The Question is from neet-physics laws-of-motion motion-of-a-body-on-the-inclined-plane, and it was created by the author ABDUL JILANI BASHA

Question

A block of mass $m$ is placed on a rough inclined plane of angle $	heta$. The coefficient of static friction is $\mu_s$. A horizontal force $F$ is applied to the block. What is the minimum value of $F$ required to prevent the block from sliding down the plane?

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: Option A
## CONCEPT FRAMEWORK
	**Primary Concept**: Static Equilibrium and Forces on an Inclined Plane
	**Related Topics**: Static Friction, Free Body Diagrams, Resolving Forces
	**Prerequisites**: Understanding of trigonometry, Newton's Laws of Motion, concept of static friction
## SOLUTION PATHWAY
**Given Information**:
	* Mass of the block: $m$
	* Angle of the inclined plane: $	heta$
	* Coefficient of static friction: $\mu_s$
	* Horizontal force applied: $F$
	* Condition: Block is at rest (not sliding down)
**Method & Steps**:
	1. Draw a free body diagram of the block. The forces acting on the block are:
		* Gravitational force ($mg$) acting vertically downward.
		* Normal force ($N$) acting perpendicular to the inclined plane.
		* Static frictional force ($f_s$) acting up the inclined plane (since we want to prevent sliding down).
		* Applied horizontal force ($F$)
	2. Resolve all forces into components parallel and perpendicular to the inclined plane:
		* $mg$: Component parallel to the plane: $mg\sin	heta$ (down the plane), Component perpendicular to the plane: $mg\cos	heta$ (into the plane).
		* $F$: Component parallel to the plane: $F\cos	heta$ (up the plane), Component perpendicular to the plane: $F\sin	heta$ (out of the plane).
		* $N$: perpendicular to the plane
		* $f_s$: parallel to the plane
	3. Apply Newton's first law (sum of forces equals zero) for static equilibrium:
		* Perpendicular to the plane: $N - mg\cos	heta - F\sin	heta = 0 \Rightarrow N = mg\cos	heta + F\sin	heta$
		* Parallel to the plane: $F\cos	heta + f_s - mg\sin	heta = 0$
	4. Since we are looking for the minimum force $F$ to prevent sliding down, the friction force will be at its maximum static value, i.e., $f_s = \mu_s N$
	5. Substitute $f_s$ and $N$ into the parallel force equation:
		* $F\cos	heta + \mu_s (mg\cos	heta + F\sin	heta) - mg\sin	heta = 0$
	6. Solve for $F$:
		* $F\cos	heta + \mu_s mg\cos	heta + \mu_s F\sin	heta - mg\sin	heta = 0$
		* $F(\cos	heta + \mu_s \sin	heta) = mg(\sin	heta - \mu_s \cos	heta)$
		* $F = mg(\frac{\sin	heta - \mu_s \cos	heta}{\cos	heta + \mu_s \sin	heta})$

**Result**: The minimum force F is: $F = mg(\frac{\sin	heta - \mu_s \cos	heta}{\cos	heta + \mu_s \sin	heta})$
**Verification**: The answer is consistent with the forces at play and the condition of minimum force to prevent sliding down.
 ## OPTION ANALYSIS
[A] ✓: This is the correct expression for the minimum force $F$ obtained by the force analysis.
[B] :x: This option has sine and cosine interchanged in the numerator and the denominator, leading to an incorrect expression.
[C] :x: This option has the wrong signs in the numerator and the denominator, leading to an incorrect expression.
[D] :x: This option has the wrong signs in the numerator and the denominator, leading to an incorrect expression.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrectly resolving forces: Make sure you correctly resolve forces into components parallel and perpendicular to the plane. Use trigonometry correctly.
		2. Confusing static and kinetic friction: Remember that we are working with static friction here, and it is equal to $\mu_s N$ only at the limit of motion.
	**Quick Tips**:
		1. Always draw a free body diagram first to visualize all forces.
		2. Remember to consider the condition for static equilibrium (net force = 0).

ABDUL JILANI BASHA · Answer

The correct Option is "$mg(\frac{sin	heta - \mu_s cos	heta}{cos	heta + \mu_s sin	heta})$"

ABDUL JILANI BASHA · Answer

$mg(\frac{cos	heta - \mu_s sin	heta}{sin	heta + \mu_s cos	heta})$

ABDUL JILANI BASHA · Answer

$mg(\frac{sin	heta + \mu_s cos	heta}{cos	heta - \mu_s sin	heta})$

ABDUL JILANI BASHA · Answer

$mg(\frac{cos	heta + \mu_s sin	heta}{sin	heta - \mu_s cos	heta})$

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