Test: Simple Harmonic Motion - UPSC MCQ

Simple Harmonic Motion (SHM) refers to the back and forth motion of an object along a line about a stable equilibrium position.

In simple harmonic motion (SHM), amplitude is not involved in the description of the period and frequency. 

The period and frequency of SHM are independent of amplitude and the initial phase of the motion. the period is determined by the mass and the restoring force. The period is the time it takes for one complete cycle of the motion.

The restoring force in SHM is directly proportional to the displacement from the equilibrium position.

The time period of a simple pendulum depends on the length of the pendulum and the acceleration due to gravity.

In simple harmonic motion (SHM), the frequency and time period are inversely proportional to each other:

• Frequency: The number of oscillations per unit time, typically measured in hertz (Hz)
• Time period: The time it takes for one oscillation

 An object executing SHM experiences maximum potential energy at the extreme points of its motion.

The maximum displacement of an object from its mean position in SHM is called the amplitude.

Total mechanical energy remains constant during Simple Harmonic Motion (SHM).

The restoring force in SHM is always directed towards the mean position or equilibrium position.

The motion of a mass-spring system is an example of simple harmonic motion.

The displacement-time relationship in SHM is represented by a sine wave graph.

The force constant of a spring is a measure of its elasticity.

In SHM, the phase difference between the displacement and velocity is 90 degrees.

Acceleration is directly proportional to displacement, but with a negative direction. When an object is at its maximum displacement, the force is also maximum, which results in maximum acceleration.

The motion of a simple pendulum is approximately SHM when the angle of oscillation is small.

Frequency = 1 / Time period = 1 / 2 seconds = 0.5 Hz.

To find the ratio of maximum acceleration to maximum velocity of a particle performing Simple Harmonic Motion (S.H.M.), we can follow these steps:

Step 1: Understand the equations of S.H.M.
The displacement y of a particle in S.H.M. can be expressed as:
y=Asin(ωt)
where:
- A is the amplitude,
- ω is the angular frequency,
- t is time.

Step 2: Find the maximum velocity.
The velocity v of the particle is the derivative of displacement with respect to time:
v=dy/dt=d / dt(Asin(ωt))=Aωcos(ωt)
The maximum velocity vmax occurs when cos(ωt)=1:
v_max=Aω

Step 3: Find the maximum acceleration.
The acceleration a is the derivative of velocity with respect to time:
a=dv/dt=d/dt(Aωcos(ωt))=−Aω²sin(ωt)
The maximum acceleration amax occurs when sin(ωt)=1:
a_max=Aω²

Step 4: Calculate the ratio of maximum acceleration to maximum velocity.
Now, we can find the ratio of maximum acceleration to maximum velocity:

Conclusion:
Thus, the ratio of maximum acceleration to maximum velocity of a particle performing S.H.M. is equal to ω.

The angular frequency of an oscillating system is given by 2π times the square root of the force constant divided by the mass.

Hooke's law is applicable only for solids (not for liquids or gases). Hooke's law is not a universal law because it works up to the proportionality limit only.

The quality factor (Q-factor) of an oscillating system is a measure of its damping.