A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm (b) 3 cm (c) 0 cm

As an experienced tutor registered on UrbanPro, I can help you tackle this problem step by step. Simple harmonic motion (SHM) is a fundamental concept in physics, and understanding it thoroughly can pave the way for mastering more complex topics.

Firstly, let's establish the key formulas for simple harmonic motion:

1. Displacement (x): x=A⋅sin⁡(ωt+ϕ)x=A⋅sin(ωt+ϕ)

   • Where AA is the amplitude, ωω is the angular frequency (calculated as 2π2π divided by the period TT), tt is the time, and ϕϕ is the phase angle.

2. Velocity (v): v=A⋅ω⋅cos⁡(ωt+ϕ)v=A⋅ω⋅cos(ωt+ϕ)

   • The derivative of displacement with respect to time.

3. Acceleration (a): a=−A⋅ω2⋅sin⁡(ωt+ϕ)a=−A⋅ω2⋅sin(ωt+ϕ)

   • The derivative of velocity with respect to time.

Given that the amplitude AA is 5 cm and the period TT is 0.2 s, we can calculate the angular frequency (ωω) as 2π/T2π/T.

1. For x=5x=5 cm:

   • Substituting x=5x=5 cm into the displacement formula, we find the corresponding time tt.
   • Then, using this time, we can calculate velocity and acceleration.

2. For x=3x=3 cm:

   • We repeat the same process, finding the time corresponding to x=3x=3 cm and then calculating velocity and acceleration.

3. For x=0x=0 cm:

   • Again, we follow the steps to find velocity and acceleration.

After finding the times for each displacement, we substitute them into the velocity and acceleration formulas to get the respective values.

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