### Key Concepts

- The simple harmonic motion
- The factors on which SHM of a pendulum depend

**Introduction**:

When an object is kept in the equilibrium position, the net force acting on it is always zero. Whenever the object is pulled away from its equilibrium position, the net force on the object becomes non-zero and the non-zero force or restoring pulls the object back towards its equilibrium position. If the force that restores the object to its equilibrium position is directly proportional to the displacement of the object, the motion that results is called simple harmonic motion.

**Explanation**:

We describe simple harmonic motion using some variables such as:

- Time period

- Frequency
- Displacement
- Amplitude

**Simple Harmonic Motion of a Simple Pendulum:**

The SHM 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 simple harmonic 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.

**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.

**The necessary and essential conditions for SHM:**

- A simple pendulum executes a simple harmonic motion if the amplitude of the oscillation is very small.
- If the force that restores the object to its equilibrium position is directly proportional to the displacement of the pendulum the body executes SHM.

**Explanation:**

When the angular displacement is small (less than about 15 degrees), the restoring force is proportional to the displacement, so the movement is simple harmonic.

Restoring force = F 𝝰 θ

(F= 0 when θ = 0 and F = Maximum when θ = Maximum)

The period of a pendulum is given by the following equation:

Time period of a pendulum = T = 2π

Lg−−√Lg

Thus, the time period of a pendulum only depends upon:

- Length of the pendulum (L)
- Acceleration due to gravity (g)

The time period doesn’t depend upon:

- The mass of the bob (m)

- The amplitude of the oscillations (θ)

**Proof:**

Consider an experimental setup of two pendulums of the same length but with bobs of different masses. The length is measured from the point of suspension to the center of mass of the bob.

**Observations:1**

When both bobs were pulled aside by the same displacement and released at the same time, each pendulum would complete one oscillation at the same time.

** Observation:2**

On changing the amplitude of one of the pendulums, and releasing them at the same time we found that they still have the same time period.

**Conclusion:**

The time period of a pendulum doesn’t depend on the mass or on the amplitude of the oscillations.

**Question:1**

Why does the period of a pendulum depend on the pendulum length and the gravitational acceleration?

**Answer:1**

When two pendulums have different lengths (L1, L2) but the same amplitude (θ), the shorter pendulum will have a smaller arc to travel through, as shown in the figure.

Because the distance from maximum displacement to equilibrium is less while the acceleration caused by the restoring force remains the same, the shorter pendulum will have a shorter time period.

**Question:2**

Why doesn’t the mass of a bob affect the time period of a pendulum?

**Explanation:**

When the angular displacement is small (less than about 15 degrees), the restoring force is proportional to the displacement, so the movement is simple harmonic.

Restoring force = F 𝝰 θ

(F= 0 when θ = 0 and F = Maximum when θ = Maximum)

The period of a pendulum is given by the following equation:

Time period of a pendulum = T = 2π

Lg−−√Lg

Thus, the time period of a pendulum only depends upon:

- Length of the pendulum (L)
- Acceleration due to gravity (g)

The time period doesn’t depend upon:

- The mass of the bob (m)

- The amplitude of the oscillations (θ)

**Proof:**

Consider an experimental setup of two pendulums of the same length but with bobs of different masses. The length is measured from the point of suspension to the center of mass of the bob.

**Observations:1**

When both bobs were pulled aside by the same displacement and released at the same time, each pendulum would complete one oscillation at the same time.

** Observation:2**

On changing the amplitude of one of the pendulums, and releasing them at the same time we found that they still have the same time period.

**Conclusion:**

The time period of a pendulum doesn’t depend on the mass or on the amplitude of the oscillations.

**Question:1**

Why does the period of a pendulum depend on the pendulum length and the gravitational acceleration?

**Answer:1**

When two pendulums have different lengths (L1, L2) but the same amplitude (θ), the shorter pendulum will have a smaller arc to travel through, as shown in the figure.

Because the distance from maximum displacement to equilibrium is less while the acceleration caused by the restoring force remains the same, the shorter pendulum will have a shorter time period.

**Question:2**

Why doesn’t the mass of a bob affect the time period of a pendulum?

**Answer:2**

When the bobs of two pendulums differ in mass, the heavier mass provides a larger restoring force, but it also needs a larger force to achieve the same acceleration.

It is similar to the situation for objects in freefall, where the acceleration for objects remain the same regardless of their mass. Because the acceleration of both pendulums is the same, the time period for both pendulums is also the same.

**Question:3**

Why doesn’t the amplitude of the bob affect the time period of a pendulum?

**Answer:3**

When the amplitude is small (<15 degrees) the pendulum executes a simple harmonic motion. For small angles (between 0 to 15 degrees), when the amplitude of a pendulum increases, the restoring force also increases proportionally.

Because force is proportional to acceleration, the initial acceleration will be greater, However, the distance this pendulum must cover is also greater. For small angles, the effects of the two increasing quantities (acceleration and distance to travel) cancel and the pendulum’s period remains the same.

Graphs to study SHM of a simple pendulum:

The figures show the position-time and velocity-time graphs for the simple harmonic motion of a simple pendulum.

From the position-time and velocity-time graphs for the simple harmonic motion of a simple pendulum, it is clear that the displacement and the velocity of the bob from equilibrium can be represented by the sine or cosine function.

**The energy of a simple pendulum executing SHM:**

The figure shows how a pendulum’s mechanical energy charges as the pendulum oscillate. At maximum displacement from equilibrium, a pendulum’s energy is entirely gravitational potential energy. As the pendulum swings towards equilibrium, it gains kinetic energy and loses potential energy. At the equilibrium position, its energy becomes entirely kinetic.

As the pendulum swings past its equilibrium position, the kinetic energy decreases while the potential energy increases. At maximum displacement from equilibrium, the pendulum’s energy is once again entirely gravitational potential energy.

**Question:4**

Two simple pendulums of lengths 4 m and 16 m respectively are given small linear displacement in one direction at the same time. They will again be in the phase when the pendulum of shorter length has completed …….. oscillations.

**Answer: 4**

The two pendulums will be in phase when the shorter pendulum has made n1 oscillations and the bigger pendulum n2 oscillations. So,

The total time elapsed =T1×n1=T2×n2

Also T = 2π√(𝐿/𝑔) so T1 / T2 = √(𝐿1)/√(𝐿2)

n1 / n2 = T1 / T2 = 4/16 = 1/4

They will again be in phase for the first time when the shorter pendulum has made 4 oscillations and the longer pendulum has made 1 oscillation.

### Summary

- The necessary and essential conditions for SHM:
- A simple pendulum executes a simple harmonic motion if the amplitude of the oscillation is very small.
- If the force that restores the object to its equilibrium position is directly proportional to
- the displacement of the pendulum.
- The period of a pendulum is given by the following equation
- Time period of a pendulum = T = 21
- Thus, the time period of a pendulum only depends upon:
- Length of the pendulum (L)
- Acceleration due to gravity (g) The time period doesn’t depend upon:
- The mass of the bob (m)
- The amplitude of the oscillations (0)

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