Derivation Energy ( Potential and Kinetic ) in Simple Harmonic Motion (SHM)

Energy in Simple Harmonic Motion (SHM)

A particle executing Simple Harmonic Motion (SHM) possesses both Kinetic Energy (K.E.) and Potential Energy (P.E.). During the motion, these two forms of energy continuously transform into each other. When one increases, the other decreases by the same amount. In the absence of non-conservative forces, the total mechanical energy remains constant.

At the mean position, the particle has maximum velocity; therefore, its kinetic energy is maximum and potential energy is zero. At the extreme positions, the velocity becomes zero; hence kinetic energy is zero while potential energy becomes maximum.

The total mechanical energy of a particle executing SHM is the sum of its kinetic and potential energies.

$$E=K+U$$

• E = Total Mechanical Energy
• K = Kinetic Energy
• U = Potential Energy

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Potential Energy in SHM

Potential Energy is the energy possessed by a particle due to its position. In SHM, it is equal to the work done against the restoring force in displacing the particle from the mean position to any position.

Derivation of Potential Energy

Consider a particle of mass m executing SHM along a straight line.

If the particle is displaced by a distance x from its mean position, its acceleration is

$$a=-\omega^2x$$

According to Newton's Second Law,

$$F=ma$$

Substituting the value of acceleration,

$$F=-m\omega^2x$$

The negative sign indicates that the restoring force always acts towards the mean position.

Suppose the particle is displaced by a very small distance dx against the restoring force.

The small amount of work done is

$$dW=-Fdx$$

Substituting the value of restoring force,

$$dW=-(-m\omega^2x)dx$$

$$dW=m\omega^2x\,dx$$

The total work done in moving the particle from the mean position (x = 0) to any position x is

$$W=\int_0^x m\omega^2x\,dx$$

Taking the constant outside the integral,

$$W=m\omega^2\int_0^x x\,dx$$

Integrating,

$$W=m\omega^2\left[\frac{x^2}{2}\right]_0^x$$

$$W=\frac12m\omega^2x^2$$

This work done is stored as the Potential Energy of the particle.

Therefore,

$$\boxed{U=\frac12m\omega^2x^2}$$

Since

$$k=m\omega^2$$

the above equation can also be written as

$$\boxed{U=\frac12kx^2}$$

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Potential Energy as a Function of Time

The displacement of a particle executing SHM is

$$x=A\cos(\omega t+\phi)$$

Substituting this value in the potential energy equation,

$$U=\frac12k(A\cos(\omega t+\phi))^2$$

Therefore,

$$\boxed{U=\frac12kA^2\cos^2(\omega t+\phi)}$$

This equation shows that the potential energy varies periodically with time.

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Variation of Potential Energy

At Mean Position

When

$$x=0$$

Then

$$U=\frac12k(0)^2=0$$

Therefore, the potential energy is zero at the mean position.

At Extreme Position

When

$$x=\pm A$$

Then

$$U=\frac12kA^2$$

Therefore, the potential energy is maximum at the extreme positions.

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Important Characteristics of Potential Energy

• Potential energy is zero at the mean position.
• Potential energy is maximum at the extreme positions.
• Potential energy is directly proportional to the square of displacement.
• Potential energy varies as cos²(ωt + φ).
• The time period of potential energy is T/2.
• Potential energy is always positive or zero.
• As the particle moves from the mean position towards the extreme position, potential energy continuously increases.

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Key Points

• A particle executing SHM always possesses both kinetic and potential energies.
• The total mechanical energy remains constant throughout the motion.
• Potential energy depends only on the displacement from the mean position.
• The restoring force stores energy in the form of potential energy.
• Maximum potential energy occurs at the amplitude.

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Kinetic Energy in Simple Harmonic Motion (SHM)

Kinetic Energy (K.E.) is the energy possessed by a particle due to its motion. Since the velocity of a particle executing SHM continuously changes, its kinetic energy also changes continuously.

