Question:

A particle moves with simple harmonic motion in a straight line. In first τs, after starting from rest, it travels a distance a, and in next τs it travels 2a, in the same direction, then:

Amplitude of motion is 4a

Time period of oscillations is 8τ

Time period of oscillations is 6τ

Amplitude of motion is 3a

A(1-cosωτ)=a
A(1-cos2ωτ)=3a
cosωτ=(1-a/A)
cos2ωτ=(1-3a/A)
2(1-a/A)²-1=(1-3a/A)
Solving the equation
a/A=1/2
A=2a
cosωτ=1/2
ωτ=π/3
T=2πω
T=6τ