"Simple Harmonic Motion"

About

From the swinging of a pendulum to the vibration of atoms in a solid, oscillatory motion is everywhere. This chapter introduces simple harmonic motion (SHM) — the purest form of oscillation. You will learn the mathematical description of SHM, derive expressions for displacement, velocity, acceleration, and energy, and study practical systems like the simple pendulum and floating cylinder.

Key Concepts

13.1 Periodic and Oscillatory Motion

Periodic motion: Any motion that repeats itself at regular intervals of time.

Oscillatory motion: A special type of periodic motion in which a body moves to and fro about a fixed mean position.

Every oscillatory motion is periodic, but NOT every periodic motion is oscillatory.

13.2 Simple Harmonic Motion (SHM)

SHM is the simplest form of oscillatory motion where:

1. Restoring force is proportional to displacement: $F\propto -x$
2. Force is always directed toward the mean position: $F=-kx$

Where $k$ is the force constant (spring constant).

Equation of SHM: $\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0$

Where $\omega$ = angular frequency.

General solution: $x=A\sin (\omega t+\varphi )\text{or}x=A\cos (\omega t+\varphi )$

Where $A$ = amplitude, $\varphi$ = initial phase.

13.3 Characteristics of SHM

Key relations:

• Velocity is maximum at the mean position ($x=0$)
• Acceleration is maximum at the extreme positions ($x=\pm A$)
• Acceleration is zero at the mean position

Time period and frequency: $T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{k}}$ $\nu =\frac{1}{T}=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

13.4 Energy in SHM

Kinetic Energy: $K=\frac{1}{2}m{v}^2=\frac{1}{2}m{\omega }^2(A^2-x^2)$

Potential Energy: $U=\frac{1}{2}k{x}^2=\frac{1}{2}m{\omega }^2{x}^2$

Total Energy (constant): $E=K+U=\frac{1}{2}k{A}^2=\frac{1}{2}m{\omega }^2{A}^2$

• At mean position ($x=0$): KE max, PE = 0
• At extreme ($x=\pm A$): PE max, KE = 0
• Total energy is conserved and proportional to $A^2$

13.5 Simple Pendulum

For small amplitudes ($\theta$ small, $\sin \theta \approx \theta$):

Restoring force: $F=-mg\sin \theta \approx -mg\theta$

Time period: $T=2\pi \sqrt{\frac{L}{g}}$

• Independent of mass of the bob
• Independent of amplitude (for small angles — isochronous)
• Depends only on length $L$ and $g$

13.6 Other SHM Systems

Ball in a spherical bowl: $T=2\pi \sqrt{\frac{r}{g}}$ (where $r$ = bowl radius)

Floating cylinder: $T=2\pi \sqrt{\frac{l}{g}}$ (where $l$ = immersed length)

Spring combinations:

• Parallel: $k_{\text{eff}}=k_1+k_2$
• Series: $\frac{1}{k_{\text{eff}}}=\frac{1}{k_1}+\frac{1}{k_2}$

13.7 Damped and Forced Oscillations

Damped oscillations: Amplitude decreases with time due to friction/dissipative forces.

Forced oscillations: System driven by an external periodic force. When driving frequency equals natural frequency → resonance (large amplitude).

INTEXT QUESTIONS 13.1

Q1. What is the difference between a periodic motion and an oscillatory motion?

Ans: Periodic motion is any motion that repeats at regular time intervals — it does NOT necessarily involve back-and-forth movement (e.g., Earth revolving around the Sun). Oscillatory motion is a specific type of periodic motion where an object moves to and fro about a fixed mean position (e.g., pendulum, spring-mass system). Every oscillatory motion is periodic, but not every periodic motion is oscillatory.

Q2. Which of the following examples represent a periodic motion?

(i) A bullet fired from a gun — No (ii) An electron revolving round the nucleus in an atom — Yes (iii) A vehicle moving with a uniform speed on a road — No (iv) A comet moving around the Sun — Yes (v) Motion of an oscillating mercury column in a U-tube — Yes

Q3. Give an example of (i) an oscillatory periodic motion and (ii) a non-oscillatory periodic motion.

Ans: (i) Oscillatory periodic motion: Swinging of a pendulum — it moves back and forth about its mean position at regular intervals.

(ii) Non-oscillatory periodic motion: Motion of the Earth around the Sun — repeats in a fixed time (one year) but does not move to and fro about a mean position.

INTEXT QUESTIONS 13.2

Q1. A small spherical ball of mass m is placed in contact with the surface on a smooth spherical bowl of radius r a little away from the bottom point. Calculate the time period of oscillations of the ball.

