A steel ball initially at a pressure of 1.0 × 105 Pa is heated from 20°C to 120°C keeping its volume constant. Find the pressure inside the ball. Coefficient of linear expansion of steel = 12 × 10–6 °C–1 and bulk modulus of steel = 1.6 × 1013 N m–1.
Given:
Initial pressure on the steel ball : P = 1.0 × 105 Pa
T1= 20° C
T2 = 120° C
Change in temperature : Δ T = 100° C
Volume is Constant.
Coefficient of linear expansion of steel : α = 12 × 10–6 °C–1
Bulk modulus of steel : B = 1.6 × 1011 N m–1.
Formula used:
When a body is subject to pressure, its volume decreases and retains its volume when pressure is removed.
Bulk modulus is the ratio of pressure and Strain.
Formula for bulk modulus is:
Here B is the Bulk Modulus, P is the Pressure inside the ball.
We know that the formula for volume expansion of a body is
Where V’ is the final or changed Volume when temperature is increased to T2, V is the initial volume at T1 and ΔV is the Change in Volume.
ΔV = V’ – V.
Substituting value of ΔV in the equation of B, we get:
But γ = 3× α
The Pressure inside the ball is 5.76 × 108 Pa.