Angular Momentum and its Significance in the Question:

In this question, the concept of angular momentum is used to determine the minimum initial velocity required for the particle to just graze the spherical shell. Angular momentum is a conserved quantity in the absence of external torques, meaning it remains constant throughout the motion of a system.

1. Angular momentum definition:
Angular momentum (L) is defined as the cross product of the particle's linear momentum (p) and its position vector (r) relative to a chosen axis of rotation.
L = r x p

2. Conservation of angular momentum:
If no external torques act on a system, the total angular momentum of the system remains constant. Mathematically, this can be expressed as:
L_initial = L_final

3. Application in the given question:
The particle is projected towards the fixed spherical shell, and we want to find the minimum initial velocity required for it to just graze the shell. When the particle just grazes the shell, its distance from the center of the shell is equal to the radius of the shell (R).

4. Angular momentum conservation during motion:
As the particle moves towards the shell, its distance from the center decreases, resulting in a decrease in the magnitude of its position vector (r). To conserve angular momentum, the particle's linear momentum (p) must increase to compensate for the decrease in r.

5. Angular momentum at initial and final states:
At the initial state, before the particle reaches the shell, its angular momentum is given by:
L_initial = r_initial x p_initial

At the final state, when the particle just grazes the shell, its angular momentum is given by:
L_final = R x p_final

6. Using conservation of angular momentum:
Since angular momentum is conserved, we can equate the initial and final angular momenta:
L_initial = L_final
r_initial x p_initial = R x p_final

7. Solving for the minimum initial velocity:
To find the minimum initial velocity required for the particle to just graze the shell, we need to determine the final linear momentum (p_final) at the moment of grazing.

8. Conclusion:
By using the concept of conservation of angular momentum, we can establish an equation relating the initial and final states of the particle. Solving this equation allows us to find the minimum initial velocity (v0) required for the particle to just graze the non-conducting fixed spherical shell.