Experiment 7: Simple Harmonic Motion - Examples and Applications

Experiment 7: Simple Harmonic Motion — Worked Examples

These examples progress from basic timing calculations to pendulum analysis, spring-constant determination, effective-mass interpretation, energy calculations, and experimental error assessment.

Example 1: Determine period and frequency from 10 oscillations

A pendulum completes $10$ oscillations in $18.6\,\text{s}$ . Determine its period $T$ , frequency $f$ , and angular frequency $\omega$ .

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Cycle-counting mistake

One complete cycle requires the object to return to the same position while moving in the same direction. Counting a one-way trip as a complete cycle makes the calculated period too small.

Example 2: Predict the period of a simple pendulum

A simple pendulum has length $L=0.750\,\text{m}$ . Using $g=9.81\,\text{m/s}^2$ , determine its theoretical period and frequency for a small release angle.

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Example 3: Estimate gravitational acceleration from pendulum data

A pendulum of length $L=0.900\,\text{m}$ completes $20$ oscillations in $38.2\,\text{s}$ . Determine the experimental value of $g$ and its percent error relative to $9.81\,\text{m/s}^2$ .

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Pendulum length reference

Measure $L$ from the pivot point to the center of the bob. Measuring only the string or measuring to the bottom of the bob causes a systematic error in the calculated period and value of $g$ .

Example 4: Predict the effect of changing pendulum length

A pendulum has period $T_1=1.20\,\text{s}$ at length $L_1=0.360\,\text{m}$ . The length is increased to $L_2=0.810\,\text{m}$ . Determine the new period without recalculating from $g$ .

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Example 5: Check whether a pendulum release is within the small-angle range

A pendulum has length $L=0.800\,\text{m}$ and is released from an approximate horizontal displacement $A=0.100\,\text{m}$ . Estimate the release angle in radians and degrees, then evaluate whether it satisfies a practical $10^\circ$ small-angle limit.

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Amplitude interpretation

The approximation $\theta\approx A/L$ is most accurate when $A$ represents a small arc displacement. When $A$ is measured horizontally, the estimate remains useful only for small release angles.

Example 6: Compare measured periods for different bob masses

Two pendulum bobs have masses $0.0500\,\text{kg}$ and $0.200\,\text{kg}$ . With the same length and release angle, their measured times for $10$ cycles are $16.4\,\text{s}$ and $16.5\,\text{s}$ . Determine each period and the percent difference between them.

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Example 7: Determine spring period and frequency

A vertical spring has spring constant $k=24.0\,\text{N/m}$ and supports an effective oscillating mass $M_{\text{eff}}=0.150\,\text{kg}$ . Determine $T$ , $f$ , and $\omega$ .

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Example 8: Determine spring constant from measured period

A spring supports an effective oscillating mass $M_{\text{eff}}=0.250\,\text{kg}$ . The measured time for $15$ cycles is $12.0\,\text{s}$ . Determine the spring constant $k$ .

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Use effective mass, not only added mass

The spring period depends on all mass participating in the motion. Ignoring the pan and effective spring mass causes the calculated $k$ to be systematically incorrect.

Example 9: Include pan and spring mass in the effective mass

A spring carries an added mass $M=0.180\,\text{kg}$ on a pan of mass $m_{\text{pan}}=0.0400\,\text{kg}$ . The spring mass is $m_{\text{s}}=0.0300\,\text{kg}$ . Assume $\eta=1/3$ and $k=20.0\,\text{N/m}$ . Determine the effective mass and theoretical period.

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Example 10: Determine spring constant from a graph of period squared versus mass

A best-fit graph places $T^2$ on the vertical axis and added mass $M$ on the horizontal axis. Its slope is $1.80\,\text{s}^2/\text{kg}$ . Determine $k$ .

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Example 11: Interpret a graph of added mass versus period squared

A best-fit graph places added mass $M$ on the vertical axis and $T^2$ on the horizontal axis. The fitted equation is

M=(0.620\,\text{kg/s}^2)T^2-0.0520\,\text{kg}

Determine the spring constant and the combined effective mass of the pan and spring.

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Graph-axis reversal

Do not use the same slope formula after reversing the axes. A $T^2$ -versus- $M$ graph has slope $4\pi^2/k$ , while an $M$ -versus- $T^2$ graph has slope $k/(4\pi^2)$ .

Example 12: Determine spring energy and speed at a specified displacement

A spring oscillator has $k=40.0\,\text{N/m}$ , mass $M_{\text{eff}}=0.200\,\text{kg}$ , and amplitude $A=0.0800\,\text{m}$ . Determine the total mechanical energy and the speed when $x=0.0500\,\text{m}$ .

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Example 13: Quantify the effect of a timing-count error

A student records $16.0\,\text{s}$ while incorrectly counting $20$ half-cycles as $20$ complete cycles. Determine the reported period, the correct period, and the percent error in the reported period.

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Recommended laboratory calculation sequence

For every timing trial, first confirm the number of complete cycles, then calculate $T$ , $f$ , and any required squared quantity. For graph analysis, write the linearized equation and identify both axes before interpreting the slope or intercept.

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