7.2 Frequency and Period of SHM

Learning objectives

7.2A—Describe the frequency and period of an object exhibiting SHM. TOPIC 7.2 Frequency and Period of SHMDescribe the frequency and period of an object exhibiting SHM. TOPIC 7.2 Frequency and Period of SHM• The period of SHM is related to the frequency f of the object’s motion by the following equation: 1T= f- i. The period of an object–ideal-spring oscillator is given by the equation- ii. The period of a simple pendulum displaced by a small angle is given by the equation AP Physics 1: Algebra-Based Course and Exam Description Oscillations UNIT 7

Connect cycle time and cycle rate

The period $T$ is the time for one complete cycle, measured in seconds. The frequency $f$ is the number of cycles per second, measured in hertz.

$\begin{gathered}T=\frac1f\\ f=\frac1T\end{gathered}$

Choose the oscillator model

SHM model	Period	Parameters that set the period
Object–ideal-spring oscillator	$T_s=2\pi\sqrt{m/k}$	Oscillating mass $m$ ; spring constant $k$
Small-angle simple pendulum	$T_p=2\pi\sqrt{\ell/g}$	Pendulum length $\ell$ ; gravitational field strength $g$

Calculate spring period and frequency

Spring example: $m=0.50\,\text{kg}$ and $k=200\,\text{N}\,\text{m}^{-1}$ .

$T_s=2\pi\sqrt{\dfrac{0.50}{200}}=2\pi(0.050)=0.314\,\text{s}$

$f=\dfrac1T=\dfrac1{0.314}=3.18\,\text{Hz}$ .

The unit check works because $m/k$ has units of $\text{s}^2$ .

Predict square-root changes

Parameter change	Period change
Spring mass $m$ doubled	$T_s$ multiplied by $\sqrt2$
Spring constant $k$ multiplied by $4$	$T_s$ divided by $2$
Pendulum length $\ell$ multiplied by $4$	$T_p$ multiplied by $2$
Field strength $g$ multiplied by $4$	$T_p$ divided by $2$

Respect each model's conditions

Match the equation to the physical oscillator. The pendulum result assumes a small angular displacement; the spring result assumes an ideal spring. Frequency changes inversely with period, so a longer period means a lower frequency.

Next Topic

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