Circular Eddy Current Brakes

Classical Mechanics Level 3

Eddy current brakes are brakes that use eddy currents generated by electromagnetic induction to slow down an object. Eddy current brakes are widely used in high speed trains and roller coasters. Since there is no physical contact, eddy current brakes are much quieter and last much longer before wearing down as compared to frictional brakes.

The diagram above shows the working of a circular eddy current brake. To slow down the rotating disk, the electromagnet is switched on. This causes circular loops of currents, called eddy currents, to be formed in the disk. Since the disk has resistance, and current flows through it, heat is generated and the kinetic energy of the disk decreases.

Initially, the disk D has rotates with angular velocity ω \omega . After the brakes are applied for some time, the angular velocity of the disk becomes ω / 10 \omega /10 .

Find K final K initial \dfrac{K_{\text{final}}}{K_{\text{initial}}} , the ratio of final rotational kinetic energy to the initial rotational kinetic energy of the disk.

Image Credit: Chetvorno Wikimedia Commons


The answer is 0.01.

Pranshu Gaba by Pranshu Gaba , 22, India

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1 solution

Pranshu Gaba
Feb 20, 2016

The rotational kinetic energy of a rotating object is given by 1 2 I ω 2 \frac{1}{2} I \omega ^{2} , where I I is the moment of inertia of the object about the axis of rotation. The moment of inertia of the disk does not change, therefore the kinetic energy of the disk is directly proportional to the square of angular velocity.

K ∝ ω 2 K \propto \omega^{2}

∴ K final K initial = ( ω final ω initial ) 2 = ( ω 10 ω ) 2 = ( 1 10 ) 2 = 1 100 = 0.01 □ \begin{aligned} \therefore \quad \dfrac{K_{\text{final}}}{K_{\text{initial}}} & = \left( \dfrac{\omega_{\text{final}}}{\omega_{\text{initial}}} \right)^{2} \\ & = \left( \frac{\frac{\omega}{10}}{\omega}\right)^{2} \\ & = \left( \frac{1}{10}\right)^{2} \\ & = \frac{1}{100} \\ & = \boxed{0.01} & _\square \end{aligned}

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The problem was nice, thanks man!!

Fahim Abid - 5 years, 3 months ago

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