Question

A force of \(10\text{ N}\) acting at an angle on a particle produces a displacement of \[ (3\hat{i}-4\hat{j})\text{ m} \] due to this force. If the kinetic energy of the particle is decreased by \(25\text{ J}\), then the angle between the force and the displacement is

• A. \(\cos^{-1}\left(\frac13\right)\)
• B. \(30^\circ\)
• C. \(120^\circ\)
• D. \(\cos^{-1}\left(\frac34\right)\)

Hint:

If work done is negative, the angle between force and displacement is obtuse because \[ \cos\theta\lt 0. \]

Answer 1

The Correct Option is C

Step 1: Apply the work-energy theorem.
Net work done $= \Delta KE$. Since KE decreases by 25 J: $W = -25$ J.
Step 2: Find the magnitude of the displacement vector.
$\vec{s} = 3\hat{i} - 4\hat{j}$ m: \[ |\vec{s}| = \sqrt{9 + 16} = 5 \text{ m} \]
Step 3: Use $W = F s \cos\theta$.
\[ -25 = 10 \times 5 \times \cos\theta \Rightarrow \cos\theta = -\frac{1}{2} \]
Step 4: Solve for $\theta$.
\[ \theta = \cos^{-1}\!\left(-\frac{1}{2}\right) = 120^\circ \] The obtuse angle confirms the force has a component opposing displacement, decreasing KE.
Step 5: Verify physically.
Force component along displacement: $10\cos 120^\circ = -5$ N. Work $= -5 \times 5 = -25$ J. Confirmed.
Step 6: State the final answer.
\[ \boxed{\theta = 120^\circ} \]