# How Does Adding Masses Affect the Angular Velocity of a Rotating Cylinder?

• brotherbobby
In summary, a cylinder with two small masses glued to its axis of rotation will have the same angular velocity as a bare cylinder, assuming the masses have negligible radius and the moment of inertia remains constant. However, the problem is incomplete as it does not specify what remains constant.
brotherbobby
Problem :

A cylinder of mass ##M## and radius ##R## rotates with an angular velocity ##\omega_1## about an axis passing through its centre of symmetry. Two small masses each of mass ##m## (small in comparison to the radius of the cylinder) are glued to either of the two circular faces of the cylinder right on its axis of rotation. The cylinder-mass system now rotates with an angular velocity ##\omega_2##.
Is ##\omega_2## greater than, less than or equal to ##\omega_1##?

Relevant equations :

The moment of inertia of the solid cyliner is ##\frac{1}{2} MR^2##. My attempt :

The two small masses do not increase the moment of inertia of the cylinder, being point masses. Hence ##\omega_2## = ##\omega_1##

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Your problem is ill defined. What else is given in the problem? Obviously you can rotate either setup with whatever angular velocity you want, it just depends on how much angular momentum you put into the system. You need to specify what other assumptions should be taken, such as same angular momentum.

The exercise is incomplete: they don't tell you what remains constant. However small, the extra masses definitely do not decrease the moment of inertia and do contribute to it. There is a good argument to claim ##\omega_2 < \omega_1##.

Will the moment of inertia of the combined system be more than that of the bare cylinder?

Check the definition

If the two masses lie along the axis of the cylinder, then they won't contribute to the moment of inertia (about that axis).

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Yes, but these are point masses. They have no radius of their own. They are put along the axis of the cylinder.

point masses don't exist and a small mass is not a point mass

brotherbobby said:
Yes, but these are point masses. They have no radius of their own. They are put along the axis of the cylinder.
Correct. They have a radius, but it is considered negligible compared to that of the cylinder.
brotherbobby said:
small in comparison to the radius of the cylinder
In this case, negligible means that ##MR^2 \gg mr^2##, so it is actually an issue of both mass and radius ...

True, if you insist that the masses have a radius, they will contribute to I. But the question is about point masses.

Then you won't need much glue

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It would be fair to call this entire argumentation ...(drumrolls please)... pointless ...

Anyway, I think it is clear from the formulation of the problem that the additional masses should be considered to give a negligible contribution to the moment of inertia. What is missing, as already stated, is the assumption about what is supposed to be considered constant.

## 1. What is the concept of rotation of a solid cylinder?

The rotation of a solid cylinder refers to the circular movement of the cylinder around its central axis. This results in all points on the cylinder moving in a circular path, with the same angular velocity.

## 2. What is the difference between rotational and translational motion?

Rotational motion involves movement around a fixed point or axis, while translational motion involves movement in a straight line from one point to another. In the case of a solid cylinder, rotational motion occurs around its central axis, while translational motion would involve the cylinder moving from one location to another in a straight line.

## 3. How is the rotational motion of a solid cylinder measured?

The rotational motion of a solid cylinder is typically measured in terms of angular displacement, which is the angle through which the cylinder has rotated from its original position. It can also be measured in terms of angular velocity, which is the rate of change of angular displacement over time.

## 4. What factors affect the rotational motion of a solid cylinder?

The rotational motion of a solid cylinder can be affected by factors such as the mass and shape of the cylinder, the applied torque or force, and any external frictional forces. These factors can influence the angular velocity and acceleration of the cylinder.

## 5. How is the rotational motion of a solid cylinder applied in real-world situations?

The rotational motion of a solid cylinder has various practical applications, such as in machinery and vehicles. For example, the rotation of a car's wheels allows it to move forward, and the rotation of gears in a machine can transfer and amplify torque. It is also used in sports, such as spinning a basketball on a finger or throwing a discus in track and field.

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