The velocity of a particle executing SHM varies with displacement $$(x)$$ as $$4v^2 = 50 - x^2$$. The time period of oscillations is $$\frac{x}{7}$$ s. The value of $$x$$ is ______. [Take $$\pi = \frac{22}{7}$$]

Solution & Explanation

1. Understand the Velocity-Displacement Relation in SHM

For a particle executing Simple Harmonic Motion (SHM), the standard mathematical relation between its instantaneous velocity ($$v$$) and its displacement ($$x$$) from the equilibrium position is given by:

$$v^2 = \omega^2 (A^2 - x^2)$$

Where:

• $$\omega$$ is the angular frequency of oscillation
• $$A$$ is the amplitude of the motion

2. Rearrange the Given Equation

We are given the specific particle velocity equation in the problem statement:

$$4v^2 = 50 - x^2$$

To align this perfectly with our standard SHM form, we divide both sides of the equation by $$4$$:

$$v^2 = \frac{50 - x^2}{4}$$

$$v^2 = \frac{1}{4} (50 - x^2)$$

3. Compare and Determine Angular Frequency ($$\omega$$)

By comparing our rearranged equation directly with the standard formula $$v^2 = \omega^2 (A^2 - x^2)$$, we match the coefficients sitting outside the parentheses:

$$\omega^2 = \frac{1}{4}$$

Taking the square root of both sides gives the angular frequency:

$$\omega = \sqrt{\frac{1}{4}} = \frac{1}{2} \,\, \text{rad/s}$$

4. Calculate the Time Period ($$T$$) and Find the Value of $$x$$

The standard relationship linking the total time period of oscillation ($$T$$) to the angular frequency ($$\omega$$) is:

$$T = \frac{2\pi}{\omega}$$

Substitute our calculated value of $$\omega = \frac{1}{2}$$ into this relationship:

$$T = \frac{2\pi}{\frac{1}{2}} = 4\pi \,\, \text{s}$$

We are instructed to take the approximation value $$\pi = \frac{22}{7}$$:

$$T = 4 \times \left(\frac{22}{7}\right) = \frac{88}{7} \,\, \text{s}$$

The problem states that the time period of oscillation is expressionally equal to $$\frac{x}{7} \,\, \text{s}$$. Comparing the two fraction representations directly:

$$\frac{x}{7} = \frac{88}{7} \implies x = 88$$

Correct Numerical Answer: 88