Question

A small block of mass m is attached at the bottom end of an elastic massless rod of length L, area A and Young’s modulus Y. Elastic energy stored in the rod is A

Answer 1

Given,

A small block of mass m is attached at the bottom end of an elastic massless rod of length L, area A and Young's modulus Y.

To find,

Elastic energy stored in the rod.

cut an element of thickness dx at height x from the bottom.

so weight of rod of length x, F = (Mg/L)x .....(1)

from Young's modulus concept,

Y = Fdx/AL

from equation (1) we get,

Y = (Mg/L)x dx/AdL

⇒∫dL = (Mg/ALY ) ∫x dx

⇒∆L = (Mg/ALY) [L²/2]

⇒∆L = MgL/2AY ...........(2)

now energy density = 1/2 × stress × strain

⇒ energy / volume = 1/2 × (stress)²/Y

⇒energy / A dx = 1/2 × F²/A²Y

⇒energy = 1/2 [(Mg/L)x]²/AY × dx

⇒energy = 1/2 × M²g²x²/AL²Y dx

⇒energy = 1/2 M²g²/AL²Y ∫x² dx

⇒energy = M²g²L/6AY

therefore energy stored in the rod is M²g²L/6AY