Animated Solution for Physics - Laws of Motion: A block of mass $0.1\text{ kg}$ is held against a wall applying a horizontal force of $5\text{ N}$ on the block. If the coefficient of friction between the block and the wall is $0.5$, the magnitude of the frictional force acting on the block is

Welcome, future physicists and engineers! Today, we are embarking on a thrilling journey into the heart of classical mechanics. We are going to dissect a problem that seems deceptively simple but hides one of the most beautiful and misunderstood concepts in physics: the self-adjusting nature of static friction.

Imagine you are holding a heavy textbook against a vertical wall. You push it horizontally. If you don't push hard enough, the book slides down. If you push just right, it stays perfectly still. What is happening at the microscopic level? How do the forces balance out? This is exactly what we will explore in this problem.

Let us start by visualizing the physical reality of our problem. We have a block of mass $m=0.1\text{ kg}$ held against a vertical wall.

To keep it there, a horizontal force of $5\text{ N}$ is applied.

Whenever you face a mechanics problem, your first instinct should always be to draw a Free Body Diagram (FBD). It is the ultimate tool that translates physical reality into mathematical equations.

In our FBD, we have four distinct forces acting on the block.

First, the applied force $F$ pushing the block into the wall.

Second, the wall pushes back. According to Newton's Third Law, every action has an equal and opposite reaction. This pushback is the normal force $N$.

Third, gravity is relentlessly pulling the block downwards. This is the weight of the block, $W=mg$.

Finally, we have friction. Since gravity wants to pull the block down, friction acts upwards along the surface of the wall to oppose this impending motion.

Horizontal Equilibrium

Let us break the problem down into two independent dimensions: horizontal and vertical.

In the horizontal direction, the block is completely stationary. It is not accelerating into the wall, nor is it flying off the wall.

This means the block is in perfect horizontal equilibrium.

According to Newton's Second Law, if the acceleration is zero, the net force must be zero. Therefore, the forces pointing left must perfectly balance the forces pointing right.

Since we know the applied force is $5\text{ N}$, we can immediately determine the normal force.

This is a critical milestone. The normal force is the gateway to understanding friction. It tells us how tightly the microscopic irregularities of the block and the wall are pressed together.

Limiting Friction

Now, let us turn our attention to the vertical direction. The force trying to hold the block up is static friction.

But how much friction can the wall actually provide?

Friction is not infinite. It has a breaking point. This maximum possible value of static friction is called the limiting friction, denoted as $f_{\max }$.

The formula for limiting friction is one of the most fundamental equations in your physics arsenal:

Here, ${\mu }_s$ is the coefficient of static friction, a dimensionless number that represents the roughness of the two surfaces in contact.

We are given ${\mu }_s=0.5$. We just calculated $N=5\text{ N}$. Let us substitute these values into our master equation.

This result is profoundly important. It tells us that the wall is capable of providing up to $2.5\text{ N}$ of upward force. If the downward pull exceeds this value, the block will inevitably slide.

So, what is the downward pull? It is simply the weight of the block.

The weight is calculated using the equation:

We know the mass $m=0.1\text{ kg}$.

For the acceleration due to gravity, $g$, we must be strategic. In JEE problems, unless $10{\text{ m/s}}^2$ is explicitly stated, it is highly recommended to use $9.8{\text{ m/s}}^2$. Furthermore, looking at the options provided in the question ($0.98\text{ N}$, $4.9\text{ N}$, etc.), they are clear multiples of $9.8$. This is a massive hint!

Let us perform the calculation:

Gravity is pulling the block down with a force of $0.98\text{ N}$.

Now we arrive at the climax of our problem. This is where the trap is set, and this is where countless students lose precious marks.

Many students, in the heat of an exam, will calculate $f_{\max }=2.5\text{ N}$ and immediately select option (a).

They assume that the frictional force is always equal to ${\mu }_{s}N$.

The formula ${\mu }_{s}N$ only gives you the maximum possible static friction. It is a ceiling, a limit. It is not necessarily the actual force acting at that moment.

Think about it physically. If the wall applied an upward force of $2.5\text{ N}$ while gravity only pulled down with $0.98\text{ N}$, the net force would be upwards! The block would spontaneously accelerate up the wall. That violates the laws of physics and common sense.

The Self-Adjusting Force

Static friction is a highly intelligent, self-adjusting force.

It only exerts exactly as much force as is necessary to prevent motion, up to its maximum limit.

It is like a lazy but strong defender. It won't use its full strength of $2.5\text{ N}$ if it doesn't have to.

We must compare the required force (the weight) with the available force (the limiting friction).

Since $W<f_{\max }$, the block is perfectly safe. It will not slide.

Because it does not slide, it is in vertical equilibrium. The upward frictional force must exactly balance the downward weight.

And there we have it! The actual frictional force acting on the block is $0.98\text{ N}$.

This problem is a beautiful reminder that physics is not just about blindly plugging numbers into formulas. It is about understanding the physical reality behind the mathematics. It is about visualizing the forces, understanding their nature, and applying logic before applying algebra.

Keep this lesson close to your heart. Whenever you deal with static friction, always ask yourself: "Is the applied force trying to cause motion greater than the limiting friction?" If not, friction is simply equal to the applied force.

Stay curious, keep visualizing, and never fall for the trap of limiting friction again!