Consider three masses \(M_1\), \(M_2\) and \(M_3\), where \(M_1 \gt M_2 \gt M_3\), are at rest on a horizontal plane as shown in the figure. Now the angle of inclination \(\theta\) of the plane is gradually increased until the masses just begin to slide. Assume the coefficient of static friction between the masses and the surface is constant. Then the correct statement is
Show Hint
For a body on a rough inclined plane, the angle of repose is given by
\[
\tan\theta=\mu_s.
\]
It depends only on the coefficient of static friction and not on the mass of the body.
\(M_3\) begins to slide at a higher inclination angle than \(M_1\) and \(M_2\)
\(M_3\) begins to slide at a lower inclination angle than \(M_1\) and \(M_2\)
\(M_1\), \(M_2\) and \(M_3\) begin to slide at the same inclination angle
\(M_2\) begins to slide at a higher inclination angle than \(M_1\) and \(M_3\)
Show Solution
The Correct Option isC
Solution and Explanation
Step 1: Write the sliding condition for a block on an inclined plane. A block slides when \( mg\sin\theta \geq \mu_s\,mg\cos\theta \), i.e., \( \tan\theta \geq \mu_s \). The mass m cancels from both sides completely.
Step 2: Apply to all three masses. Since \( \mu_s \) is the same for all and mass does not appear in the sliding condition, all three masses begin sliding at the same critical angle \( \theta_c = \arctan(\mu_s) \). \[ \boxed{M_1,\,M_2\text{ and }M_3\text{ begin to slide at the same inclination angle}} \]
