Simple Harmonic Motion

Simple Harmonic Motion

1. Periodic Motion

Any motion that repeats itself after a regular interval of time is called periodic or harmonic motion.

Examples:

• Motion of the hands of a clock

• Motion of the Earth around the Sun

• Motion of the needle of a sewing machine

• Motion of a pendulum

2. Oscillatory Motion

If a particle repeats its motion after a regular time interval about a fixed point, motion is said to be oscillatory or vibratory, i.e., oscillatory motion is a constrained periodic motion between precisely fixed limits.

Examples:

• Motion of the piston in an automobile engine

• Motion of the balance wheel of a watch

3. Time Period

• Time taken in one complete oscillation is called a time period.

• Or, the time after which motion is repeated is called the time period.

4. Frequency

Frequency is the number of oscillations completed by an oscillating body in a unit time interval.

• The frequency SI unit is hertz.

• If n = frequency and T = time period, then nT = 1

5. Simple Harmonic Motion

If a particle repeats its motion about a fixed point after a regular time interval in such a way that at any moment the acceleration of the particle is directly proportional to its displacement from the fixed point at that moment and is always directed towards the fixed point, then the motion of the particle is called simple harmonic motion (SHM).

• The fixed point is called the mean point.

Characteristics of SHM

When a particle executing SHM passes through the mean position:

1. No force acts on the particle.

2. The acceleration of the particle is zero.

3. Velocity is maximum.

4. Kinetic energy is maximum.

5. Potential energy is zero.

When a particle executing SHM is at the end:

1. The acceleration of the particle is maximum.

2. The restoring force acting on the particle is maximum.

3. The velocity of the particle is zero.

4. The kinetic energy of the particle is zero.

5. Potential energy is maximum.

6. Simple Pendulum

If a point mass is suspended from a fixed support with the help of a massless and inextensible string, the arrangement is called a simple pendulum.

• This is an ideal definition.

• Practically, a simple pendulum is made by suspending a small ball (called a bob) from a fixed support with the help of a light string.

Motion of a Simple Pendulum

If the bob of a simple pendulum is slightly displaced from its mean position and released, it starts oscillating in simple harmonic motion.

• The time period of oscillation is given as

where l is the effective length of the pendulum and g is the acceleration due to gravity.

Interesting Observations

• If a person on a swing stands up, his center of gravity rises, the length of the pendulum decreases, and as a result, the time period of the swing decreases.

• On the moon, the value of g is 1/6th the value on Earth. Hence, if a pendulum clock that keeps correct time on Earth is taken to the moon, the time period becomes 6 times the value on Earth. As a result, the clock will run 6 times slower.

Final Thoughts

Periodic motion repeats after regular intervals, like clock hands or Earth’s orbit. Oscillatory motion occurs around a fixed point, as seen in pistons or watch balance wheels.

The time period is the duration of one oscillation, and frequency counts oscillations per second. In simple harmonic motion (SHM), acceleration is proportional to displacement toward the mean point.

At the mean, velocity and kinetic energy are maximum; at extremes, potential energy peaks. A simple pendulum demonstrates SHM, with its period depending on string length and gravity. Everyday examples, from swings to pendulum clocks, show how rhythmic motion governs the world around us.