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The motion of a simple pendulum is approximately SHM when:
- a)The angle of oscillation is small
- b)The angle of oscillation is large
- c)The length of the pendulum is long
- d)The mass of the pendulum is large
Correct answer is option 'A'. Can you explain this answer?
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The motion of a simple pendulum is approximately SHM when:a)The angle ...
The motion of a simple pendulum is approximately SHM when the angle of oscillation is small.
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The motion of a simple pendulum is approximately SHM when:a)The angle ...
Understanding Simple Harmonic Motion (SHM) in a Pendulum
A simple pendulum exhibits simple harmonic motion (SHM) primarily under specific conditions related to the angle of oscillation. Here's why the correct answer is option 'A':
Small Angle Approximation
- When the angle of oscillation is small (typically less than 15 degrees), the motion of the pendulum can be approximated as SHM.
- In this range, the restoring force acting on the pendulum is directly proportional to the displacement from the equilibrium position. This relationship is a key characteristic of SHM.
Mathematical Justification
- For a pendulum, the restoring force is derived from gravity. The formula for the angular displacement (θ) leads to the equation of motion resembling that of SHM when θ is small.
- Specifically, sin(θ) approximates to θ (in radians), allowing the motion to be simplified.
Limitations at Large Angles
- When the angle of oscillation is large, the approximation fails. The sin(θ) no longer equals θ, and the restoring force becomes nonlinear, deviating from SHM.
- This nonlinearity introduces complexities, such as varying periods and increased amplitude effects.
Other Factors
- The length of the pendulum and mass do not influence the approximation of SHM's fundamental nature. While they affect the period of oscillation, they do not change the condition under which the motion behaves as SHM.
Conclusion
- Therefore, for a simple pendulum to exhibit SHM, the angle of oscillation must be small (option A), ensuring a linear relationship between restoring force and displacement. This fundamental understanding is crucial for studying pendulum dynamics in physics.
A simple pendulum exhibits simple harmonic motion (SHM) primarily under specific conditions related to the angle of oscillation. Here's why the correct answer is option 'A':
Small Angle Approximation
- When the angle of oscillation is small (typically less than 15 degrees), the motion of the pendulum can be approximated as SHM.
- In this range, the restoring force acting on the pendulum is directly proportional to the displacement from the equilibrium position. This relationship is a key characteristic of SHM.
Mathematical Justification
- For a pendulum, the restoring force is derived from gravity. The formula for the angular displacement (θ) leads to the equation of motion resembling that of SHM when θ is small.
- Specifically, sin(θ) approximates to θ (in radians), allowing the motion to be simplified.
Limitations at Large Angles
- When the angle of oscillation is large, the approximation fails. The sin(θ) no longer equals θ, and the restoring force becomes nonlinear, deviating from SHM.
- This nonlinearity introduces complexities, such as varying periods and increased amplitude effects.
Other Factors
- The length of the pendulum and mass do not influence the approximation of SHM's fundamental nature. While they affect the period of oscillation, they do not change the condition under which the motion behaves as SHM.
Conclusion
- Therefore, for a simple pendulum to exhibit SHM, the angle of oscillation must be small (option A), ensuring a linear relationship between restoring force and displacement. This fundamental understanding is crucial for studying pendulum dynamics in physics.
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