Problem 372 - JEE Physics

Question

A plate of mass \(\mathrm{M}\) is placed on a horizontal frictionless surface and a body of mass \(m\) is placed on this plate, The coefficient of dynamic friction between this body and the plate is \(\mu\). If a force \(2 \mu \mathrm{mg}\). is applied to the body of mass \(\mathrm{m}\) along the horizontal direction the acceleration of the plate will be (A) \((\mu \mathrm{m} / \mathrm{M}) \mathrm{g}\) (B) \([\mu \mathrm{m} /(\mathrm{M}+\mathrm{m})] \mathrm{g}\) (C) \([(2 \mu \mathrm{m}) / \mathrm{M}] \mathrm{g}\) (D) \([(2 \mu \mathrm{m}) /(\mathrm{M}+\mathrm{m})] \mathrm{g}\)

Step-by-Step Solution

Answer

The acceleration of the plate is (A) \((\mu m / M) g\).

Step 1: Determining the Maximum Force of Static Friction

Calculate the maximum force of static friction between the plate and the body of mass m. The maximum force of static friction is given by: \( F_{s_{\max}} = \mu m g \)

Step 2: Analyzing Forces on Body m

The horizontal force applied on body m is given as 2μmg. Since this force is greater than the maximum force of static friction, the body will move. The net force acting on body m (Fm) can be calculated as: \( F_m = 2 \mu m g - \mu m g = \mu m g \)

Step 3: Applying Newton's Second Law to Body m

Now that we know the net force acting on body m, we can apply Newton's second law of motion to find its acceleration (am): \( F_{m} = m a_m \) Substitute the values and solve for am: \( a_m = \frac{F_{m}}{m} = \frac{\mu m g}{m} = \mu g \)

Step 4: Analyzing Forces on Plate M

The only horizontal force acting on plate M is the static friction force between the plate and the body of mass m. Since body m is moving to the right, the frictional force will be acting to the left. Therefore, the net force acting on plate M (FM) can be written as: \( F_{M} = -\mu m g \)

Step 5: Applying Newton's Second Law to Plate M

Now that we know the net force acting on plate M, we can apply Newton's second law of motion to find its acceleration (aM): \( F_{M} = M a_{M} \) Substitute the values and solve for aM: \( a_M = \frac{F_{M}}{M} = \frac{-\mu m g}{M} \) The negative sign indicates that the plate accelerates to the left while the body m accelerates to the right. Thus, the acceleration of the plate is (A) \((\mu m / M) g\).

Key Concepts

Dynamic Friction

When two objects are in contact and moving relative to each other, a force called **dynamic friction** (or kinetic friction) comes into play. This force acts in the opposite direction of the motion and tries to slow down the moving object.
This friction is defined by the equation:\[ F_d = \mu_d \cdot N \]- **\( F_d \)** represents the dynamic friction force.- **\( \mu_d \)** stands for the dynamic (or kinetic) coefficient of friction.- **\( N \)** is the normal force, which is typically equal to the weight of the object if the surface is horizontal.In the given problem, the body of mass \( m \) experiences dynamic friction as it moves across the plate. When a force is applied that surpasses the maximum static friction, dynamic friction takes over, ensuring that the body continues to move relative to the plate. This results in an ongoing resistance against the movement driven by the coefficient \( \mu \). Dynamic friction is crucial because it provides stability and prevents unchecked acceleration of moving masses.

Net Force Calculation

Calculating the **net force** is a key step in understanding motion, particularly when discussing Newton's Second Law. According to this law, force is the product of mass and acceleration:\[ F = ma \]To determine how an object will behave under various forces, identify all the forces acting on it and combine them to find the net force.In this scenario:- The applied force on body \( m \) is \( 2 \mu mg \).- The maximum potential static friction force resisting motion is \( \mu mg \).Since the applied force exceeds the force of static friction, the object starts to move. The net force (\( F_m \)) acting on body \( m \) becomes:\[ F_m = 2 \mu mg - \mu mg = \mu mg \]This net force is what causes the object \( m \) to accelerate across the plate. Understanding net force is fundamental for predicting how objects react under different conditions, highlighting Newton's law's practical application.

Static Friction

**Static friction** is the force that keeps an object at rest when a force is applied, preventing any initial movement. It acts only until a certain threshold, known as the maximum static friction force, is surpassed.The formula to calculate the maximum static friction is:\[ F_{s_{\max}} = \mu_s \cdot N \]- **\( F_{s_{ ext{max}}} \)** is the maximum static friction force.- **\( \mu_s \)** is the static coefficient of friction.- **\( N \)** represents the normal force.In the exercise, this static friction needs to be exceeded by the applied force for motion to occur. Given that \( \mu mg \) is the maximum static friction, the applied force of \( 2 \mu mg \) is more than enough to overcome it, causing the body \( m \) to slide over the plate. Static friction is crucial because it determines the start of motion and assures an initial resistance that depends on how tightly surfaces grip each other.