The Question is from neet-physics laws-of-motion , and it was created by the author P.NARASIMHA

Question

A block of mass $m$ is placed on a horizontal rotating disc at a distance $r$ from the center. The coefficient of static friction between the block and the disc is $\mu_s$. What is the maximum angular velocity of the disc so that the block does not slide?

P.NARASIMHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [Option C]
## CONCEPT FRAMEWORK
	**Primary Concept**: Circular Motion and Static Friction
	**Related Topics**: Centripetal Force, Frictional Force
	**Prerequisites**: Understanding of circular motion, static friction, and Newton's laws of motion.
## SOLUTION PATHWAY
**Given Information**:
	* Mass of the block: $m$
	* Distance from the center: $r$
	* Coefficient of static friction: $\mu_s$
	* Block is not sliding.
**Method & Steps**:
	1. Identify the forces acting on the block:
		Reasoning: The block experiences a frictional force towards the center (providing the centripetal force), its weight downwards, and a normal force upwards.
		Formula/Rule:  Forces: Friction, Weight, Normal Force
	2. Relate the static frictional force to the centripetal force:
		Working: The maximum static frictional force is $f_s = \mu_s N$, where N is the normal force. In this case, N = mg. The centripetal force is $F_c = m\omega^2 r$. For the block not to slide, the frictional force must equal the centripetal force:  $\mu_s m g = m \omega^2 r$.
		Key Point: We consider the *maximum* static frictional force, as this corresponds to the *maximum* angular velocity before sliding occurs.
	3. Solve for the maximum angular velocity, $ \omega$:
		Result: From  $\mu_s m g = m \omega^2 r$, we have $\omega^2 = \frac{\mu_s g}{r}$. Thus, $\omega = \sqrt{\frac{\mu_s g}{r}}$.
		Verification: The units of the answer are consistent with angular velocity (rad/s).
 ## OPTION ANALYSIS
[A] :x: This option is not dimensionally correct for angular velocity. It has units of acceleration times length.
[B] :x: This option is not dimensionally correct for angular velocity. It has units of acceleration over length.
[C] ✓: This option correctly calculates the maximum angular velocity using the relationship between centripetal force and static friction and has correct units.
[D] :x: This option is not dimensionally correct for angular velocity. It has units of sqrt(acceleration times length).
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing centripetal force with another force. Centripetal force is a net force caused by other forces.
			[How to avoid]: Always identify the forces acting on the body and then relate them to the centripetal force.
		2.[Error 2]: Forgetting that the maximum static friction force is involved.
			[How to avoid]: Remember that the maximum angular velocity corresponds to the maximum static frictional force.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember the centripetal force formula $F_c = m\omega^2 r$
		2.[Memory aid]: Think about how increasing the angular velocity makes the block want to slide outwards, requiring a greater friction force to keep it in place.

P.NARASIMHA · Answer

$\mu_s g r$

P.NARASIMHA · Answer

$\frac{\mu_sg}{r}$

P.NARASIMHA · Answer

The correct Option is "$\sqrt{\frac{\mu_sg}{r}}$"

P.NARASIMHA · Answer

$\sqrt{\mu_s g r}$

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