Question

The ratio of force constants of two springs is \(1: 5\). The equal mass suspended at the free ends of both springs are performing S.H.M. If the maximum acceleration for both springs are equal, the ratio of amplitudes for both springs is \(\ldots \ldots\) (A) \((1 / \sqrt{5})\) (B) \((1 / 5)\) (C) \((5 / 1)\) (D) \((\sqrt{5} / 1)\)

Short answer

The ratio of amplitudes for both springs is (C) (5/1).

Step-by-step solution

Write down the known information

We know that the ratio of force constants is 1:5 and the masses on both springs are equal. We're given that their maximum accelerations are equal.

Write down the relation between force constant and maximum acceleration

In simple harmonic motion, the maximum acceleration (a_max) is given by: \(a_{max} = k \cdot A/m\) where \(k\) is the spring constant, \(A\) is the amplitude, and \(m\) is the mass.

Write down the ratio of amplitudes considering the equality of maximum acceleration

Let the force constants of the springs be \(k_1\) and \(k_2\), and their respective amplitudes be \(A_1\) and \(A_2\). Given that the maximum accelerations of both springs are equal, we can write: \( \frac{k_1 \cdot A_1}{m} = \frac{k_2 \cdot A_2}{m} \) Since the masses are equal, they will cancel out: \( k_1 \cdot A_1 = k_2 \cdot A_2\)

Substitute the ratio of force constants into the equation

Given that the ratio of the force constants is 1:5, we can write that \(k_2 = 5k_1\). Let's substitute this into the equation from step 3: \( k_1 \cdot A_1 = (5k_1) \cdot A_2\) Now, divide both sides of the equation by \(k_1\): \( A_1 = 5A_2\)

Find the ratio of amplitudes and the correct answer

The ratio of the amplitude of the first spring to the amplitude of the second spring is: \( \frac{A_1}{A_2} = 5 \) So, the correct answer is (C) (5/1).