Practice Questions Simple Harmonic Motion - AP Physics 1 - Grade 11 PDF

SECTION I: MULTIPLE CHOICE

Directions

Answer the following 20 multiple-choice questions. Each question has four answer choices labeled A through D. Select the one best answer for each question. You may use a calculator and reference the equation sheet provided for AP Physics 1.

Question 1

A mass-spring system oscillates horizontally on a frictionless surface with simple harmonic motion. The mass is initially displaced 0.20 m from equilibrium and released from rest. The period of oscillation is 2.0 s. What is the maximum speed of the mass during one complete oscillation?

1. 0.10 m/s
2. 0.20 m/s
3. 0.31 m/s
4. 0.63 m/s

Question 2

A simple pendulum consists of a small bob of mass m attached to a string of length L. The pendulum is displaced by a small angle and released, executing simple harmonic motion. Which of the following changes would result in doubling the period of the pendulum?

1. Doubling the mass of the bob
2. Doubling the length of the string
3. Quadrupling the length of the string
4. Reducing the initial displacement angle by half

Question 3

Based on the graph in Figure 1, at which of the following times is the magnitude of the acceleration of the mass at its maximum value?

1. t = 0 s
2. t = 1.0 s
3. t = 2.0 s
4. t = 3.0 s

Question 4

A block of mass 0.50 kg is attached to a horizontal spring with spring constant 20 N/m. The block is displaced 0.10 m from equilibrium and released from rest. What is the maximum kinetic energy of the block during its motion?

1. 0.050 J
2. 0.10 J
3. 0.20 J
4. 1.0 J

Question 5

How does the period of System B compare to the period of System A?

1. The period of System B is four times the period of System A.
2. The period of System B is twice the period of System A.
3. The period of System B is half the period of System A.
4. The period of System B is one-fourth the period of System A.

Question 6

A mass oscillates on a vertical spring with period T and amplitude A. At what position(s) during the oscillation is the net force on the mass equal to zero?

1. At the equilibrium position only
2. At the maximum displacement positions only
3. At both the equilibrium position and the maximum displacement positions
4. The net force is never zero during the oscillation

Question 7

Data Table

A student measures the period of oscillation for a mass-spring system using different masses attached to the same spring.

Which of the following graphs would best linearize the data to determine the spring constant?

1. Period vs. Mass
2. Period vs. Square root of Mass
3. Period squared vs. Mass
4. Period squared vs. Mass squared

Question 8

A mass-spring system undergoes simple harmonic motion with amplitude A. At what displacement from equilibrium is the kinetic energy of the mass equal to the elastic potential energy stored in the spring?

1. \( x = \frac{A}{4} \)
2. \( x = \frac{A}{2} \)
3. \( x = \frac{A}{\sqrt{2}} \)
4. \( x = A \)

Question 9

What is the amplitude of oscillation?

1. 0.12 m
2. 0.24 m
3. 0.48 m
4. 0.75 m

Question 10

A simple pendulum of length 1.0 m oscillates with small amplitude on Earth where \( g = 10 \, \text{m/s}^2 \). The same pendulum is transported to a planet where the acceleration due to gravity is \( 2.5 \, \text{m/s}^2 \). How does the period of the pendulum on the planet compare to its period on Earth?

1. The period on the planet is one-fourth the period on Earth.
2. The period on the planet is half the period on Earth.
3. The period on the planet is twice the period on Earth.
4. The period on the planet is four times the period on Earth.

Question 11

A horizontal mass-spring system has total mechanical energy E when oscillating with amplitude A. If the amplitude is increased to 2A while the mass and spring constant remain unchanged, what is the new total mechanical energy of the system?

1. \( \frac{E}{2} \)
2. \( E \)
3. \( 2E \)
4. \( 4E \)

Question 12

What is the amplitude of the resulting oscillation?

1. 0.05 m
2. 0.10 m
3. 0.15 m
4. 0.20 m

Question 13

A mass oscillates on a horizontal spring with period T. The position of the mass as a function of time is given by \( x(t) = A \cos(\omega t) \), where A is the amplitude and \( \omega \) is the angular frequency. What is the acceleration of the mass at time \( t = \frac{T}{4} \)?

