Collection of Solved Problems in Physics

Diameter Increase of Cylinder When Being Heated

Task number: 1809

• Hint

  What relationship must apply between the measured quantity and the accuracy of the measurement device, so that we can use the device to measure the quantity?

• Analysis

  To measure the increase of the diameter of the cylinder using the given micrometer, the change of the diameter must be greater than the accuracy of the micrometer.

  The change in diameter the cylinder can be determined using the formula for thermal linear expansion.

• Given values

  t0 = 5 °C	initial temperature of the cylinder
  t1 = 35 °C	final temperature of the cylinder
  d0 = 20.45 mm	diameter of the cylinder at initial temperature
  Δd = ?	change in diameter of the cylinder

  From The Handbook of Chemistry and Physics:

  α = 18.7·10−6 °C−1

  coefficient of thermal expansion of brass

• Solution

  We must first determine the change in diameter Δd which is caused by heating of the cylinder. We use the well-known formula for linear expansion.

  It is true that:

  Δd = d₀αΔt = d₀α(t₁ − t₀),

  where d₀ is the initial diameter of the cylinder, t₀ the initial temperature, t₁ the final temperature, and α is the coefficient of thermal expansion of brass.

• Numerical solution

  \[\mathrm{\Delta} d=d_{0}\alpha(t_{1}-t_{0})=20.45\cdot{18.7}\cdot{10^{-6}}\cdot{(35-5)}\, \mathrm{mm}\] \[\mathrm{\Delta} d\,\dot=\, 0.011\, \mathrm{mm}\]

• Answer

  The increase in diameter of a brass cylinder can be measured using a micrometer. This measurement, however, is not very precise.