AP Physics C Mechanics - 2.8 Spring Forces- Exam Style questions- FRQs

Spring Forces AP  Physics C Mechanics FRQ

Unit 2: Force and Translational Dynamics

Weightage : 20-15%

AP Physics C Mechanics Exam Style Questions – All Topics

Question

A student conducts an investigation to determine the relationship between the period of oscillation \(T\) of a system consisting of a block and \(N\) attached springs. The student starts with a block of mass \(m\) attached to a single ideal spring of spring constant \(k\), as shown in Figure 1. The student holds the block so that the spring is neither stretched nor compressed at a vertical height \(1.00\,\text{m}\) above a motion detector. The student releases the block from rest and records the period of oscillation for the system consisting of the single spring and block. An additional identical spring is attached in parallel, as shown in Figure 2, and the procedure is repeated for \(N=2\) springs. This procedure is repeated through \(N=10\) springs.
(a) Derive an expression for \(T\) as a function of \(N\). Express your answer in terms of \(m\), \(k\), \(N\), and physical constants as appropriate.
(b) On the following axes, sketch a graph of \(T\) as a function of \(N\) for \(N \ge 1\).
(c) The student plots the data for \(T^2\) as a function of \(N^{-1}\) as shown.
i. Draw the best-fit line for the data.
ii. The mass of the block is measured to be \(m=1.5\,\text{kg}\). Using the graph, calculate an experimental value for the spring constant \(k\) for a single spring.
iii. The student finds that the value given by the manufacturer for the spring constant is larger than the value determined experimentally in part (c)(ii). Determine a single source of experimental error that could result in the observed difference in the value for \(k\). Briefly justify your answer.
(d) The student conducts a similar investigation to determine the relationship between the period of oscillation \(T\) of a system consisting of a block and a number \(N\) of identical springs but arranges the block-spring system horizontally on a table, as shown in Figure 3. Frictional forces between the table and the block are negligible.
In each trial, the block is displaced the same horizontal distance from equilibrium and released from rest.
i. The student plots \(T^2\) as a function of \(N^{-1}\) for this new data. Would the slope of the best-fit line from this new investigation be greater than, less than, or the same as the slope of the best-fit line in part (c)(i)?
_____ greater than     _____ less than     _____ the same as
Briefly justify your answer.
ii. When \(N=1\), the maximum speed of the block is found to be \(v_{\text{max}}\). When \(N\) increases, will \(v_{\text{max}}\) increase, decrease, or stay the same?
_____ increase     _____ decrease     _____ stay the same
Justify your answer.

Most-appropriate topic codes (AP Physics C: Mechanics):

• Topic 2.8 — Spring Forces (Parts a, d)
• Topic 7.2 — Frequency and Period of SHM (Parts a, b, c, d)
• Topic 7.4 — Energy of Simple Harmonic Oscillators (Part d)
▶️ Answer/Explanation

(a)
When identical springs are arranged in parallel, the equivalent spring constant is simply the sum of the individual constants.
\( k_{\text{eq}} = \sum_{i=1}^{N} k_i \)
\( k_{\text{eq}} = Nk \)
The period of a mass on a spring is derived from the standard equation:
\( T = 2\pi\sqrt{\dfrac{m}{k_{\text{eq}}}} \)
Substituting the equivalent spring constant gives the final expression:
\( \boxed{T = 2\pi\sqrt{\dfrac{m}{Nk}}} \)

(b)


Using the formula \( T = \dfrac{2\pi\sqrt{m/k}}{\sqrt{N}} \), we see that the period \( T \) is proportional to \( N^{-1/2} \).
Therefore, the graph should be a decreasing curve that is concave up.

(c)(i)


Draw an appropriate straight line of best fit that closely matches the trend of the plotted data, balancing the points above and below the line.

(c)(ii)
Select two points directly from your line of best fit to determine the slope.
\( \text{slope} = \dfrac{\Delta(T^2)}{\Delta(N^{-1})} \)
Using typical data points (e.g., \( (0.3, 3.5) \) and \( (0.8, 9.5) \)):
\( \text{slope} = \dfrac{9.5\,\text{s}^2 – 3.5\,\text{s}^2}{0.8 – 0.3} = 12\,\text{s}^2 \)
By squaring the expression from part (a), we get \( T^2 = 4\pi^2\dfrac{m}{k} (N^{-1}) \), which means the slope is equal to \( 4\pi^2\dfrac{m}{k} \).
\( k = \dfrac{4\pi^2m}{\text{slope}} \)
\( k = \dfrac{4\pi^2(1.5\,\text{kg})}{12\,\text{s}^2} \)
\( \boxed{k \approx 4.93\,\text{N/m}} \)

(c)(iii)
A possible source of error is that the true mass of the oscillating system was larger than \( 1.5\,\text{kg} \) because the mass of the springs wasn’t accounted for.
A larger mass will yield a larger measured period \( T \) for the oscillation. Since the calculated \( k \) is inversely proportional to the slope (which depends on \( T^2 \)), a larger period leads to a smaller calculated experimental value for \( k \).

(d)(i)
Correct choice: the same as
The period of a horizontal or vertical spring-block system depends exclusively on the mass of the block and the spring constant.
Because the period is completely independent of the gravitational force, altering the orientation of the setup does not affect the period, keeping the slope unchanged.

(d)(ii)
Correct choice: increase
Adding more springs increases the effective spring constant \( k_{\text{eq}} \).
For the same displacement \( x \), the elastic potential energy stored in the system \( U_s = \dfrac{1}{2}k_{\text{eq}}x^2 \) will be greater.
Through the conservation of mechanical energy, this extra potential energy converts entirely into kinetic energy at the equilibrium position, resulting in a higher maximum speed \( v_{\text{max}} \).

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