1. \( -\omega^2 A \)
2. \( 0 \)
3. \( \omega^2 A \)
4. \( \omega A \)

Question 14

At what position is the speed of the mass equal to half its maximum speed?

1. 0.07 m
2. 0.10 m
3. 0.12 m
4. 0.14 m

Question 15

A student models a swinging chandelier as a simple pendulum. The chandelier has a length of 4.0 m and swings with small amplitude. Which of the following is the best estimate of the period of oscillation? (Use \( g = 10 \, \text{m/s}^2 \))

1. 1.3 s
2. 2.0 s
3. 4.0 s
4. 8.0 s

Question 16

What is the period of oscillation for this system? (Use \( g = 10 \, \text{m/s}^2 \))

1. 0.45 s
2. 0.63 s
3. 0.89 s
4. 1.26 s

Question 17

Two mass-spring systems, System 1 and System 2, oscillate horizontally on a frictionless surface. System 1 has mass m and spring constant k. System 2 has mass 2m and spring constant 2k. Both are displaced by the same amount and released from rest. How do the maximum speeds of the two systems compare?

1. System 2 has half the maximum speed of System 1.
2. System 2 has the same maximum speed as System 1.
3. System 2 has \( \sqrt{2} \) times the maximum speed of System 1.
4. System 2 has twice the maximum speed of System 1.

Question 18

Which of the following statements about this system is correct?

1. The kinetic energy is constant at all positions.
2. The elastic potential energy is constant at all positions.
3. The amplitude of oscillation is 0.20 m.
4. The spring constant cannot be determined from the given information.

Question 19

A mass on a horizontal spring oscillates with simple harmonic motion. At a certain instant, the mass is at position \( x = +0.10 \, \text{m} \) (to the right of equilibrium) and has velocity \( v = -0.30 \, \text{m/s} \) (moving to the left). Which of the following correctly describes the direction of the acceleration and the direction the mass is moving at this instant?

1. Acceleration is to the left; mass is moving to the left.
2. Acceleration is to the left; mass is moving to the right.
3. Acceleration is to the right; mass is moving to the left.
4. Acceleration is to the right; mass is moving to the right.

Question 20

Which of the following would best support the student's hypothesis?

1. The measured periods are all identical within experimental uncertainty.
2. The measured periods increase linearly with increasing mass.
3. The measured periods decrease as mass increases.
4. The measured periods are proportional to the square root of the mass.

SECTION II: FREE RESPONSE

Directions

Answer both of the following free-response questions. Show all your work clearly, including equations, substitutions with units, and a box around your final answer where appropriate. Partial credit may be awarded for incomplete work. The suggested time for answering both questions is 50 minutes.

FRQ 1: Mathematical Routines (25 minutes)

1. Begin your derivation by writing a fundamental physics principle or equation. Derive an expression for the speed v of the block when it first passes through the equilibrium position in terms of m, k, A, μ, and physical constants as appropriate. Show all steps in your derivation.
2. A student performs an experiment with m = 0.50 kg, k = 25 N/m, A = 0.20 m, and μ = 0.10. Calculate the numerical value of the speed when the block first passes through equilibrium. Use \( g = 10 \, \text{m/s}^2 \).
3. After passing through equilibrium the first time, the block continues to oscillate, but with decreasing amplitude due to friction. Explain, using physics principles, why the motion is not considered simple harmonic motion even though a spring is involved.
4. Suppose the experiment is repeated on a surface where friction is negligible (μ ≈ 0). Sketch a graph of the block's kinetic energy versus position for one complete oscillation, starting from maximum displacement. Clearly label the axes, including maximum values and the equilibrium position.

