A body B lies on a smooth horizontal table and another body A is placed on B. The coefficient of friction between A and B is `mu`. What acceleration given to B will cause slipping to occur between A and B

Options

A. `mug`
B. `(g)/(mu)`
C. `(mu)/(g)`
D. `sqrt(mug)`

Explanation

To solve the problem, we need to analyze the forces acting on both bodies A and B and apply Newton's second law. Here’s a step-by-step solution:

### Step 1: Understand the System
- Body A is placed on body B, which is on a smooth horizontal table.
- The coefficient of friction between A and B is `μ`.
- We need to find the acceleration of body B (denoted as \( a_B \)) that will cause body A to slip on body B.

### Step 2: Free Body Diagram for Body B
- For body B, the forces acting on it are:
- A force \( F \) applied in the direction of acceleration.
- A frictional force \( F_f \) acting in the opposite direction due to body A.
- According to Newton's second law, we can write the equation for body B:
\[ F - F_f = m_B a_B \]

### Step 3: Determine the Frictional Force
- The maximum static frictional force \( F_f \) that can act between A and B is given by:
\[ F_f = \mu N_A \]
where \( N_A \) is the normal force on body A.
- Since body A is resting on body B, the normal force \( N_A \) is equal to the weight of body A:
\[ N_A = m_A g \]
- Thus, we can express the frictional force as:
\[ F_f = \mu m_A g \]

### Step 4: Substitute into the Equation for Body B
- Substituting \( F_f \) into the equation for body B gives:
\[ F - \mu m_A g = m_B a_B \]
- Rearranging this equation, we find:
\[ a_B = \frac{F - \mu m_A g}{m_B} \]

### Step 5: Free Body Diagram for Body A
- For body A, the frictional force \( F_f \) is the only force causing it to accelerate. Thus, we can write:
\[ F_f = m_A a_A \]
- Substituting for \( F_f \) gives:
\[ \mu m_A g = m_A a_A \]
- Simplifying this, we find:
\[ a_A = \mu g \]

### Step 6: Condition for Slipping
- For slipping to occur, the acceleration of body B must be greater than the acceleration of body A:
\[ a_B > a_A \]
- Substituting \( a_A \) into this inequality gives:
\[ a_B > \mu g \]

### Step 7: Critical Point of Slipping
- The critical point where slipping just begins is when:
\[ a_B = a_A \]
- Therefore, at the point of slipping:
\[ a_B = \mu g \]

### Final Answer
- The acceleration given to body B that will cause slipping to occur between A and B is:
\[ a_B = \mu g \]