Analysis of the Problem:
To determine the time at which the block will start sliding on the plate, we need to consider the forces acting on the block and the frictional force between the block and the plate. The block will start sliding when the frictional force reaches its maximum value, which can be determined using the equation:

Frictional Force:
The frictional force between the block and the plate can be calculated using the equation:

F_friction = μ * F_normal

where μ is the coefficient of friction and F_normal is the normal force acting on the block. Since the block is resting on the plate, the normal force is equal to the weight of the block:

F_normal = m * g

where m is the mass of the block and g is the acceleration due to gravity.

Forces Acting on the Block:
There are two forces acting on the block: the frictional force and the force exerted by the moving plate. The force exerted by the moving plate can be calculated using Newton's second law:

F_plate = m * a

where m is the mass of the block and a is the acceleration of the block.

Equating Forces:
At the point when the block starts sliding, the frictional force will be equal to the force exerted by the moving plate:

F_friction = F_plate

μ * F_normal = m * a

μ * m * g = m * a

Simplifying the Equation:
We can cancel out the mass of the block from both sides of the equation:

μ * g = a

Velocity as a Function of Time:
The velocity of the plate is given by the equation:

v = b * t^2

where b is a constant and t is time.

The acceleration of the block can be calculated by taking the derivative of the velocity equation with respect to time:

a = d(v)/d(t) = 2 * b * t

Substituting Values:
Now we can substitute the value of acceleration into the equation:

μ * g = 2 * b * t

Solving for Time:
To find the time at which the block starts sliding, we can solve the equation for t:

t = (μ * g) / (2 * b)

Conclusion:
The block will start sliding on the plate at a time given by the equation t = (μ * g) / (2 * b). This equation shows that the time at which the block starts sliding depends on the coefficient of friction, the acceleration due to gravity, and the constant b which determines the velocity of the plate.