To determine when the block will start sliding on the plate, we need to analyze the forces acting on the block as the plate accelerates. The key here is to understand the relationship between the frictional force and the forces due to the plate's motion.

Understanding the Forces Involved

When the plate moves, it exerts a force on the block due to its acceleration. The block will remain stationary relative to the plate as long as the static friction can counteract this force. The static friction force can be calculated using the formula:

• F_friction = μ_s * N

Where:

• μ_s is the coefficient of static friction (1/2 in this case).
• N is the normal force, which equals the weight of the block if we assume it is resting on a horizontal surface (N = mg).

Calculating the Normal Force

Assuming the mass of the block is m, the normal force can be expressed as:

• N = mg

Thus, the maximum static friction force becomes:

• F_friction = (1/2) * mg

Acceleration of the Plate

The plate's velocity is given by v = 2t^2. To find the acceleration of the plate, we differentiate the velocity with respect to time:

• a = dv/dt = d(2t^2)/dt = 4t

Force on the Block

The force exerted on the block due to the plate's acceleration is:

• F_block = ma = m(4t) = 4mt

Setting Up the Inequality

The block will start sliding when the force exerted by the plate exceeds the maximum static friction force:

• 4mt > (1/2)mg

We can simplify this by canceling out m (assuming it is not zero):

• 4t > 1/2

Solving for Time

Now, we can solve for t:

• t > 1/8

This means the block will start sliding when time t is greater than 1/8 seconds. However, we need to check if this time aligns with the provided options.

Evaluating the Options

The options given are 5 sec, 3 sec, 4 sec, and "depends on the mass of the block." Since we found that the block starts sliding at approximately 0.125 seconds, none of the options directly match this time. Therefore, the correct answer is that the sliding time does not depend on the mass of the block, but rather on the acceleration of the plate and the coefficient of friction.

In conclusion, the block will start sliding almost immediately after the plate begins to accelerate, specifically after 1/8 seconds, which is much less than any of the provided options. Thus, the answer is that it does not depend on the mass of the block.