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Question
Two identical steel cubes (masses 50 g, side 1 cm) collide head-on face to face with a speed of 10 cm/s each. Find the maximum compression of each. Young’s modulus for steel = Y = 2 × 1011 N/m2.
Solution
Let m = 50 g = 50 × 10–3 kg
Side = L = 1 cm = 0.01 m
Speed = v = 10 cm/s = 0.1 m/s
Young's modulus = Y = 2 × 1011 N/m2
Maximum compression ΔL = ?
In this case, all KE will be converted to PE
By Hooke's law, `F/A = Y (ΔL)/L`
Where A is the surface area and L is the length of the side of the cube. If k is spring or compression constant, then
Force F = kΔL
∴ k = `Y A/L = YL`
Initial KE = `2 xx 1/2 mv^2 = 5 xx 10^-4` J
Final PE = `2 xx 1/2 k (ΔL)^2`
∴ ΔL = `sqrt((KE)/k)`
= `sqrt((KE)/(YL))`
= `sqrt((5 xx 10^-4)/(2 xx 10^11 xx 0.1))`
= 1.58 × 10–7 ......[∵ PE = KE]
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