# Find the temperature increase of a rotating disk

• Edward258
In summary, the problem involves a beam of wavelength λ, causing an absorbent disk with mass m, specific heat C, and moment of inertia I to have an angular velocity Ω. The goal is to show that the temperature increases by a certain amount, given by the equation ΔT=(IΩ/mC)(2πc/λ-Ω/2). The solution involves using the equations Q=CmΔT and Q=U2-U1, and justifying the values of U1 and U2 as the rotational kinetic energy and photon energy, respectively. This can be found in a reference on classical mechanics or by using the fact that each circularly polarized photon has an angular momentum of ħ.
Edward258

## Homework Statement

A beam of wavelength λ, in the state of right circular polarization, leads to an absorbent disk.The mass of the disk is m, it's specific heat is C, and its moment of inertia is I .The disk is initially at rest, but after a lapse of time has an angular velocity Ω. Show that the temperature increases in:

ΔT=(IΩ/mC)(2πc/λ-Ω/2)

Q=CmΔT

## The Attempt at a Solution

From the equation;
Q=CmΔT
I can get ΔT
ΔT=Q/Cm
And
Q=U2-U1
So
ΔT=(U2-U1)/Cm
But I don't know how to justify that;
U1=IΩ2/2 ------- Rotational kinetic energy
U2=(IΩ2πc)/λ --------- con E=hc/λ Photon energy

Any good reference on classical mechanics should tell you the relation between angular velocity and rotational kinetic energy. Otherwise, you need to use the fact that each circularly polarized photon has an angular momentum of ħ.

## 1. What is the formula for finding the temperature increase of a rotating disk?

The formula for finding the temperature increase of a rotating disk is ΔT = (ω^2 * R^2 * ρ * c_p) / (4 * k), where ΔT is the temperature increase, ω is the angular velocity, R is the radius of the disk, ρ is the density of the material, c_p is the specific heat capacity, and k is the thermal conductivity.

## 2. How does the rotational speed affect the temperature increase of a rotating disk?

The higher the rotational speed of the disk, the greater the temperature increase will be. This is because the formula for temperature increase includes the square of the angular velocity, meaning that a small increase in rotational speed can lead to a significant increase in temperature.

## 3. What material properties affect the temperature increase of a rotating disk?

The material properties that affect the temperature increase of a rotating disk are density, specific heat capacity, and thermal conductivity. A higher density and specific heat capacity will result in a larger temperature increase, while a higher thermal conductivity will result in a smaller temperature increase.

## 4. Can the temperature increase of a rotating disk be accurately predicted?

Yes, the temperature increase of a rotating disk can be accurately predicted using the formula mentioned in the first question. However, there may be slight variations due to factors such as friction and heat dissipation.

## 5. Is the temperature increase of a rotating disk affected by its size?

Yes, the temperature increase of a rotating disk is affected by its size. The larger the disk, the greater the surface area and volume, which can lead to a larger temperature increase. Additionally, a larger disk may also have a larger thermal mass, which can affect the rate at which it heats up.

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