Problem Analysis:
- We have two spheres joined by a light rod of length 2a.
- The center of the spheres, P and Q, are at a distance 4a apart.
- The spheres are initially at rest on a smooth horizontal table.
- An impulse is given to sphere P along a line through P at right angles to PQ.
- The magnitude of the impulse is given as (10k 8) kg m/s.
- The initial speed of sphere Q is u.
- Given that the coefficient of friction between the spheres and the table is mu=4.

Solution:
1. Impulse and Momentum:
- Impulse is defined as the change in momentum of an object.
- Momentum is the product of mass and velocity.
- Impulse is given by the equation: Impulse = Change in momentum = m * (final velocity - initial velocity)

2. Applying Impulse-Momentum Principle:
- The impulse given to P causes a change in its momentum.
- The momentum gained by P is equal to the momentum lost by Q, as there is no external force acting on the system.
- Let the mass of each sphere be m.

3. Calculating the Initial Velocity of P:
- The initial velocity of P is zero as the spheres are at rest on the table.
- The final velocity of P can be calculated using the impulse equation: Impulse = m * (final velocity - initial velocity)
- Given that the magnitude of the impulse is (10k 8) kg m/s, we can write the equation as: (10k 8) = m * final velocity

4. Calculating the Final Velocity of Q:
- The final velocity of Q can be calculated using the impulse equation: Impulse = m * (final velocity - initial velocity)
- Given that the magnitude of the impulse is (10k 8) kg m/s and the initial velocity of Q is u, we can write the equation as: (10k 8) = m * (0 - u)
- Solving for u, we get: u = -(10k 8) / m

5. Calculating the Change in Momentum:
- The change in momentum of P can be calculated using the equation: Change in momentum = m * (final velocity - initial velocity)
- The change in momentum of Q can be calculated using the equation: Change in momentum = m * (final velocity - initial velocity)
- As the spheres are initially at rest, the initial velocities of both P and Q are zero.
- The final velocity of P is given by: final velocity = (10k 8) / m
- The final velocity of Q is given by: final velocity = u = -(10k 8) / m

6. Applying Coefficient of Friction:
- The coefficient of friction between the spheres and the table is mu=4.
- The frictional force acting on each sphere is given by: Frictional force = mu * Normal force
- The normal force is the force exerted by the table on each sphere and is equal to the weight of the sphere.
- The frictional force opposes the motion of the spheres.

7. Finding the Value of k:
- To find the value of k, we need to consider the motion of the spheres after the impulse is given.
- We can analyze the motion using the equations of motion and the concept of conservation