)

So an experimental value for the mass of the meterstick is \( \boxed{1.0\ \mathrm{kg}} \) \( (\text{approximately}) \).

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