Angular momentum of a rolling sphere

To find the magnitude of the angular momentum of the sphere, we need to consider the properties of a rolling object and the formula for angular momentum.

Properties of a rolling object:
1. The sphere is rolling without slipping. This means that the linear velocity of the sphere at the point of contact with the surface is equal to the angular velocity multiplied by the radius of the sphere.

Formula for angular momentum:
The angular momentum (L) of an object is given by the equation L = Iω, where I is the moment of inertia and ω is the angular velocity.

The moment of inertia (I) for a solid sphere about its own axis is (2/5)mr^2, where m is the mass of the sphere and r is the radius of the sphere.

The angular velocity (ω) can be related to the linear velocity (v) using the equation ω = v/r, where v is the linear velocity of the sphere and r is the radius of the sphere.

Therefore, the angular momentum can be calculated as follows:

L = Iω
= (2/5)mr^2 * (v/r)
= (2/5)mvr

Magnitude of the angular momentum:
Since we are interested in finding the magnitude of the angular momentum with respect to a point along the path of the sphere on the surface, we need to consider the distance between the chosen point and the center of mass of the sphere.

Let's assume the chosen point is a distance d from the center of mass of the sphere.

The distance between the chosen point and the center of mass of the sphere can be calculated as follows:

d = r - (a/2)

Substituting this value into the equation for the angular momentum, we get:

L = (2/5)mvr
= (2/5)m(r - (a/2))r
= (2/5)m(r^2 - (a/2)r)

Simplifying further, we get:

L = (2/5)m(r^2 - (a/2)r)
= (2/5)m(r^2 - (a/2)(2r/2))
= (2/5)m(r^2 - ar + ar/2)
= (2/5)m(r^2 - ar/2)
= (2/5)m(r(r - a/2))

Since we are only interested in the magnitude of the angular momentum, we can ignore the negative sign and obtain:

|L| = (2/5)m|r(r - a/2)|

Given that the mass of the sphere is 2m and the radius of the sphere is a/2, we can substitute these values into the equation:

|L| = (2/5)(2m)|r(r - (a/2))|
= (4/5)m|r(r - (a/2))|

Simplifying further, we get:

|L| = (4/5)m|r^2 - (ar/2)|

Now, let's analyze the expression inside the absolute value:

r^2 - (ar/2) = r(r - (a/2))

Since r is positive, the expression inside the absolute value will be positive if (r - (a/2)) > 0, and negative if (r - (a/2