Ans: Restoring force: $F=mg\sin \theta \approx mg\theta$ (for small $\theta$)

For small $\theta$, $\theta =x/r$, so $F=\frac{mg}{r}x$

Since $F\propto -x$, this is SHM.

$\omega =\sqrt{\frac{g}{r}},T=2\pi \sqrt{\frac{r}{g}}$

Q2. A cylinder of mass m floats vertically in a liquid of density ρ. The length of the cylinder inside the liquid is l. Obtain an expression for the time period of its oscillations.

Ans: Upthrust = $yA\rho g$, where $y$ is additional immersion.

Mass displaced $m=Al\rho$ (law of floatation).

Equation of motion: $F=-A\rho gy=-m{\omega }^{2}y$

${\omega }^2=\frac{A\rho g}{m}=\frac{g}{l},T=2\pi \sqrt{\frac{l}{g}}$

Q3. Calculate the frequency of oscillation of the mass m connected to two rubber bands as shown. The force constant of each band is k.

Ans: Two identical bands in parallel → combined force constant = $2k$

Restoring force = $-(2k)x$

$\omega =\sqrt{\frac{2k}{m}},\nu =\frac{1}{2\pi }\sqrt{\frac{2k}{m}}$

Terminal Exercise

1. Define periodic motion and oscillatory motion. Give two examples of each. Are all oscillatory motions periodic?

2. Derive the differential equation of SHM: $\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0$. Show that $x=A\sin (\omega t+\varphi )$ is its solution.

3. Derive expressions for velocity and acceleration of a particle executing SHM. Show that $v=\omega \sqrt{A^2-x^2}$.

4. Show that the total energy of a particle executing SHM is constant and proportional to the square of the amplitude.

5. Derive the expression for the time period of a simple pendulum: $T=2\pi \sqrt{L/g}$. State the assumptions made.

6. A simple pendulum has a time period of 2 s on Earth. What will be its time period on the Moon? ($g_{\text{moon}}=g/6$)

7. A particle executes SHM of amplitude 5 cm and time period 2 s. Find: (a) maximum velocity, (b) velocity at displacement 3 cm, (c) acceleration at extreme position.

8. A spring of force constant $k$ is cut into two equal halves. One piece is attached to a mass $m$. Compare the time period with the original.

9. What is a second's pendulum? Calculate its length. ($g=9.8$ m/s²)

10. Distinguish between free, damped, and forced oscillations. What is resonance? Give one harmful and one useful effect of resonance.

11. A body of mass 0.2 kg executes SHM with amplitude 0.1 m and time period $\pi /5$ seconds. Find: (a) force constant, (b) maximum force, (c) total energy.

12. Two springs of force constants $k_1$ and $k_2$ are connected in (a) series, (b) parallel to a mass $m$. Find the time period in each case.

Worked Examples

Example 1: SHM Parameters

Problem: A particle executes SHM with amplitude 10 cm and period 4 s. Find maximum velocity and maximum acceleration.

Solution: $\omega =\frac{2\pi }{T}=\frac{2\pi }{4}=\frac{\pi }{2}\text{ rad/s}$ $v_{\text{max}}=A\omega =0.1\times \frac{\pi }{2}=0.157\text{ m/s}$ $a_{\text{max}}=A{\omega }^2=0.1\times {\left(\frac{\pi }{2}\right)}^2=0.247{\text{ m/s}}^2$

Example 2: Pendulum

Problem: Find the length of a simple pendulum whose time period is 2 s. ($g=9.8$ m/s²)

Solution: $T=2\pi \sqrt{\frac{L}{g}}⟹L=\frac{g{T}^2}{4{\pi }^2}=\frac{9.8\times 4}{4{\pi }^2}$ $=\frac{9.8}{{\pi }^2}\approx 0.993\text{ m}\approx 99.3\text{ cm}$

Example 3: Energy

Problem: A 0.5 kg mass on a spring ($k=200$ N/m) is stretched 5 cm and released. Find total energy.

Solution: $E=\frac{1}{2}k{A}^2=\frac{1}{2}\times 200\times (0.05{)}^2=0.25\text{ J}$

Common Mistakes

1. Confusing frequency $\nu$ with angular frequency $\omega$: $\omega =2\pi \nu$.
2. Forgetting that $T=2\pi \sqrt{L/g}$ is valid only for small amplitudes: Large amplitudes give longer periods.
3. Thinking acceleration is zero when velocity is zero: At extremes, $v=0$ but $a=a_{\text{max}}$.
4. Using $T=2\pi \sqrt{m/k}$ for the wrong $k$: Use effective spring constant for combinations.
5. Assuming the pendulum's time period depends on mass: It doesn't — only $L$ and $g$ matter.

Quick Revision