FRQ 2: Experimental Design and Analysis (25 minutes)

1. Describe an experimental procedure the student could use to collect data necessary to determine the spring constant. Your description should include:
   • What quantities should be measured
   • How each quantity should be measured
   • A step-by-step procedure including what should be varied
   • What steps ensure accuracy and precision
   You may include a simple diagram if it helps clarify your procedure.
2. Describe how the student should graph the collected data to determine the spring constant. Your description should include:
   • What quantity should be graphed on the vertical axis
   • What quantity should be graphed on the horizontal axis
   • What the slope of the best-fit line represents in terms of physical quantities
   • How the spring constant k can be calculated from the slope
3. The student collects the following data: Complete the data table below by calculating the appropriate quantities based on your answer to part (b). Show a sample calculation for one row.
4. On the grid below, plot your calculated data points from part (c) and draw a best-fit line. [You would be provided with a grid in the actual exam]
5. The slope of the best-fit line from the student's graph is \( 2.0 \, \text{s}^2/\text{kg} \). Calculate the spring constant k. Show all work, including the equation you use and substitution with units.

ANSWER KEY

Part A: Multiple Choice Answer Table

Part B: Free Response Detailed Answers

FRQ 1 - Answer Key

Part A: Derivation of speed at equilibrium

Fundamental principle: Conservation of energy, accounting for work done by friction.

The initial state has the block at rest at displacement A from equilibrium. The final state has the block at equilibrium position with speed v.

Step 1: Write the energy conservation equation with work done by friction:
\[ E_{\text{initial}} + W_{\text{friction}} = E_{\text{final}} \]

Step 2: Express each term:
Initial energy: \( E_{\text{initial}} = \frac{1}{2}kA^2 + 0 \) (elastic PE + KE)
Final energy: \( E_{\text{final}} = 0 + \frac{1}{2}mv^2 \) (elastic PE + KE)
Work by friction: \( W_{\text{friction}} = -f_k \cdot d = -\mu mg \cdot A \) (negative because friction opposes motion; distance traveled is A)

Step 3: Substitute into energy equation:
\[ \frac{1}{2}kA^2 - \mu mg A = \frac{1}{2}mv^2 \]

Step 4: Solve for v:
\[ \frac{1}{2}mv^2 = \frac{1}{2}kA^2 - \mu mg A \]
\[ v^2 = \frac{kA^2}{m} - 2\mu g A \]
\[ v = \sqrt{\frac{kA^2}{m} - 2\mu g A} \]

Final answer:

\[ v = \sqrt{\frac{kA^2}{m} - 2\mu g A} \]

Alternative acceptable form:

\[ v = A\sqrt{\frac{k}{m} - \frac{2\mu g}{A}} \]

Part B: Numerical calculation

Given: m = 0.50 kg, k = 25 N/m, A = 0.20 m, μ = 0.10, g = 10 m/s²

Step 1: Substitute values into the derived equation:
\[ v = \sqrt{\frac{(25 \, \text{N/m})(0.20 \, \text{m})^2}{0.50 \, \text{kg}} - 2(0.10)(10 \, \text{m/s}^2)(0.20 \, \text{m})} \]

Step 2: Calculate the first term:
\[ \frac{kA^2}{m} = \frac{(25)(0.04)}{0.50} = \frac{1.0}{0.50} = 2.0 \, \text{m}^2/\text{s}^2 \]

Step 3: Calculate the second term:
\[ 2\mu g A = 2(0.10)(10)(0.20) = 0.40 \, \text{m}^2/\text{s}^2 \]

Step 4: Subtract and take square root:
\[ v = \sqrt{2.0 - 0.40} = \sqrt{1.6} = 1.26 \, \text{m/s} \]

Final answer: \( v = 1.3 \, \text{m/s} \) (or 1.26 m/s with more precision)

Part C: Explanation of why motion is not SHM

Simple harmonic motion requires that the net force on the object is proportional to the displacement from equilibrium and directed toward equilibrium, with no other forces doing work on the system. Mathematically, this means \( F_{\text{net}} = -kx \) only.

In this scenario, friction does negative work continuously as the block moves. This means:

• The total mechanical energy decreases over time
• The amplitude of oscillation decreases with each cycle
• The restoring force is not the only force acting (friction also acts)
• Energy is not conserved; it is dissipated as thermal energy

Because the amplitude is not constant and mechanical energy is not conserved, the motion does not satisfy the requirements for simple harmonic motion. The motion is damped oscillation, not SHM.

