Question

The following figure depicts a circular motion. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated on the figure. The simple harmonic motion of the x-projection of the radius vector of the rotating particle P can be shown as: 

• A. x(t)=acos((2π t)/(4)+(π)/(4))
• B. x(t)=acos((π t)/(4)+(π)/(4))
• C. x(t)=asin((2π t)/(4)+(π)/(4))
• D. x(t)=acos((π t)/(3)+(π)/(2))

Hint:

Projection of uniform circular motion on a diameter always executes SHM with x=acos(ω t+φ)

Answer 1

The Correct Option is A

Step 1: Given period: T=4s ⟹ ω=(2π)/(T)=(π)/(2) 

Step 2: At t=0, the radius vector makes an angle of 45^∘ with the x-axis: φ=(π)/(4) 

Step 3: Equation of SHM for x-projection: x(t)=acos(ω t+φ) x(t)=acos((π)/(2)t+(π)/(4)) = acos((2π t)/(4)+(π)/(4))