The kinetic energy of a particle is given by

$$K=\frac12mv^2$$

where

• m = Mass of the particle
• v = Instantaneous velocity

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Derivation of Kinetic Energy

The velocity of a particle executing SHM at displacement x is

$$v=\omega\sqrt{A^2-x^2}$$

Squaring both sides,

$$v^2=\omega^2(A^2-x^2)$$

Substituting this value into the kinetic energy equation,

$$K=\frac12m\omega^2(A^2-x^2)$$

Hence, the kinetic energy of a particle executing SHM is

$$\boxed{K=\frac12m\omega^2(A^2-x^2)}$$

Since

$$k=m\omega^2$$

the equation can also be written as

$$\boxed{K=\frac12k(A^2-x^2)}$$

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Kinetic Energy as a Function of Time

The velocity of a particle executing SHM is

$$v=\omega A\sin(\omega t+\phi)$$

Substituting this value into the kinetic energy equation,

$$K=\frac12m(\omega A\sin(\omega t+\phi))^2$$

Therefore,

$$\boxed{K=\frac12kA^2\sin^2(\omega t+\phi)}$$

This equation shows that the kinetic energy varies periodically with time.

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Variation of Kinetic Energy

At Mean Position

When

$$x=0$$

Then

$$K=\frac12kA^2$$

Therefore, the kinetic energy is maximum at the mean position.

At Extreme Position

When

$$x=\pm A$$

Then

$$K=0$$

Therefore, the kinetic energy is zero at the extreme positions.

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Important Characteristics of Kinetic Energy

• Kinetic energy is maximum at the mean position.
• Kinetic energy is zero at the extreme positions.
• Kinetic energy decreases as the particle moves away from the mean position.
• Kinetic energy varies as sin²(ωt + φ).
• The time period of kinetic energy is T/2.
• Kinetic energy is always positive or zero.

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Total Mechanical Energy in SHM

The total mechanical energy of a particle executing SHM is the sum of its kinetic and potential energies.

$$E=K+U$$

Substituting the expressions for kinetic and potential energies,

$$E=\frac12kA^2\sin^2(\omega t+\phi)+\frac12kA^2\cos^2(\omega t+\phi)$$

Taking the common term,

$$E=\frac12kA^2\left[\sin^2(\omega t+\phi)+\cos^2(\omega t+\phi)\right]$$

Using the identity

$$\sin^2\theta+\cos^2\theta=1$$

we obtain

$$\boxed{E=\frac12kA^2}$$

Since

$$k=m\omega^2$$

Therefore,

$$\boxed{E=\frac12m\omega^2A^2}$$

Also,

$$\omega=2\pi\nu$$

Hence,

$$\boxed{E=2\pi^2m\nu^2A^2}$$

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Variation of Total Energy

• Total mechanical energy remains constant throughout the motion.
• As kinetic energy increases, potential energy decreases by the same amount.
• As potential energy increases, kinetic energy decreases by the same amount.
• The total energy depends only on the amplitude of oscillation.
• Total mechanical energy is independent of displacement and time.

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Energy Conversion in SHM

During SHM, energy is continuously transformed between kinetic energy and potential energy.

• At the mean position, the entire energy is kinetic.
• As the particle moves away from the mean position, kinetic energy is gradually converted into potential energy.
• At the extreme position, the entire energy becomes potential energy.
• During the return journey, potential energy converts back into kinetic energy.
• This continuous conversion repeats every cycle while the total energy remains constant.

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Graph Interpretation

Energy vs Time

• Total mechanical energy is represented by a horizontal straight line because it remains constant.
• Kinetic energy follows a sin² curve.
• Potential energy follows a cos² curve.
• When kinetic energy is maximum, potential energy is zero.
• When potential energy is maximum, kinetic energy is zero.
• Both curves repeat after every T/2.

Energy vs Displacement

• Potential energy increases parabolically with displacement.
• Kinetic energy decreases parabolically from the mean position to the extreme position.
• The total energy remains constant for all values of displacement.

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NCERT Observations

• Both kinetic energy and potential energy are always positive.
• Both kinetic and potential energies have a time period of T/2.
• Total mechanical energy remains constant throughout the motion.
• At the mean position, energy is completely kinetic.
• At the extreme positions, energy is completely potential.
• The continuous interchange between kinetic and potential energy is the characteristic feature of SHM.

