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, you need to consider the rotational motion of the stainless steel annular disk. You can use the principle of conservation of angular momentum to relate the fall distance of the hanging mass to the rotation of the disk.

First, let's calculate the initial angular momentum of the system. Since the hanging mass starts from rest, the initial angular momentum is zero.

Next, we need to find the final angular momentum of the system after the hanging mass has fallen for 4.4 seconds. To do this, we can use the principle of conservation of angular momentum:

Initial angular momentum = Final angular momentum

Since the initial angular momentum is zero, we have:

0 = I * ω

where I is the moment of inertia of the disk and ω is the angular speed of the disk after the hanging mass has fallen.

The moment of inertia of a solid disk about its central axis is given by:

I = (1/2) * m * R²

where m is the mass of the disk and R is the radius of the disk.

In this case, the mass of the disk is given as 1349 grams. We should convert it to kilograms by dividing by 1000:

m = 1349 g / 1000 = 1.349 kg

The outer radius of the disk is 68 mm, while the inner radius is 7.4 mm. We can use these values to calculate the moment of inertia:

I = (1/2) * m * (Rₒ² - Rᵢ²)

where Rₒ is the outer radius and Rᵢ is the inner radius.

I = (1/2) * 1.349 kg * ((0.068 m)² - (0.0074 m)²)

Now that we have the moment of inertia, we can solve the conservation of angular momentum equation for ω:

ω = 0 / I = 0

Therefore, the angular speed of the disk after the hanging mass has fallen for 4.4 seconds is zero. This means that the disk is not rotating.

Since the disk is not rotating, the distance that the hanging mass has fallen is equal to the length of the string, which is the radius of the cylinder. In this case, the radius of the cylinder is given as 12.5 mm. We need to convert it to meters:

Radius of the cylinder = 12.5 mm = 0.0125 m

Therefore, the distance that the hanging mass has fallen after 4.4 seconds is 0.0125 meters or 12.5 mm.

In conclusion, the hanging mass has fallen a distance of 12.5 mm by the time of 4.4 seconds.