A block of mass M is in contact with a cart of mass M. The coefficient of static friction between block and cart is `mu` . To prevent the block from sliding on cart, acceleration a should be

Options

A. `agtfrac{mg}{(m+M)mu}`
B. `agefrac{g}{mu}`
C. `altfrac{g}{mu}`
D. `altfrac{mg}{Mmu}`

Answer

To solve the problem, we need to determine the condition under which the block of mass \( M \) does not slide on the cart of mass \( M \) when the cart is accelerating with an acceleration \( a \).

### Step-by-Step Solution:

1. **Identify Forces Acting on the Block**:
- The block experiences a gravitational force downward, which is \( Mg \).
- It also experiences a normal force \( N \) exerted by the cart, acting upward.
- When the cart accelerates, a pseudo force acts on the block in the opposite direction of the acceleration of the cart. This pseudo force is given by \( F_{\text{pseudo}} = Ma \).

2. **Static Friction Force**:
- The static friction force \( F_s \) between the block and the cart must be sufficient to prevent the block from sliding. The maximum static friction force can be expressed as:
\[ F_s \leq \mu N \]
- Here, \( \mu \) is the coefficient of static friction, and \( N \) is the normal force. Since the block is not moving vertically, the normal force \( N \) equals the weight of the block:
\[ N = Mg \]
- Therefore, the maximum static friction force becomes:
\[ F_s \leq \mu Mg \]

3. **Setting Up the Equation**:
- For the block to not slide, the static friction force must be equal to the pseudo force acting on the block:
\[ F_s = Ma \]
- Substituting the expression for maximum static friction, we get:
\[ Ma \leq \mu Mg \]

4. **Solving for Acceleration \( a \)**:
- Dividing both sides of the inequality by \( M \) (assuming \( M \neq 0 \)):
\[ a \leq \mu g \]
- This implies that to prevent the block from sliding, the acceleration \( a \) of the cart must satisfy:
\[ a \leq \mu g \]

5. **Conclusion**:
- The condition for the block not to slide on the cart is:
\[ a \leq \mu g \]

### Final Answer:
To prevent the block from sliding on the cart, the acceleration \( a \) should be:
\[ a \leq \mu g \]