A solid cylinder is rolling down on an inclined plane of angle $ \theta , $ The coefficient of static friction between the plane and cylinder is $ \mu _{s}$ . Then condition for the cylinder not to slip is
- $ \tan \,\theta \ge \,3\mu_{s}$
- $ \tan \,\theta >3\mu _{s}$
- $ \tan \,\theta \le 3\mu _{s}$
- $ \tan \,\theta < 3\mu_{s}$
The Correct Option is C
Solution and Explanation
$a=\frac{g}{3} \sin \theta$
$\therefore$ The condition for the cylinder to remain in equilibrium
$m a \leq \mu_{s} R$
$\Rightarrow \frac{1}{3} M g \sin \theta \leq M g \cos \theta \cdot \mu_{s}$
or $\mu_{s} \geq \frac{1}{3} \tan \theta$
or $\tan \theta \leq 3\, \mu_{s}$
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Concepts Used:
Inclined planes
An inclined plane is a simple machine also known as a ramp. It is a simple machine that consists of a sloping surface. Its purpose is to reduce the force that is applied to raise a load. When we need to raise a body vertically, similarly we need to apply a force that is equal to the body’s weight. An incline is like a surface that sets a different angle part from the right one.
Some of the examples of ramps are Roads, vertically, Sloping, wedges, Chisels, Plows, etc.,
Normal Force:
An inclined plane is directed upwards to the opposite direction with the force of gravity, known as the Normal Force. Simply, A normal force is not always directed in the direction that we are accustomed to. But they are not always upwards.
Gravity Force:
The component of the force of gravity is directed opposite to the normal force. It balances the normal force but can’t balance the parallel component of the force of gravity with the help of any other force. It can be resolved into two components- first is directed parallel towards the inclined surfaces, and the other one is directed perpendicular towards the inclined surface.
Read More: Acceleration on Inclined Plane
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