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### Key Concepts

- Terms Related to Oscillatory Motion
- Oscillatory Motion of a Pendulum
- Oscillatory motion of a spring and Mass System

**Introduction**:

The motion of a mass attached to a string and a pendulum are also examples of periodic motion but of different types. All these objects execute to and fro, or back and forth motion periodically about a mean position such a motion is termed an oscillatory motion.

We describe an oscillatory motion using some variables such as:

- Time Period
- Frequency

- Displacement
- Amplitude

**Explanation**:

**Oscillations of a Simple Pendulum:**

The oscillation of a simple pendulum can be demonstrated by the swing of a pendulum. A simple pendulum consists of a massive object called the bob, suspended by an unstretched string of length L. After the bob is pulled to one side and released, it swings back and forth as shown in the figure.

- A pendulum bob executes an oscillator motion about an equilibrium position.
- The starting position of the pendulum is called the mean position or equilibrium position labeled as A.
- The bob moves to and fro about its mean position labeled as A and rises to extreme positions on both sides labeled as C and B and repeats its motion.

**For one complete oscillation, the bob can follow anyone the following paths:**

**Time period:**

The smallest interval of time after which the motion is repeated is called its time period.

The time taken to complete one oscillation is known as the time period.

The time period is denoted by the letter T.

The S.I. unit of time period is second (s).

**Frequency:**

The reciprocal of the time period or the number of oscillations a pendulum performs in one second is called the frequency of the periodic motion.

It is represented by the symbol “ν ” or “f “.

The relation between ν and T is:

ν = 1/ T

The S.I. unit of ν is thus (1/sec) or hertz (Hz).

**Displacement:**

The motion of a simple pendulum can be described in terms of the angle “θ” it makes with the vertical as a function of time.

It is convenient to measure the angular displacement of the bob from its equilibrium position.

**Amplitude:**

The maximum angular displacement of the bob from its equilibrium position, i.e., when it moves from A→B or A→C is called its amplitude.

The S.I. unit of amplitude is radian or meter.

**Oscillations of Spring and Mass System:**

**Horizontal Oscillatory motion of a Spring and a Mass System:**

The horizontal oscillatory motion of a spring and a block system is one of the simplest types of back-and-forth periodic motion. Let us assume that a mass moves on a frictionless horizontal surface. When the spring is stretched or compressed and then released, it oscillates back and forth about its unstretched position.

**Observations:**

- The direction of the elastic force acting on the mass is always opposite to the direction of the block’s displacement from equilibrium (x = 0).
- When the spring is stretched to the right, the spring force pulls the mass to the left.
- When the spring is unstretched, the spring force is zero.
- When the spring is compressed to the left the spring force is directed to the right.

**Conclusion:**

The amplitude of the mass-spring system is the maximum displacement amount the spring is stretched or compressed from its equilibrium position.

Amplitude = maximum displacement from the equilibrium = X

Time period = Time it takes to move from 0 →(+X) → 0 → (-X)→ 0

Frequency = Number of oscillations it makes in one second.

Frequency = 1/Time period

**Vertical oscillatory motion of a spring and a block system:**

**Observation:**

The direction of the elastic force acting on the mass is always opposite to the direction of the block’s displacement from equilibrium (Y = 0).

**Observation:**

When the spring is stretched down (Y = -Y), the spring force pulls the mass up.

When the spring is unstretched (Y = 0), the spring force is zero.

When the spring is compressed up (Y = +Y), the spring force pulls the mass down.

**Conclusion:**

The amplitude of the mass-spring system is the maximum displacement amount the spring is stretched or compressed from its equilibrium position.

Amplitude = maximum displacement from the equilibrium = X

Time period = Time it takes to move from 0 →(+X) → 0 → (-X) → 0

Frequency = Number of oscillations it makes in one second.

Frequency = 1/Time period

**Question:**

A simple pendulum takes 52 seconds to complete 20 oscillations. What is the time period of the pendulum?

**Answer: **

We know that time period is the time required to complete one oscillation:

Time period = Total Time taken for oscillations / No. of oscillations

Time period = 52 / 20 = 2.6 Second

The time period of the pendulum is 2.6 seconds

### Summary

- We describe oscillatory motion using some variables such as: Time Period, Frequency, Displacement Amplitude
- Time period: The smallest interval of time after which the motion is repeated is called its time period.
- Frequency: The reciprocal of the time period or the number of oscillations a system performs in one second is called the frequency of the periodic motion. S.I. unit of frequency is Hertz.
- Amplitude: The maximum angular displacement or displacement of a body from its equilibrium position is called amplitude. The S.I. unit of amplitude is radian or meter.

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