Question

The displacement of a particle is given at time t, by: x = Asin(-2ω t) + Bsin² ω t Then,

• A. the motion of the particle is SHM with an amplitude of √(A² + (B²)/(4))
• B. the motion of the particle is not SHM, but oscillatory with a time period of T = π/ω
• C. the motion of the particle is oscillatory with a time period of T = π/2ω
• D. the motion of the particle is periodic.

Hint:

If displacement can be written as a combination of sin and cos terms of the same angular frequency: • The motion is SHM. • Amplitude is found using root-sum-square method.

Answer 1

The Correct Option is A

Step 1: Simplify the given expression: x = -Asin(2ω t) + Bsin²ω t Using identity sin²ω t = (1-cos 2ω t)/(2), x = -Asin(2ω t) + (B)/(2) - (B)/(2)cos(2ω t)
Step 2: Rearranging, x - (B)/(2) = -Asin(2ω t) - (B)/(2)cos(2ω t) This represents SHM with angular frequency 2ω.
Step 3: Amplitude of SHM: Aₑff = √(A² + ((B)/(2))²)
Step 4: Since the motion repeats after equal intervals of time, it is also periodic.