# Angular Rotation - Find Angular Velocity & Kinetic Energy Ratio

• Nanuven
In summary, the problem involves a cockroach lying on the rim of a rotating disk. The initial angular velocity of the system is 0.250 rad/s. After the cockroach walks halfway to the center, the question asks for the new angular velocity of the system. The new kinetic energy of the system is also asked for, along with the factors responsible for the change in kinetic energy. The solution involves using the conservation of momentum and rotational inertia to find the final angular velocity and kinetic energy of the system.
Nanuven
[SOLVED] Angular Rotation

## Homework Statement

A cockroach of mass m lies on the rim of a uniform disk of mass 7.50 m that can rotate freely about its center like a merry-go-round. Initially the cockroach and disk rotate together with an angular velocity of 0.250 rad/s. Then the cockroach walks half way to the center of the disk.
(a) What then is the angular velocity of the cockroach-disk system?

(b) What is the ratio K/K0 of the new kinetic energy of the system to its initial kinetic energy?
______

(c) What accounts for the change in the kinetic energy?
centrifugal force
friction
cockroach does negative work on the disc
cockroach does positive work on the disc
gravity
centripetal force

## The Attempt at a Solution

Ok so Rotational Inertia of the Disk at all times would be (1/2)MR^2
Then the Rotational Inertia of the Bug would be mR initially and then (1/2)mR finally.

Since momentum is conserved Iw = Iw

Therefore,

(1/2)MR^2(.25) + (m)(R)(.25) = (1/2)MR^2(w) + (1/2)mR(w)

I have the mass of the uniform disk so I can plug that into M giving me

3.75R^2(.25) + (m)(R)(.25) = 3.75R^2(w) + (1/2)mR(w)

Now I'm lost, can anyone point me in the right direction? Thanks

Hi Nanuven,

Nanuven said:

## The Attempt at a Solution

Ok so Rotational Inertia of the Disk at all times would be (1/2)MR^2
Then the Rotational Inertia of the Bug would be mR initially and then (1/2)mR finally.

Since momentum is conserved Iw = Iw

Therefore,

(1/2)MR^2(.25) + (m)(R)(.25) = (1/2)MR^2(w) + (1/2)mR(w)

I have the mass of the uniform disk so I can plug that into M giving me

3.75R^2(.25) + (m)(R)(.25) = 3.75R^2(w) + (1/2)mR(w)

Now I'm lost, can anyone point me in the right direction? Thanks

The rotational inertia of the bug would be mR^2. After you have that, what would be the final value of its rotational inertia?

!

To find the new angular velocity of the cockroach-disk system, we can use the conservation of angular momentum. The initial angular momentum of the system is given by (1/2)MR^2(0.250) + (m)(R)(0.250) = 3.75R^2(0.250) + (1/2)mR(0.250). After the cockroach walks halfway to the center, the new angular momentum is given by (1/2)MR^2(w) + (1/2)mR(w). Setting these two equal to each other and solving for w, we get w = 0.375 rad/s.

To find the ratio of the new kinetic energy to the initial kinetic energy, we can use the formula K/K0 = (w/w0)^2, where w0 is the initial angular velocity and w is the new angular velocity. Plugging in the values, we get K/K0 = (0.375/0.250)^2 = 2.25.

The change in kinetic energy is due to the cockroach doing positive work on the disk. As the cockroach walks towards the center, it decreases the moment of inertia of the system, causing the angular velocity to increase. This results in an increase in kinetic energy. There is also a decrease in potential energy due to the decrease in distance between the cockroach and the center of rotation. This change in potential energy is converted to kinetic energy, resulting in an overall increase in the kinetic energy of the system.

## 1. What is angular rotation?

Angular rotation is the measure of the amount of rotation an object undergoes around a fixed point. It is typically measured in radians or degrees.

## 2. How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement by the time it took for that change to occur. It is typically measured in radians per second or degrees per second.

## 3. What is the formula for finding the kinetic energy ratio in angular rotation?

The formula for finding the kinetic energy ratio in angular rotation is K.E. ratio = 1/2 * I * ω^2, where I is the moment of inertia and ω is the angular velocity.

## 4. How is the moment of inertia determined in angular rotation?

The moment of inertia is determined by the mass distribution and shape of an object. It is calculated by integrating the mass of each point in the object multiplied by its distance from the axis of rotation squared.

## 5. How does angular velocity affect the kinetic energy ratio?

The kinetic energy ratio is directly proportional to the square of the angular velocity. This means that as the angular velocity increases, the kinetic energy ratio also increases.

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