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Question
A circular disc made of iron is rotated about its axis at a constant velocity ω. Calculate the percentage change in the linear speed of a particle of the rim as the disc is slowly heated from 20°C to 50°C, keeping the angular velocity constant. Coefficient of linear expansion of iron = 1.2 × 10–5 °C–1.
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Solution
Let initial radius of the circular disc at 20 ₒC = r20
Let final radius of the circular disc at 50 ₒC = r50
Coefficient of linear expansion of iron, α = 1.2 × 10–5 °C–1.
change in temperature,ΔT = 30°C
Let R' and R be the radius of the paricle at 50 ₒC and 20 ₒC respectively.
If v and v' be the linear speed of the particle at 50 ₒC and 20 ₒC respectively, as the angular velocity remains (ω) constant.
Therefore,
ω =`v/R =(v')/(R')` ...(1)
Now,
R' = R (1+αΔT)
⇒ R' =R +R × 1.2 ×10-5 ° C-1 × Δ T
⇒ R' = 1.00036R
Using equation(1) we have,
`v/R =(v')/(R')`
`=> v/R = (v')/(1.00036R)`
⇒ v' = 1.00036v
Percentage change in linear speed will be,
= `(v'-v)/v xx 100`
=` (1.00036v -v)/v xx 100`
= 3.6 × 10-2
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