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Important Points

• The total mechanical energy of a particle executing SHM always remains constant.
• Potential energy is zero at the mean position and maximum at the extreme positions.
• Kinetic energy is maximum at the mean position and zero at the extreme positions.
• Kinetic energy and potential energy continuously transform into each other.
• The sum of kinetic energy and potential energy always remains constant.
• Kinetic energy and potential energy are always positive or zero.
• Both kinetic energy and potential energy repeat after every T/2.
• Total energy is directly proportional to mass.
• Total energy is directly proportional to the square of amplitude.
• Total energy is directly proportional to the square of angular frequency.

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Frequently Asked Questions (FAQs)

1. Why is the kinetic energy maximum at the mean position?

Because the velocity of the particle is maximum at the mean position.

2. Why is the potential energy maximum at the extreme position?

Because the displacement from the mean position is maximum.

3. Does the total energy change during SHM?

No. The total mechanical energy remains constant throughout the motion.

4. Can kinetic energy and potential energy be negative?

No. Both are always positive or zero.

5. On which factors does total energy depend?

Total energy depends on the mass, amplitude and angular frequency of the particle.

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Multiple Choice Questions (MCQs)

1. The total mechanical energy of a particle executing SHM is
   • (A) Variable
   • (B) Zero
   • (C) Constant ✓
   • (D) Infinite
2. Potential energy is maximum at the
   • (A) Mean position
   • (B) Extreme position ✓
   • (C) Both
   • (D) None
3. Kinetic energy is maximum at the
   • (A) Mean position ✓
   • (B) Extreme position
   • (C) Both
   • (D) None
4. The time period of kinetic energy is
   • (A) T
   • (B) T/2 ✓
   • (C) 2T
   • (D) T/4
5. The expression for total energy is
   • (A) $kA^2$
   • (B) $\frac12kA^2$ ✓
   • (C) $2kA^2$
   • (D) $\frac14kA^2$

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Assertion and Reason

Q1.

Assertion (A): Total mechanical energy remains constant during SHM.

Reason (R): Kinetic energy continuously converts into potential energy and vice versa.

Answer: Both A and R are true and R is the correct explanation of A.

Q2.

Assertion (A): Kinetic energy is zero at the extreme positions.

Reason (R): Velocity becomes zero at the extreme positions.

Answer: Both A and R are true and R correctly explains A.

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True or False

1. Total mechanical energy remains constant during SHM. (True)
2. Potential energy is maximum at the mean position. (False)
3. Kinetic energy becomes zero at the extreme positions. (True)
4. Kinetic energy and potential energy are always positive. (True)
5. Total energy depends upon displacement. (False)

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Fill in the Blanks

1. Potential energy is maximum at the extreme position.
2. Kinetic energy is maximum at the mean position.
3. Total mechanical energy remains constant.
4. The time period of kinetic energy is T/2.
5. The restoring force always acts towards the mean position.

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Very Short Answer Questions

1. Define potential energy in SHM.
2. Write the expression for kinetic energy.
3. State the expression for total mechanical energy.
4. At which position is kinetic energy maximum?
5. At which position is potential energy zero?

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Short Answer Questions

1. Derive the expression for potential energy in SHM.
2. Derive the expression for kinetic energy in SHM.
3. Explain the variation of kinetic and potential energy during SHM.
4. Why does the total mechanical energy remain constant?
5. Explain the energy conversion taking place during SHM.

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Long Answer Questions

1. Derive the expressions for kinetic energy, potential energy and total mechanical energy of a particle executing SHM.
2. Explain the variation of kinetic energy, potential energy and total energy with displacement and time using suitable graphs.
3. Discuss the conservation of mechanical energy in SHM.

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Chapter Summary

• A particle executing SHM possesses both kinetic energy and potential energy.
• Potential energy is proportional to the square of displacement.
• Kinetic energy decreases as displacement increases.
• At the mean position, all the energy is kinetic.
• At the extreme positions, all the energy is potential.
• The total mechanical energy remains constant throughout the motion.
• Kinetic energy and potential energy interchange continuously during SHM.
• The total energy depends only on the mass, amplitude and angular frequency of the particle.