80. Consider a block of mass ‘m’ hanging using a string and pulley arrangement, as shown in the figure. The weight ‘mg’ and tension ‘T’ are working on the block in such a way that the block is not moving and the string is parallel to the perfectly vertical wall. If the block is just in contact (but not attached/fixed) with the wall and the coefficient of static friction is ‘μ’, then the static frictional force acting on the block is
(A) 0
(B) μmg
(C) μT
(D) μ(mg + T)/2
Static Friction on a Block Just Touching a Vertical Wall: Complete Force Analysis and Solution
Correct Answer
Option (A): 0
Understanding the Physical Situation
The most important statement in the question is that the string is parallel to the perfectly vertical wall. This means the tension in the string acts completely in the vertical direction. At the same time, the weight of the block also acts vertically downward due to gravity.
Therefore, both the major forces acting on the block—its weight and the tension in the string—are vertical. There is no horizontal component of either force that can push the block against the wall.
The question also states that the block is just in contact with the wall. This means the wall merely touches the block but does not exert any compressive force unless some horizontal force presses the block into it.
Free Body Diagram (FBD) Analysis
To solve any friction problem correctly, the first step is always to draw the Free Body Diagram.
The forces acting on the block are:
- Weight (mg) acting vertically downward.
- Tension (T) acting vertically upward.
- No horizontal external force acting on the block.
Since there is no horizontal force pushing the block toward the wall, the wall cannot exert any normal reaction on the block.
Therefore,
Normal Reaction (N) = 0
Relationship Between Normal Reaction and Friction
Static friction is not an independent force. It always depends on the normal reaction between two surfaces.
The maximum value of static friction is given by
fs(max) = μN
If the normal reaction is zero, then
fs(max) = μ × 0 = 0
Since the maximum possible static friction itself is zero, the actual static friction must also be zero.
Why Does the Block Remain at Rest?
The block is already in equilibrium because the upward tension balances the downward weight.
T = mg
As these two forces cancel each other, there is no tendency for the block to move either upward or downward. Consequently, friction is not required to maintain equilibrium.
This is a very important concept in Mechanics: friction appears only when it is required to oppose impending or actual relative motion. If there is no tendency for motion and no normal reaction, friction does not develop.
Why Option (A) is Correct
Since the block experiences no horizontal force, the wall does not exert a normal reaction. Because friction depends entirely on the normal reaction, the frictional force becomes zero.
Thus,
Static Friction = 0
Hence, the correct answer is Option (A).
Why Option (B) is Incorrect
Option (B): μmg
This option incorrectly assumes that the normal reaction equals the weight of the block. Weight acts vertically downward, whereas the normal reaction always acts perpendicular to the surface. Since the wall is vertical, its normal reaction would be horizontal. Because there is no horizontal force pressing the block into the wall, the normal reaction cannot be equal to mg.
Why Option (C) is Incorrect
Option (C): μT
This option assumes that the tension itself acts as the normal reaction. However, the tension is also vertical because the string is parallel to the wall. A vertical force cannot produce a horizontal normal reaction. Therefore, this option is physically incorrect.
Why Option (D) is Incorrect
Option (D): μ(mg + T)/2
This expression has no physical basis. The normal reaction is never calculated by taking the average of weight and tension. Since neither mg nor T acts perpendicular to the wall, their average cannot represent the normal force. Therefore, this option is also incorrect.
Important Concept to Remember
Students often memorize the formula
f = μN
without understanding that this equation depends entirely on the value of the normal reaction. Before applying any friction formula, always determine the normal force by drawing the Free Body Diagram. If there is no normal reaction, friction cannot exist regardless of the value of the coefficient of friction.
Exam-Oriented Insight
This question is an excellent example of a conceptual trap. The examiner intentionally provides the coefficient of static friction μ to tempt students into using the formula directly. However, the coefficient becomes irrelevant because the normal reaction itself is zero. In competitive examinations, always identify the forces producing the normal reaction before calculating friction. This habit helps avoid many common mistakes in Mechanics.
Key Takeaways
A block merely touching a wall does not necessarily experience friction. Friction requires two conditions: contact between surfaces and a normal reaction. In this problem, although contact exists, there is no horizontal force to produce a normal reaction. As a result, the static frictional force is zero.
Final Answer
Static Frictional Force = 0
Correct Option: (A)
