Question

If the equation for a particle performing S.H.M. is given by $\mathrm{y}=\sin 2 \mathrm{t}+\sqrt{3} \cos 2 \mathrm{t}\(, its periodic time will be \)\ldots \ldots .$ s. (A) 21 (B) \(\pi\) (C) \(2 \pi\) (D) \(4 \pi\).

Short answer

The periodic time of the given S.H.M. is \(2\pi\) seconds. The correct answer is (C) \(2\pi\).

Step-by-step solution

Determine the period of each term

In the given equation, y(t) = sin(2t) + √3 * cos(2t). The two individual trigonometric functions are sin(2t) and cos(2t). Both of these functions have a period of \(2\pi\), because any multiple of their argument will give the same value and they only repeat at those values. For example: sin(x) = sin(x+2π) cos(x) = cos(x+2π)

Determine the period of their sum

To find the period of the sum, we're looking for the smallest positive value of t for which both sin(2t) and cos(2t) repeat. Since both functions have a period of \(2\pi\), their sum will also have the same period. So, the period of the given equation y(t) = sin(2t) + √3 * cos(2t) is \(2\pi\).

Convert the period to periodic time

The periodic time is the time taken to complete one oscillation or cycle. It is represented by the variable T. As per the given equation, when the period of the S.H.M. is \(2\pi\), the periodic time T is also \(2\pi\). So the periodic time of the given S.H.M. is \(2\pi\) seconds. The correct answer is (C) \(2\pi\).