Question

When a particle executes S.H.M., the restoring force F varies with displacement y as

(a) F ∝ y (b) F ∝ y2

(c) F ∝ 1/y (d) F ∝ 1/y2​

Answer 1

The particle executing SHM like any other oscillatory ... Illustration 1: Find the period of the function, y sin t sin2 t sin3 t. = ω + ... on which a restoring force F acts to impart an acceleration.

Answer 2

The answer to this quesiton is option (a) F ∝ y

Explanation:

Let us consider a particle executing S.H.M motion.

Now any S.H.M motion can be represented by a sinusoidal displacement motion.

Let displacement of the particle from it's mean position be given by,

y-y₀=Asin(ωt+ϕ) --------(1)

where y₀ is the mean/equlibrium position of the particle.

Thus, velocity is given by v=dx/dt=d(Asin(ωt+ϕ))/dt

​⇒v= Aωcos(ωt+ϕ)--------(2)

Similarly, acceleration for the particle is given by, a= dv/dt

⇒a=−Aω²sin(ωt+ϕ)-------(3)

⇒ On putting the eq (1) in eq (3), we get,

⇒ a=−ω²(y-y₀)

⇒Net restoring force F=ma

⇒F=m*(-w²(y-y₀))

Now since w² and y₀ are constant values

⇒F∝ -y

⇒F∝y

Hence option (a) is correct.