The Question is from neet-physics laws-of-motion friction-on-vertical-surface, and it was created by the author ABDUL JILANI BASHA

Question

A block of mass $m$ is pressed against a vertical wall with a force $F$ acting at an angle $	heta$ above the horizontal. The coefficient of static friction between the block and the wall is $\mu_s$.  If the block is just about to slip downwards, which of the following expressions correctly relates $F$, $m$, $	heta$, and $\mu_s$?

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Static Equilibrium and Friction
	**Related Topics**: Forces, Free Body Diagrams, Trigonometry
	**Prerequisites**: Understanding of static friction, resolving forces into components, Newton's first law
## SOLUTION PATHWAY
**Given Information**:
	* Mass of the block: $m$
	* Applied force: $F$ at angle $	heta$ above the horizontal
	* Coefficient of static friction: $\mu_s$
	* Block is just about to slip downwards (static friction is at its maximum value)
**Method & Steps**:
	1. Draw a free body diagram of the block.
		Reasoning: This helps visualize all the forces acting on the block.
		Formula/Rule: None
	2. Resolve the applied force $F$ into horizontal and vertical components.
		Working: Horizontal component $F_x = F\cos(	heta)$, vertical component $F_y = F\sin(	heta)$
		Key Point: These components will be used in force balance.
	3. Apply Newton's first law (equilibrium condition) in both horizontal and vertical directions.
		Reasoning: Since the block is about to slip, it is still in static equilibrium.
		Formula/Rule: $\Sigma F_x = 0$, $\Sigma F_y = 0$
	4. Horizontal equilibrium: The normal force $N$ from the wall balances the horizontal component of $F$.
		Working:  $N = F_x = F\cos(	heta)$
		Key Point: The normal force is needed to calculate friction.
	5. Vertical equilibrium: The gravitational force (weight) $mg$ is balanced by the upward forces, which are the vertical component of $F$ and the static friction $f_s$.
		Working: $mg = F_y + f_s = F\sin(	heta) + \mu_s N$
		Key Point: Since the block is about to slip, the static friction is at its maximum value: $f_s = \mu_s N$
	6. Substitute $N$ from step 4 into the vertical equilibrium equation.
		Working: $mg = F\sin(	heta) + \mu_s (F\cos(	heta)) = F(\sin(	heta) + \mu_s \cos(	heta))$
		Key Point: Solving for F gives the final equation.
	7. Solve for $F$.
		Result: $F = \frac{mg}{\sin(	heta) + \mu_s \cos(	heta)}$
		Verification: The derived formula matches option B.
 ## OPTION ANALYSIS
[A] :x: Incorrect because the denominator is incorrect. The vertical component of the applied force is $F\sin(	heta)$ and the friction is $\mu_s F\cos(	heta)$, so the denominator should be $\sin(	heta)+\mu_s\cos(	heta)$.
[B] ✓: Correct. The applied force is balanced by gravity and static friction when the block is about to slip downwards. This gives us the correct expression for F.
[C] :x: Incorrect because the expression should be a fraction where $mg$ is divided by the terms involving $	heta$ and $\mu_s$.
[D] :x: Incorrect because the expression should be a fraction where $mg$ is divided by the terms involving $	heta$ and $\mu_s$, and the trigonometric terms are incorrect.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Forgetting to resolve the applied force into its components. Failing to account for both vertical forces acting upwards. [How to avoid]: Always start with a free-body diagram and resolve all forces into x and y components.
		2.[Error 2]: Using the wrong direction for the friction force. Friction always opposes the motion or the tendency of motion.[How to avoid]: Draw the free body diagram with correct direction of the friction force.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember that when an object is on the verge of slipping, the static friction is at its maximum value, $f_s = \mu_s N$.
		2.[Memory aid]: Visualize the forces acting on the block to ensure you include all relevant forces in the free-body diagram.

ABDUL JILANI BASHA · Answer

F = mg / (cos(θ) + μ_s sin(θ))

ABDUL JILANI BASHA · Answer

The correct Option is "F = mg / (sin(θ) + μ_s cos(θ))"

ABDUL JILANI BASHA · Answer

F = mg * (sin(θ) + μ_s cos(θ))

ABDUL JILANI BASHA · Answer

F = mg * (cos(θ) + μ_s sin(θ))

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