To solve the problem of a bead sliding on a rod that is rotating with constant angular acceleration, we need to analyze the forces acting on the bead and how they change over time. Let's break this down step by step.

Understanding the Forces at Play

When the rod rotates about point A, the bead experiences a centripetal force due to the circular motion. This force is provided by the friction between the bead and the rod. The key factors to consider are:

• Angular Acceleration (α): This is the rate at which the angular velocity of the rod increases.
• Distance (L): The initial distance of the bead from point A.
• Frictional Force (f): This is what keeps the bead from slipping. It is proportional to the normal force and the coefficient of friction (μ).

Calculating the Forces

The centripetal force required to keep the bead moving in a circular path is given by:

F_c = m * ω² * r

Where:

• m: Mass of the bead
• ω: Angular velocity of the rod
• r: Distance from point A to the bead (which is L initially)

As the rod accelerates, the angular velocity (ω) increases over time according to:

ω = α * t

Frictional Force and Its Role

The maximum static frictional force that can act on the bead before it starts slipping is:

f_max = μ * N

Where N is the normal force. In this case, since the rod is horizontal and gravity is neglected, the normal force is simply the weight of the bead, which we can consider as zero for this scenario. Thus, the frictional force is:

f_max = μ * m * g

However, since gravity is neglected, we can focus on the relationship between the forces without needing to calculate the weight explicitly.

Setting Up the Equation

The bead will start to slip when the required centripetal force exceeds the maximum static frictional force:

m * ω² * L > μ * m * g

Since we are neglecting gravity, we can simplify this to:

ω² * L > μ * 0

This indicates that the bead will start slipping when the angular velocity becomes sufficiently large, which will depend on the angular acceleration and time.

Finding the Time Until Slipping Occurs

Substituting ω into the inequality gives us:

(α * t)² * L > μ * 0

As time progresses, the angular velocity increases, and thus the bead will eventually slip. The exact time can be derived from the relationship:

t = √(μ * L / α)

Final Thoughts

In conclusion, the time after which the bead starts slipping can be calculated using the derived formula. The answer will depend on the values of the coefficient of friction, the distance L, and the angular acceleration α. If we were to evaluate the possible answers provided, we would need specific values to determine which one is correct. However, the general approach outlined here provides a clear method to analyze the situation.