The ratio of maximum acceleration to maximum velocity of a particle performing S.H.M is equal toa)Square of amplitudeb)Square of angular velocityc)Amplituded)Angular velocityCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Maximum acceleration = w2A
Maximum velocity = wA
Ratio = w2A/wA = w
w = angular velocity
A = amplitude

Answer

Maximum acceleration=w^2A
Maximum velocity=wA
Ratio=w^2A/wA=w
w=angular velocity
A=amplitude

Answer

The ratio of maximum acceleration to maximum velocity in a particle performing Simple Harmonic Motion (S.H.M) is equal to the angular velocity.

Explanation:

Simple Harmonic Motion (S.H.M)
Simple Harmonic Motion is a type of periodic motion where an object oscillates back and forth around a stable equilibrium position. It can be observed in various physical systems, such as a mass-spring system, a pendulum, or an oscillating fan.

Maximum Velocity and Maximum Acceleration
In S.H.M, the maximum velocity occurs when the particle is at the equilibrium position, while the maximum acceleration occurs when the particle is at the extreme positions. The velocity and acceleration of the particle vary sinusoidally with time.

Relationship between Velocity and Acceleration
The velocity and acceleration of a particle in S.H.M are related by the equation:

a = -ω^2x

where a is the acceleration, x is the displacement from the equilibrium position, and ω is the angular velocity.

Ratio of Maximum Acceleration to Maximum Velocity
To find the ratio of maximum acceleration to maximum velocity, we consider the magnitudes of these quantities at their respective maximum values.

Maximum Velocity:
The maximum velocity occurs when the particle is at the equilibrium position. At this point, the displacement from the equilibrium position is maximum, i.e., x = amplitude (A). Therefore, the maximum velocity (Vmax) can be calculated using the equation:

Vmax = ωA

Maximum Acceleration:
The maximum acceleration occurs when the particle is at the extreme positions. At these points, the displacement from the equilibrium position is maximum, i.e., x = amplitude (A). Therefore, the maximum acceleration (amax) can be calculated using the equation:

amax = -ω^2A

Ratio of Maximum Acceleration to Maximum Velocity:
Taking the ratio of maximum acceleration to maximum velocity, we get:

amax/Vmax = (-ω^2A)/(ωA)

Simplifying the expression, we have:

amax/Vmax = -ω/ω

The negative signs cancel out, and we are left with:

amax/Vmax = 1

Hence, the ratio of maximum acceleration to maximum velocity is equal to 1, which is the same as the ratio of angular velocity (ω) to angular velocity (ω). Therefore, the correct answer is option 'D' - Angular velocity.

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