# Conservation of angular momentum and rotational kinetic energy

• Saptarshi Sarkar
In summary, the conversation discusses two attempts to solve a problem by conserving rotational kinetic energy and angular momentum. The first attempt resulted in a incorrect answer, while the second attempt using angular momentum resulted in the correct answer. It is noted that energy is being used up in the process, specifically calculated as 1/10 of the moment of inertia multiplied by the angular speed squared. The conversation concludes by emphasizing the importance of not assuming energy conservation without sufficient reason and trusting the laws of linear and angular momentum when applicable. The problem at hand is an example of coalescence, where two bodies interact and reach the same velocity. This interaction could be due to friction or other means, such as two toothed wheels engaging.

#### Saptarshi Sarkar

Homework Statement
A thin uniform disc of mass M and radius R is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity ##\omega##. Another disk of same dimensions but mass M/4 is placed on the first disk coaxially. What is the angular speed of the system is now
Relevant Equations
##K.E. = \frac 12I\omega^2##
##L= I\omega##
I first tried to get the solution by conserving the rotational kinetic energy and got ##\omega'=\frac2{\sqrt5} \omega##.

But, it was not the correct answer. Next I tried by conserving the angular momentum and got ##\omega'=\frac 45 \omega##, which is the correct answer.

Why is the rotational kinetic energy not conserved here? Where is the energy being used up?

Energy used up calculated as
$$\frac{1}{10}I\omega^2$$
is used up when two disks was getting the same angular speed in most cases by friction between the surfaces.

Delta2 and Saptarshi Sarkar
The moral here is never to assume work is conserved without a good reason to do so.
It is generally clearer whether linear or angular momentum are conserved. If those laws suffice to find the answer, trust them.
The present problem is an example of coalescence. As in a totally inelastic collision, two bodies meet and arrive at the same velocity through some interaction. As @anuttarasammyak observes, that interaction could be by friction. Equally, it could have been two toothed wheels, the teeth engaging on contact.

Saptarshi Sarkar and Delta2

## What is conservation of angular momentum?

Conservation of angular momentum is a fundamental principle in physics that states that the total angular momentum of a system remains constant as long as there are no external torques acting on it. This means that the angular momentum of a system cannot be created or destroyed, only transferred between different parts of the system.

## How is angular momentum related to rotational kinetic energy?

Angular momentum and rotational kinetic energy are closely related because they both describe the motion of rotating objects. Angular momentum is a measure of an object's tendency to continue rotating, while rotational kinetic energy is a measure of the energy required to keep an object rotating at a constant speed.

## What are some real-world examples of conservation of angular momentum?

Some common examples of conservation of angular momentum in everyday life include spinning tops, figure skaters performing spins, and the rotation of the Earth around its axis.

## Why is conservation of angular momentum important?

Conservation of angular momentum is important because it is a fundamental law of nature that helps us understand and predict the behavior of rotating objects. It is also a key principle in many fields of science, including physics, astronomy, and engineering.

## Can angular momentum be changed?

Angular momentum can be changed by applying an external torque to a rotating object. This can cause the object to speed up or slow down, or change the direction of its rotation. However, the total angular momentum of a system will always remain constant.

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