Correct Answer - Option 1 :
\(\frac{\beta ^2}{\alpha}\)
Concept:
Simple Harmonic Motion (SHM):
- The Simple Harmonic Motion is studied to discuss the periodic Motion Mathematically.
- In Simple Harmonic motion, the motion is between two extreme points, and the restoring force responsible for the motion tends to bring the object to mean position.
- The motion of a Simple pendulum and a block attached to spring are common examples of SHM.
Mathematically, SHM is Defined as:
x = A Sin (ωt + ɸ),
x is the displacement of the body from mean Position, at time t. ɸ is phase Difference.
A is Amplitude of Motion, that is the Maximum distance the body in SHM can move from mean Position.
ω is Angular Speed = \(ω = \frac{2\pi }{T}\)
T is the time period of Motion,
-
The potential Energy of the body in SHM is
P = \(\frac{1}{2}mω ^{2}x^{2}\)
- Kinetic Energy of the body in SHM is
K = \(\frac{1}{2}mω ^{2}(A^{2}-x^{2})\)
-
Total Energy of the Body in SHM (E)
E= \(\frac{1}{2}mω ^{2}A^{2}\)
- In SHM, the acceleration is directly proportional to the displacement with a negative direction.
a = - ω2 x
if A is the amplitude of the acceleration, the maximum acceleration will be when x is maximum. At x = A, the maximum acceleration is ω2 A.
Calculation:
The maximum speed is v, then the maximum kinetic energy will be \(\frac{1}{2}mv ^{2}\)
\(\frac{1}{2}mv ^{2} = \frac{1}{2}mω ^{2}(A^{2}-x^{2})\)
Maximum kinetic energy is achieved at x = 0 (mean position)
\(\frac{1}{2}mv ^{2} = \frac{1}{2}mω ^{2}A^2\)
v = ωA
Now according to the question
β = max speed = ωA -- (1)
ω = β / A
and
α = max acceleration = ω2 A -- (2)
α = ( β / A)2 × A
\(\implies \alpha = \frac{\beta ^2}{A}\)
\(\implies A = \frac{\beta ^2}{\alpha}\)
