Step-by-Step Guidance for Oscillations and Simple Harmonic Motion (Ch14)

Q14.8. Mass Hanging from a Spring: Simple Harmonic Motion

Background

Topic: Simple Harmonic Motion (SHM) and Forces in Oscillating Systems

This question tests your understanding of the forces, velocity, and acceleration at different points in the oscillation of a mass-spring system.

Key Terms and Formulas:

• Equilibrium Point: The position where the net force on the mass is zero.

• Amplitude: The maximum displacement from equilibrium.

• Velocity: Rate of change of position; zero at turning points.

• Acceleration: Rate of change of velocity; maximum at extreme positions.

• Spring Force: $F_{\text{spring}} = -k\Delta y$

• Net Force: $F_{\text{net}} = m a$

Step-by-Step Guidance

1. Consider the equilibrium position: This is where the spring is stretched just enough to balance the weight of the mass ($mg$). At this point, the net force is zero.

2. Think about the turning points (highest and lowest positions): At these points, the mass changes direction, so its velocity is momentarily zero.

3. Recall that acceleration is greatest where the restoring force is largest. This occurs at the maximum displacement from equilibrium (the highest and lowest points).

4. For the spring force to be zero, the spring must be at its natural (unstretched) length. Consider whether the oscillation ever reaches this point, given the equilibrium position is already stretched by the mass.

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Q14.10. Determining Frequency and Amplitude from a Graph

Background

Topic: Analyzing Oscillation Graphs

This question asks you to extract the frequency and amplitude of an oscillator from a position vs. time graph.

Key Terms and Formulas:

• Period ($T$): Time for one complete cycle.

• Frequency ($f$): Number of cycles per second, $f = \frac{1}{T}$

• Amplitude ($A$): Maximum displacement from equilibrium.

• Maximum Speed ($v_{\text{max}}$): $v_{\text{max}} = 2\pi f A$

Step-by-Step Guidance

1. Examine the graph and identify the period ($T$) by measuring the time between two consecutive peaks or troughs.

2. Calculate the frequency using $f = \frac{1}{T}$.

3. Determine the amplitude by measuring the maximum displacement from the equilibrium position (the highest or lowest point on the graph).

4. If maximum speed is given, use $v_{\text{max}} = 2\pi f A$ to check your amplitude calculation.

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P14.7. Air-Track Glider Oscillation

Background

Topic: Simple Harmonic Motion (SHM) of a Mass-Spring System

This question tests your ability to relate maximum speed, period, and amplitude, and to use the position function for SHM.

Key Terms and Formulas:

• Maximum Speed: $v_{\text{max}} = 2\pi \frac{A}{T}$

• Position as a function of time: $x(t) = A \cos\left(2\pi \frac{t}{T}\right)$

• Period ($T$): Time for one complete oscillation.

• Amplitude ($A$): Maximum displacement from equilibrium.

Step-by-Step Guidance

1. Identify the known values: $T = 2.0$ s, $v_{\text{max}} = 0.40$ m/s.

2. Use the formula $v_{\text{max}} = 2\pi \frac{A}{T}$ to solve for amplitude $A$.

3. For position at $t = 0.25$ s, use $x(t) = A \cos\left(2\pi \frac{t}{T}\right)$ and substitute the values for $A$, $t$, and $T$.

4. Calculate the cosine argument and set up the expression for $x(0.25)$.

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P14.18. Maximum Acceleration of Hummingbird Wing Tips

Background

Topic: Maximum Acceleration in Simple Harmonic Motion

This question asks you to find the maximum acceleration of an oscillating system using amplitude and frequency.

Key Terms and Formulas:

• Maximum Acceleration: $a_{\text{max}} = (2\pi f)^2 A$

• Frequency ($f$): Number of cycles per second.

• Amplitude ($A$): Maximum displacement.

• Acceleration in units of $g$: $a_{\text{max}} / g$

Step-by-Step Guidance

1. Read the amplitude ($A$) and period ($T$) from the graph. Convert units if necessary (e.g., mm to m, ms to s).

2. Calculate the frequency: $f = \frac{1}{T}$.

3. Plug the values into the formula $a_{\text{max}} = (2\pi f)^2 A$.

4. Set up the calculation for $a_{\text{max}}$ and for $a_{\text{max}} / g$.

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Q14.7. Maximum Kinetic Energy and Amplitude

Background

Topic: Energy in Simple Harmonic Motion

This question tests your understanding of how the maximum kinetic energy (total mechanical energy) depends on amplitude.

Key Terms and Formulas:

• Total Energy: $E = \frac{1}{2} k A^2$

• Maximum Kinetic Energy: Equal to total energy in SHM.

