In the above question the disc is rolling without slipping.I am not getting angular speed of disc with respect to z axis .This is how i did it:V(centre of mass)=wa.Since it is pure rolling.angular velocity about Z axis (Ω).Ω×l×cosθ=wa.i am getting Ω=w/4.8 but answer is w/5.

It sounds like you're working through a problem involving the rolling motion of a disc, and you're trying to relate the angular velocity of the disc to its linear velocity. Let's break down the concepts involved and clarify how to arrive at the correct relationship.

Understanding Rolling Motion

When a disc rolls without slipping, there is a direct relationship between its linear velocity and its angular velocity. The key principle here is that the point of contact between the disc and the surface is momentarily at rest. This means that the linear velocity of the center of mass (V) is related to the angular velocity (ω) by the equation:

• V = ω * r

Here, r is the radius of the disc. This relationship holds true for pure rolling motion.

Defining Angular Velocity

In your case, you mentioned that you are trying to find the angular velocity about the z-axis (Ω). When you refer to the angular velocity of the disc, it's important to clarify whether you are looking at the angular velocity of the entire disc or just a specific point on it. The angular velocity about the z-axis is typically denoted as Ω, which can be related to the angular velocity of the disc (ω) depending on the geometry of the system.

Applying the Geometry

From your equation, you have:

• Ω × l × cos(θ) = wa

Here, l represents a distance related to the system, and θ is the angle of inclination. To find the correct relationship, we need to ensure that the distances and angles are defined correctly. If you are getting Ω = ω/4.8 but the answer is ω/5, it suggests that there might be a miscalculation in how you are relating the distances or angles involved.

Steps to Resolve the Calculation

1. **Identify the Radius**: Ensure that you are using the correct radius (r) of the disc in your calculations. If the disc has a radius of r, then the linear velocity at the center of mass is V = ω * r.

2. **Consider the Geometry**: If the distance l is not simply the radius, you need to clarify what l represents in your scenario. For example, if l is the distance from the center of mass to a point on the edge of the disc, it should be equal to r.

3. **Revisit the Angles**: Make sure that the angle θ is correctly defined in relation to the motion of the disc. If θ is the angle of inclination, it may affect how you relate the linear and angular velocities.

4. **Recalculate**: With the correct definitions and relationships, recalculate Ω. If you find that Ω = ω/5, it may be due to a specific geometric factor or a trigonometric relationship that you initially overlooked.

Example Calculation

For instance, if the radius of the disc is 1 meter and the angular velocity ω is 10 rad/s, then the linear velocity V at the center of mass would be:

• V = ω * r = 10 rad/s * 1 m = 10 m/s

If you were to find the angular velocity about the z-axis, you would need to ensure that all distances and angles are correctly accounted for in your equations. If your calculations yield a different factor, it may indicate a need to reassess the geometry or the relationships you've established.

In summary, double-check your definitions and calculations, particularly focusing on the relationships between the distances and angles involved in the rolling motion. This should help clarify why you are seeing a discrepancy in your results.