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 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
Show Hint
If work done is negative, the angle between force and displacement is obtuse because
\[
\cos\theta\lt 0.
\]
Updated On: Jun 25, 2026
- \(\cos^{-1}\left(\frac13\right)\)
- \(30^\circ\)
- \(120^\circ\)
- \(\cos^{-1}\left(\frac34\right)\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Use the work-energy theorem.
According to the work-energy theorem, \[ W=\Delta K \] Since kinetic energy decreases by \(25\text{ J}\), \[ W=-25\text{ J} \]
Step 2: Find the magnitude of displacement.
Given displacement: \[ \vec{s}=3\hat{i}-4\hat{j} \] Magnitude: \[ |\vec{s}|=\sqrt{3^2+(-4)^2} \] \[ =\sqrt{9+16} \] \[ =5\text{ m} \]
Step 3: Use the work formula.
Work done by a force is \[ W=Fs\cos\theta \] Given: \[ F=10\text{ N},\quad s=5\text{ m} \] So, \[ -25=10\times 5\cos\theta \] \[ -25=50\cos\theta \] \[ \cos\theta=-\frac12 \]
Step 4: Find the angle.
\[ \theta=\cos^{-1}\left(-\frac12\right) \] \[ \theta=120^\circ \]
Step 5: Final conclusion.
Therefore, the angle between force and displacement is \[ \boxed{120^\circ} \]
According to the work-energy theorem, \[ W=\Delta K \] Since kinetic energy decreases by \(25\text{ J}\), \[ W=-25\text{ J} \]
Step 2: Find the magnitude of displacement.
Given displacement: \[ \vec{s}=3\hat{i}-4\hat{j} \] Magnitude: \[ |\vec{s}|=\sqrt{3^2+(-4)^2} \] \[ =\sqrt{9+16} \] \[ =5\text{ m} \]
Step 3: Use the work formula.
Work done by a force is \[ W=Fs\cos\theta \] Given: \[ F=10\text{ N},\quad s=5\text{ m} \] So, \[ -25=10\times 5\cos\theta \] \[ -25=50\cos\theta \] \[ \cos\theta=-\frac12 \]
Step 4: Find the angle.
\[ \theta=\cos^{-1}\left(-\frac12\right) \] \[ \theta=120^\circ \]
Step 5: Final conclusion.
Therefore, the angle between force and displacement is \[ \boxed{120^\circ} \]
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