• Amplitude ($A$): Maximum displacement.

Step-by-Step Guidance

1. Recall that maximum kinetic energy is given by $E = \frac{1}{2} k A^2$.

2. Notice that energy is proportional to the square of the amplitude.

3. If amplitude is doubled, substitute $2A$ for $A$ in the energy formula and compare to the original energy.

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P14.23. Oscillating Mass: Position Function Analysis

Background

Topic: Simple Harmonic Motion (SHM) Analysis

This question asks you to analyze the position function of an oscillating mass and determine amplitude, period, spring constant, maximum speed, total energy, and velocity at a specific time.

Key Terms and Formulas:

• Position Function: $x(t) = A \cos(\omega t)$

• Angular Frequency: $\omega = 2\pi f$

• Period: $T = \frac{2\pi}{\omega}$

• Spring Constant: $k = m \omega^2$

• Maximum Speed: $v_{\text{max}} = \omega A$

• Total Energy: $E = \frac{1}{2} k A^2$

• Velocity at time $t$: $v(t) = -\omega A \sin(\omega t)$

Step-by-Step Guidance

1. Compare the given position function to $x(t) = A \cos(\omega t)$ to identify amplitude ($A$) and angular frequency ($\omega$).

2. Calculate the period using $T = \frac{2\pi}{\omega}$.

3. Find the spring constant using $k = m \omega^2$.

4. Set up the expressions for maximum speed, total energy, and velocity at $t = 0.40$ s using the formulas above.

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P14.30. Pendulum Period on the Moon vs. Earth

Background

Topic: Pendulum Period and Gravitational Acceleration

This question tests your understanding of how the period of a pendulum depends on its length and the local gravitational acceleration.

Key Terms and Formulas:

• Pendulum Period: $T = 2\pi \sqrt{\frac{L}{g}}$

• Length ($L$): Length of the pendulum.

• Gravitational Acceleration ($g$): $g_{\text{earth}} = 9.8$ m/s$^2$, $g_{\text{moon}} = 1.62$ m/s$^2$

Step-by-Step Guidance

1. Write the period formula for both the Earth and the Moon: $T_{\text{earth}} = 2\pi \sqrt{\frac{L_{\text{earth}}}{g_{\text{earth}}}}$, $T_{\text{moon}} = 2\pi \sqrt{\frac{L_{\text{moon}}}{g_{\text{moon}}}}$.

2. Set the periods equal to each other since the question asks for matching periods.

3. Solve for $L_{\text{moon}}$ in terms of $L_{\text{earth}}$, $g_{\text{earth}}$, and $g_{\text{moon}}$.

4. Set up the calculation for $L_{\text{moon}}$ using the given values.

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P14.33. Moment of Inertia of a Lower Leg (Physical Pendulum)

Background

Topic: Physical Pendulum and Moment of Inertia

This question tests your ability to use the oscillation frequency of a physical pendulum to determine its moment of inertia.

Key Terms and Formulas:

• Frequency of Physical Pendulum: $f = \frac{1}{2\pi} \sqrt{\frac{mgd}{I}}$

• Moment of Inertia ($I$): Resistance to rotational acceleration.

• Distance to Center of Mass ($d$): Distance from pivot to center of gravity.

Step-by-Step Guidance

1. Write the formula for frequency in terms of moment of inertia: $f = \frac{1}{2\pi} \sqrt{\frac{mgd}{I}}$.

2. Rearrange the formula to solve for $I$.

3. Set up the calculation for $I$ using the given values for mass, $g$, $d$, and $f$.

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P14.50. Spring Constant, Period, and Position-Time Graph for a Hanging Ball

Background

Topic: Hooke's Law and Simple Harmonic Motion

This question tests your ability to use Hooke's law to find the spring constant, calculate the period of oscillation, and draw a position-time graph.

Key Terms and Formulas:

• Hooke's Law: $F_{\text{spring}} = k \Delta y$

• Spring Constant ($k$): $k = \frac{mg}{\Delta y}$

• Period: $T = 2\pi \sqrt{\frac{m}{k}}$

• Position-Time Graph: $y(t) = A \cos\left(2\pi \frac{t}{T}\right)$

Step-by-Step Guidance

1. Use Hooke's law to solve for the spring constant: $k = \frac{mg}{\Delta y}$, where $\Delta y$ is the stretch due to the mass.

2. Calculate the period using $T = 2\pi \sqrt{\frac{m}{k}}$.

3. Set up the position-time function for the oscillation, using the amplitude and period found above.

4. Sketch or describe the graph for three cycles, labeling axes with appropriate units and values.

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