Question

The amplitude for a S.H.M. given by the equation $\mathrm{x}=3 \sin 3 \mathrm{pt}+4 \cos 3 \mathrm{pt}\( is \)\ldots \ldots \ldots \ldots \mathrm{m}$ (A) 5 (B) 7 (C) 4 (D) \(3 .\)

Short answer

The amplitude for the given SHM equation is \(A = 5\,m\).

Step-by-step solution

Write down the given equation in general SHM form

The given equation for the SHM is \(x = 3\sin(3pt) + 4\cos(3pt)\). We aim to rewrite this equation in the general SHM form: \(x(t) = A\sin(\omega t + \phi)\).

Apply the sum-to-product trigonometric identity

We use the sum-to-product trigonometric identity to convert the sum of sine and cosine functions into a product form: \[\sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B)\] Using this identity, we rewrite our given equation as follows: \[x = R\sin(3pt + \phi)\] where \(R = \sqrt{3^2 + 4^2}\) and \(\tan{\phi} = \frac{3}{4}\).

Calculate R

Calculate the value of R using the Pythagorean theorem: \[R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]

Identify the amplitude

The amplitude, A, is equal to the value of R in our transformed equation: \[A = R = 5\]

Find the correct option

The correct option for the amplitude is: (A) 5