what will be angular momentum of a rolling disc about top most point and bottom most point

To understand the angular momentum of a rolling disc about its topmost and bottommost points, we need to consider a few key concepts in physics, particularly the definitions of angular momentum and how it applies to rolling objects. Let's break this down step by step.

Angular Momentum Basics

Angular momentum (\(L\)) is a measure of the rotational motion of an object and is defined as the product of the moment of inertia (\(I\)) and the angular velocity (\(ω\)). The formula is given by:

L = I * ω

For a disc, the moment of inertia about its center is given by:

I = \frac{1}{2} m r^2

where \(m\) is the mass of the disc and \(r\) is its radius.

Rolling Motion

When a disc rolls without slipping, it has both translational and rotational motion. The angular velocity of the disc is related to its linear velocity (\(v\)) by the equation:

ω = \frac{v}{r}

This relationship is crucial when calculating angular momentum about different points.

Angular Momentum About the Topmost Point

To find the angular momentum of the disc about its topmost point, we need to consider the distance from the center of mass to this point. The distance is equal to the radius \(r\) of the disc. The angular momentum about the topmost point can be calculated using the parallel axis theorem, which states:

L_{top} = L_{cm} + m * d * v

Here, \(L_{cm}\) is the angular momentum about the center of mass, \(d\) is the distance from the center of mass to the topmost point (which is \(2r\)), and \(v\) is the linear velocity of the center of mass.

Substituting the values, we get:

• L_{cm} = I * ω = \frac{1}{2} m r^2 * \frac{v}{r} = \frac{1}{2} m v r
• L_{top} = \frac{1}{2} m v r + m * (2r) * v = \frac{1}{2} m v r + 2 m v r = \frac{5}{2} m v r

Angular Momentum About the Bottommost Point

Now, let’s calculate the angular momentum about the bottommost point. The distance from the center of mass to the bottommost point is also \(r\). Using the same approach as before:

L_{bottom} = L_{cm} + m * d * v

In this case, \(d\) is \(r\), so:

• L_{bottom} = \frac{1}{2} m v r + m * r * v = \frac{1}{2} m v r + m v r = \frac{3}{2} m v r

Summary of Results

To summarize, the angular momentum of a rolling disc is:

• About the topmost point: L_{top} = \frac{5}{2} m v r
• About the bottommost point: L_{bottom} = \frac{3}{2} m v r

This difference arises from the varying distances from the center of mass to the points about which we are calculating the angular momentum. Understanding these concepts helps in grasping the dynamics of rolling motion and the conservation of angular momentum in various physical systems.

To understand the angular momentum of a rolling disc about its topmost and bottommost points, we need to consider a few key concepts in physics, particularly the definitions of angular momentum and the motion of rolling objects. Let's break this down step by step.

Angular Momentum Basics

Angular momentum (\(L\)) is a measure of the rotational motion of an object. It is defined as the product of the moment of inertia (\(I\)) and the angular velocity (\(\omega\)) of the object. For a disc, the moment of inertia about its center is given by:

• \(I = \frac{1}{2} m r^2\)

where \(m\) is the mass of the disc and \(r\) is its radius. When the disc rolls without slipping, it has both translational and rotational motion.

Rolling Motion

For a disc rolling on a surface, the relationship between translational velocity (\(v\)) and angular velocity is given by:

• \(v = r \omega\)

This means that as the disc rolls, every point on its edge has a velocity that depends on its angular velocity and the radius of the disc.

Angular Momentum About Different Points

Now, let's calculate the angular momentum about the topmost point and the bottommost point of the disc.

1. Angular Momentum About the Bottommost Point

When considering the bottommost point of the disc, which is in contact with the ground, we can use the parallel axis theorem to find the angular momentum. The angular momentum about the bottommost point can be expressed as:

• \(L_{bottom} = I_{cm} \omega + m v d\)

Here, \(d\) is the distance from the center of mass to the bottommost point, which is equal to the radius \(r\). Thus, we have:

• \(L_{bottom} = \frac{1}{2} m r^2 \omega + m v r\)

Substituting \(v = r \omega\) into the equation gives:

• \(L_{bottom} = \frac{1}{2} m r^2 \omega + m (r \omega) r\)
• \(L_{bottom} = \frac{1}{2} m r^2 \omega + m r^2 \omega\)
• \(L_{bottom} = \frac{3}{2} m r^2 \omega\)

2. Angular Momentum About the Topmost Point

For the topmost point, the situation is slightly different. The angular momentum about the topmost point can be calculated similarly:

• \(L_{top} = I_{cm} \omega + m v d\)

In this case, \(d\) is also equal to \(r\), but since the topmost point is above the center of mass, we need to consider the distance as \(2r\) (the radius to the center of mass plus the radius to the topmost point). Therefore:

• \(L_{top} = \frac{1}{2} m r^2 \omega + m v (2r)\)

Substituting \(v = r \omega\) gives us:

• \(L_{top} = \frac{1}{2} m r^2 \omega + m (r \omega)(2r)\)
• \(L_{top} = \frac{1}{2} m r^2 \omega + 2 m r^2 \omega\)
• \(L_{top} = \frac{1}{2} m r^2 \omega + 4 m r^2 \omega\)
• \(L_{top} = \frac{9}{2} m r^2 \omega\)

Summary of Results

In summary, the angular momentum of a rolling disc is different when calculated about the bottommost point compared to the topmost point:

• Angular momentum about the bottommost point: \(L_{bottom} = \frac{3}{2} m r^2 \omega\)
• Angular momentum about the topmost point: \(L_{top} = \frac{9}{2} m r^2 \omega\)

This difference arises due to the varying distances from the center of mass to the points of interest and the contributions from both translational and rotational motion. Understanding these concepts is crucial for analyzing rotational dynamics in various physical systems.