Question

Velocity of a particle of mass m as a function of displacement x is given by v = ɑ√x. Work done to move it from x = 0 to x = d is:

• A. \(\frac{ma^2}{2}\cdot d\)
• B. \(ma^2\cdot d\)
• C. \(3ma^2\cdot \frac{d}{2}\)
• D. \(2ma^2\cdot d\)

Answer 1

The Correct Option is A

To find the work done in moving the particle from \(x = 0\) to \(x = d\), we need to consider the relationship between velocity and displacement given in the question: \(v = \alpha\sqrt{x}\). Here, \(\alpha\) is a constant, and \(m\) is the mass of the particle.

The work done \(W\) on an object can be calculated using the work-energy principle, which states that the work done by the forces on an object is equal to the change in its kinetic energy.

The kinetic energy \(K\) of a particle with mass \(m\) and velocity \(v\) is given by:

K = \frac{1}{2} m v^2

Since \(v = \alpha\sqrt{x}\), substituting it in the kinetic energy equation gives:

K = \frac{1}{2} m (\alpha \sqrt{x})^2 = \frac{1}{2} m \alpha^2 x

The work done is the change in kinetic energy as the particle moves from \(x = 0\) to \(x = d\):

W = K_{\text{final}} - K_{\text{initial}}

At \(x = 0\), the displacement is zero, so the initial kinetic energy \(K_{\text{initial}} = \frac{1}{2} m \alpha^2 \cdot 0 = 0\).

At \(x = d\), the kinetic energy is:

K_{\text{final}} = \frac{1}{2} m \alpha^2 d

Therefore, the work done \(W\) is:

W = \frac{1}{2} m \alpha^2 d - 0 = \frac{1}{2} m \alpha^2 d

Thus, the work done to move the particle from \(x = 0\) to \(x = d\) is:

\(\frac{ma^2}{2}\cdot d\)

This matches the correct option given:

\(\frac{ma^2}{2}\cdot d\)