Question

A block of mass \( m \) is in contact with the cart C as shown in the figure. The coefficient of static friction between the block and the cart is \( \mu \). The acceleration \( a \) of the cart that will prevent the block from falling satisfies:

• A. \( a > \frac{mg}{\mu} \)
• B. \( a > \frac{g}{\mu} \)
• C. \( a \geq \frac{g}{\mu} \)
• D. \( a < \frac{g}{\mu} \)

Hint:

For blocks on carts, static friction is what prevents the block from falling. Ensure the frictional force is enough to match the acceleration of the cart.

Answer 1

The Correct Option is C

To prevent the block from falling, the frictional force must be greater than or equal to the weight of the block.
The maximum static frictional force is given by: \[ f_{\text{max}} = \mu \cdot N = \mu \cdot m \cdot g \] where \( N \) is the normal force (equal to \( m \cdot g \)). The block will move with the cart if the frictional force is sufficient to keep the block in place.
The maximum force required to prevent the block from falling is the force of acceleration, \( F = m \cdot a \).
For the block to remain in place, we need: \[ m \cdot a \leq \mu \cdot m \cdot g \] Simplifying the equation, we get: \[ a \leq \mu \cdot g \] Therefore, the condition for the block to not fall is \( a \geq \frac{g}{\mu} \).
Hence, the correct answer is (c).