Consider a stainless steel annular disk with an outer radius 68mm and inner radius7.4mm. The mass of the disk is 1349 grams.

The stainless steel annular disk is allowed to rotate on a frictionless table with the rotation axis at its center. The disk has a small cylinder rigidly mounted at the top concentrically. The cylinder's radius is 12.5mm, and the mass of the cylinder is negligible. A string is wrapped around the cylinder, and a hanging mass of19.3 g is tied at the other end of the string. When the mass falls under gravity, it causes the stainless steel annular disk to rotate. Ignoring the string's mass, and assuming that the string's motion is frictionless.

How much distance has the hanging mass been falling by this time? (Considering time is 4.4 seconds after the hanging mass is released from rest)

I am having trouble finding the formula to find the distance of the hanging mass. Am I supposed to use the angular speed or speed of the falling mass at that time?

To find the distance that the hanging mass has fallen, we need to analyze the rotational dynamics of the system. Let's break down the steps to find the solution:

Step 1: Find the moment of inertia of the annular disk.
The moment of inertia for a solid disk rotating about its axis (perpendicular to the plane of the disk) is given by the formula:

I_disk = (1/2) * m_disk * r^2

where m_disk is the mass of the disk and r is the radius of the disk.

In this case, we have an annular disk rather than a solid disk, so we need to modify the formula. The moment of inertia of an annular disk is given by the formula:

I_annular = (1/2) * m_annular * (r_outer^2 + r_inner^2)

where m_annular is the mass of the annular disk, r_outer is the outer radius of the disk, and r_inner is the inner radius of the disk.

Step 2: Find the moment of inertia of the system with the added cylinder.
Since the cylinder is rigidly mounted on top of the annular disk and they both rotate together, we can simply add the moment of inertia of the cylinder to the moment of inertia of the annular disk. Since the mass of the cylinder is negligible, its moment of inertia is also negligible.

I_system = I_annular + I_cylinder

Step 3: Obtain the angular acceleration of the system.
The angular acceleration of the system can be calculated using the torque equation:

τ = I_system * α

where τ is the torque applied to the system and α is the angular acceleration.

In this case, the torque is created by the force of gravity acting on the hanging mass. The torque can be calculated as:

τ = r_cylinder * F_gravity

where r_cylinder is the radius of the cylinder and F_gravity is the force of gravity acting on the hanging mass.

Step 4: Use the angular acceleration to find the angular displacement.
The relationship between angular acceleration (α), initial angular velocity (ω_0), angular displacement (θ), and time (t) is given by the equation:

θ = ω_0 * t + (1/2) * α * t^2

In this case, the initial angular velocity is assumed to be zero since the hanging mass was released from rest.

Step 5: Convert the angular displacement to linear distance.
The linear distance that the hanging mass has fallen can be calculated using the formula:

distance = θ * r_cylinder

where r_cylinder is the radius of the cylinder.

By following these steps, you should be able to find the distance that the hanging mass has fallen after 4.4 seconds.