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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 $\alpha$ of the cart that will prevent the block from falling satisfies
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A
$\alpha < \frac{g}{\mu}$
B
$\alpha > \frac{mg}{\mu}$
C
$\alpha > \frac{g}{\mu m}$
D
$\alpha \geq \frac{g}{\mu}$
Detailed Solution
The block is pressed against the vertical face of the cart. When the cart accelerates with $\alpha$, the cart pushes the block with a normal force N that gives the block the same acceleration: $N = m\alpha$
The weight mg acts downward and can be balanced only by static friction acting upward along the face.
Maximum static friction: $f_{max} = \mu N = \mu m\alpha$
For the block not to fall: $f_{max} \geq mg$
$\mu m\alpha \geq mg$
$\alpha \geq \frac{g}{\mu}$
The weight mg acts downward and can be balanced only by static friction acting upward along the face.
Maximum static friction: $f_{max} = \mu N = \mu m\alpha$
For the block not to fall: $f_{max} \geq mg$
$\mu m\alpha \geq mg$
$\alpha \geq \frac{g}{\mu}$