To solve the problem of a solid sphere sliding on a plank and transitioning to pure rolling, we need to analyze the forces and motions involved. Let's break this down step by step.

Understanding the Initial Conditions

We have a solid sphere of mass m and radius R placed on a plank, which also has mass m. The sphere is given an initial horizontal velocity v and starts sliding on the plank. The coefficient of friction between the sphere and the plank is denoted as μ.

Key Concepts

• Friction Force: The friction between the sphere and the plank will act to slow down the sphere's sliding motion and eventually cause it to roll.
• Acceleration: The friction force will also cause the plank to accelerate in the opposite direction.
• Pure Rolling Condition: The sphere will transition to pure rolling when the velocity of its center of mass equals the product of its angular velocity and radius.

Calculating the Forces

The frictional force f acting on the sphere can be expressed as:

f = μN, where N is the normal force. Since the sphere is on a horizontal plank, N = mg.

Thus, the frictional force becomes:

f = μmg.

Equations of Motion

As the sphere slides, it experiences a deceleration due to friction:

a_s = -f/m = -μg.

Simultaneously, the plank experiences an acceleration:

a_p = f/m = μg.

Setting Up the Equations

Let’s denote the time taken to reach pure rolling as t. The initial velocity of the sphere is v, and its final velocity when it starts rolling will be v_f.

Using the equations of motion for the sphere:

v_f = v + a_s t = v - μgt.

For the sphere to start rolling without slipping, the condition is:

v_f = ωR, where ω is the angular velocity. The angular acceleration α due to friction is:

α = f/I = μmgR/(2/5mR^2) = (5/2)(μg/R), where I is the moment of inertia of the sphere.

The relationship between linear and angular velocity gives us:

ω = αt = (5/2)(μg/R)t.

Equating the Velocities

At the point of pure rolling, we have:

v - μgt = (5/2)(μg/R)t.

Rearranging this equation leads to:

v = μgt + (5/2)(μg/R)t.

Factoring out t gives:

v = μg t \left(1 + \frac{5}{2R}\right).

Solving for Time

Now, we can solve for t:

t = \frac{v}{μg \left(1 + \frac{5}{2R}\right)}.

Final Expression

Thus, the time taken by the sphere to start pure rolling on the plank is:

t = \frac{v}{μg \left(1 + \frac{5}{2R}\right)}.

This equation gives us a clear understanding of how the initial velocity, the coefficient of friction, and the radius of the sphere influence the time taken to achieve pure rolling. Each component plays a crucial role in the dynamics of the system.