To calculate the elastic potential energy in the system, we consider two wires of identical material (both having Young's modulus
Y ) and length
L, but with different radii:
R and
2R. These wires are sequentially connected, and a weight
w is hung from them.
First, determine the elongation of each wire due to the weight
w :
For the first wire with radius
R :
∆l1=‌For the second wire with radius
2R :
∆l2=‌The elastic potential energy stored in a wire is given by:
U=‌K(∆l)2where
K=‌ is the spring constant and
A is the cross-sectional area of the wire.
Calculate the spring constants for both wires:
For the first wire
(K1) :
K1=‌For the second wire
(K2) :
K2=‌Now compute the potential energy for each wire and sum them up:
U=‌×‌×(‌)2+‌×‌×(‌)2After performing the calculations, the total elastic potential energy in the system is:
U=‌