BITSAT 2024 Shift 1 Paper

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=‌

wL

4πR2Y

For the second wire with radius 2R :
∆l2=‌

wL

πR2Y

The elastic potential energy stored in a wire is given by:
U=‌

1

2

K(∆l)2
where K=‌

YA

L

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=‌

Y×πR2

L

For the second wire (K2) :
K2=‌

Y×4πR2

L

Now compute the potential energy for each wire and sum them up:
U=‌

1

2

×‌

Y×4πR2

L

×(‌

wL

4πR2Y

)2+‌

1

2

×‌

Y×πR2

L

×(‌

wL

πR2Y

)2
After performing the calculations, the total elastic potential energy in the system is:
U=‌

5w2L

8πR2Y