A uniform disc of 2.98 kg is mounted so that it can rotate in the horizontal plane around a frictionless axis through its center of mass. The angular velocity of the disc at the start is 140 rpm. A hollow cylinder with thin walls and the same radius as the disc is released from rest just above the turntable, so that the axis of rotation also passes through its center of mass. Friction forces ensure that the two objects rotate at the same angular speed of 77.8 rpm after a short time. Determine the mass of the hollow cylinder.

angular momentum is conserved

the momentum lost by the disc is transferred to the cylinder

To determine the mass of the hollow cylinder, we will use the principle of conservation of angular momentum.

The angular momentum of an object is given by the product of its moment of inertia and its angular velocity. Mathematically, this can be represented as L = I * ω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

At the start, the disc has an initial angular velocity of 140 rpm, which we can convert to radians per second.

1 revolution = 2π radians
1 minute = 60 seconds

Thus, the initial angular velocity of the disc can be calculated as follows:

Initial angular velocity = 140 rpm * (2π radians/1 revolution) * (1 revolution/1 minute) * (1 minute/60 seconds)

Simplifying the above expression, we get:

Initial angular velocity = (140 * 2π) / 60 radians/second

Next, we need to determine the moment of inertia of the disc. The moment of inertia (I) for a uniform disc rotating about an axis through its center of mass is given by the equation:

I = (1/2) * m * r^2

where m is the mass and r is the radius of the disc.

Since the mass of the disc is given as 2.98 kg, and the radius is not provided, we will need to solve the problem using variables. Let's assume the radius of the disc is R.

Using the equation for the moment of inertia, we can write:

I_disc = (1/2) * m * R^2

The moment of inertia of the hollow cylinder can also be expressed as:

I_cylinder = m_cylinder * R^2

where m_cylinder is the mass of the hollow cylinder we want to find.

Now, we can use the conservation of angular momentum to relate the initial and final states of the system:

I_disc * initial angular velocity = I_cylinder * final angular velocity

Substituting the expressions for the moments of inertia and angular velocities, we get:

[(1/2) * m * R^2] * [(140 * 2π) / 60 radians/second] = (m_cylinder * R^2) * [(77.8 * 2π) / 60 radians/second]

Simplifying the equation, we find:

(m * π / 60) * (140 * R^2) = (m_cylinder * π / 60) * (77.8 * R^2)

Canceling out the common terms, we get:

140 * R^2 = 77.8 * R^2

Rearranging the equation, we find:

140 = 77.8

Since this equation is not true, it means that our assumption for the radius of the disc (R) is incorrect, and we need to find another way to determine the mass of the hollow cylinder.

Please provide additional information or equations if available so that we can continue solving the problem.