Part D: Graph of kinetic energy vs. position

Description of correct graph:

• Horizontal axis: Position x, ranging from -A to +A, with 0 labeled as equilibrium
• Vertical axis: Kinetic energy KE, ranging from 0 to \( \frac{1}{2}kA^2 \)
• Shape: An inverted parabola (parabola opening downward)
• Key features:
  • Maximum KE = \( \frac{1}{2}kA^2 \) at x = 0
  • KE = 0 at x = ±A
  • Symmetric about x = 0

The graph should show a smooth parabolic curve with equation \( KE = \frac{1}{2}k(A^2 - x^2) \).

FRQ 2 - Answer Key

Part A: Experimental procedure

Quantities to measure:

• Mass m of each mass used (read from labeled mass set)
• Period T of oscillation for each mass (measured with stopwatch)

Step-by-step procedure:

1. Attach the spring horizontally to a fixed support on a frictionless track.
2. Select a known mass from the mass set and attach it securely to the free end of the spring.
3. Displace the mass a small distance (approximately 5-10 cm) from its equilibrium position and release it from rest.
4. Use the stopwatch to measure the total time for 10 complete oscillations. (Starting and stopping the timer as the mass passes through the same position moving in the same direction.)
5. Calculate the period by dividing the total time by 10: \( T = \frac{\text{total time}}{10} \).
6. Repeat steps 3-5 two more times for the same mass and average the three period values to reduce measurement uncertainty.
7. Repeat steps 2-6 for at least four additional different masses, ensuring a wide range of mass values.
8. Record all mass and period values in a data table.

Steps to ensure accuracy and precision:

• Measure the time for multiple oscillations (10) rather than just one to reduce timing error
• Take multiple trials for each mass and average the results
• Ensure the amplitude is small and consistent across trials
• Ensure the track is level and friction is negligible
• Start and stop the timer at the same point in the oscillation cycle

Part B: Graphing and analysis procedure

Vertical axis: Period squared, \( T^2 \) (units: s²)

Horizontal axis: Mass, m (units: kg)

What the slope represents:

The relationship between period and mass for a mass-spring system is given by:
\[ T = 2\pi\sqrt{\frac{m}{k}} \]

Squaring both sides:
\[ T^2 = 4\pi^2 \frac{m}{k} \]

This is a linear relationship of the form \( T^2 = \left(\frac{4\pi^2}{k}\right) m \), which means:

The slope of the \( T^2 \) vs. m graph equals \( \frac{4\pi^2}{k} \).

Calculating the spring constant from the slope:

If the slope is denoted as s, then:
\[ s = \frac{4\pi^2}{k} \]

Solving for k:
\[ k = \frac{4\pi^2}{s} \]

Part C: Completed data table with sample calculation

Vertical axis quantity: \( T^2 \) (s²)

Sample calculation for m = 0.10 kg:
\( T = 0.45 \, \text{s} \)
\( T^2 = (0.45)^2 = 0.20 \, \text{s}^2 \)

Part D: Graph instructions

Students should plot the five data points from part C on a grid with:

• Horizontal axis labeled "Mass m (kg)" with scale from 0 to 0.50 kg
• Vertical axis labeled "\( T^2 \) (s²)" with scale from 0 to 1.0 s²
• Each point plotted accurately
• A best-fit straight line drawn through or near the points (not connecting dot-to-dot)
• The line should pass through or very close to the origin

Part E: Calculate spring constant from slope

Given: Slope s = 2.0 s²/kg

Equation relating slope to spring constant:
\[ k = \frac{4\pi^2}{s} \]

Substitution:
\[ k = \frac{4\pi^2}{2.0 \, \text{s}^2/\text{kg}} \]

Calculation:
\[ k = \frac{4(3.14159)^2}{2.0} \, \text{kg/s}^2 \]
\[ k = \frac{4(9.8696)}{2.0} \, \text{N/m} \]
\[ k = \frac{39.48}{2.0} \, \text{N/m} \]
\[ k = 19.7 \, \text{N/m} \]

Final answer: \( k = 20 \, \text{N/m} \) (or 19.7 N/m, both acceptable given significant figures)