All questions of Simple Harmonic Motion for UPSC CSE Exam

C

Understanding SHM
Simple Harmonic Motion (SHM) is a type of periodic motion where an object oscillates around an equilibrium position. In SHM, two key quantities are displacement and velocity.
Displacement in SHM
- Displacement (x) is the distance of the object from its equilibrium position.
- It varies sinusoidally with time, represented as x(t) = A sin(ωt), where A is the amplitude and ω is the angular frequency.
Velocity in SHM
- Velocity (v) is the rate of change of displacement with respect to time.
- The velocity in SHM can be derived by differentiating displacement: v(t) = dx/dt = Aω cos(ωt).
Phase Relationship
- Displacement and velocity are both sinusoidal functions but differ in their phase.
- Displacement is represented by a sine function, while velocity uses a cosine function.
Phase Difference Explained
- The cosine function can be expressed as a sine function with a phase shift: cos(ωt) = sin(ωt + 90 degrees).
- This means that the velocity function leads the displacement function by 90 degrees.
- In other words, when the displacement is at its maximum, the velocity is zero, and when the displacement is zero, the velocity is at its maximum.
Conclusion
- Therefore, there is a phase difference of 90 degrees between the displacement and velocity of an object in SHM.
- The correct answer to the question is indeed option 'C' – 90 degrees.

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.

Understanding the Q-Factor
The quality factor, or Q-factor, is a dimensionless parameter that characterizes the damping of oscillatory systems, such as springs, pendulums, or electrical circuits.
What is Damping?
Damping refers to the effect of energy dissipation in an oscillating system, which can be caused by friction, resistance, or other forms of energy loss.
Significance of Q-Factor
- Definition: The Q-factor is defined as the ratio of the stored energy to the energy dissipated per cycle.
- High Q: A high Q-factor indicates low damping, meaning the system oscillates for a longer time with less energy loss.
- Low Q: Conversely, a low Q-factor signals high damping, where energy is lost rapidly, resulting in quick cessation of oscillations.
Mathematical Relation
The Q-factor can be expressed mathematically as:
- Q = (Energy stored in the system) / (Energy lost per cycle)
Importance in Applications
- Mechanical Systems: In mechanical systems, a high Q-factor means efficient energy storage, leading to prolonged oscillations.
- Electrical Systems: In circuits, a high Q-factor can enhance the selectivity and sensitivity of resonant circuits, such as in radio transmitters.
Conclusion
Thus, the Q-factor serves as a critical measure of damping in oscillating systems, making option 'A' the correct answer. Understanding the Q-factor helps in designing systems with desired oscillatory behavior, whether in engineering, physics, or technology.

Understanding Simple Harmonic Motion (SHM)
In simple harmonic motion, the displacement of an object oscillating back and forth is key to understanding its motion. The displacement-time graph is crucial for visualizing this phenomenon.
Characteristics of SHM Displacement-Time Relationship
- Periodic Motion: In SHM, an object moves in a periodic manner, meaning it repeats its motion at regular intervals.
- Sine Wave Form: The displacement of the object as a function of time can be mathematically modeled using sine or cosine functions. This results in a sine wave graph.
- Maxima and Minima: The peaks and troughs of the sine wave represent the maximum positive and negative displacements, respectively. The graph oscillates symmetrically around the equilibrium position.
Why Other Options Are Incorrect
- Linear Graph: Represents uniform motion where displacement changes at a constant rate, unlike SHM, which varies periodically.
- Parabolic Graph: Indicates acceleration changes, typically associated with uniformly accelerated motion, not the oscillatory nature of SHM.
- Exponential Graph: Depicts growth or decay processes, failing to capture the oscillatory or periodic nature of SHM.
Conclusion
The correct representation of the displacement-time relationship in SHM is option 'C', the sine wave graph. This reflects the periodic nature of the motion, showcasing how displacement varies with time in a smooth, oscillatory manner. Understanding this concept is essential for grasping the principles of SHM in physics.