Question

The equation for displacement of a particle at time \(\mathrm{t}\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). The amplitude of oscillation is \(\ldots \ldots \ldots . \mathrm{cm}\). (A) 1 (B) 3 (C) 5 (D) 7

Short answer

The amplitude of oscillation is 5 cm. The correct answer is (C) 5.

Step-by-step solution

Identify the sinusoidal function

The displacement equation is given as \(y = 3 \cos 2t + 4 \sin 2t\). This equation is a sum of a cosine and a sine function, which, in general, can be expressed as \(y = A \cos(2t - \delta)\), where \(A\) represents the amplitude and \(\delta\) represents the phase difference.

Find the amplitude of the oscillation

To find the amplitude, we use the formula \(A = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the coefficients of the cosine and sine functions in the given equation, respectively. In our case, \(a = 3\) and \(b = 4\). So, the amplitude will be: \[A = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\] The amplitude of oscillation is 5 cm. The correct answer is (C) 5.