A 1 m stepped bar has area 10 cm² along part of its length and 5 cm² along the rest. Under a given axial force, strain energy is 40% of the energy in a uniform 1 m bar of area 10 cm² under the same stress as in the 5 cm² segment. If length of the 10 cm² part is 10x cm, find x.
Physics · Mixed Chapter
JEE Main Practice

Question

A 1 m stepped bar has area 10 cm² along part of its length and 5 cm² along the rest. Under a given axial force, strain energy is 40% of the energy in a uniform 1 m bar of area 10 cm² under the same stress as in the 5 cm² segment. If length of the 10 cm² part is 10x cm, find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let the 10-cm² length be ℓ cm. For force F, stepped-bar energy U=F²/(2Y)[ℓ/10+(100−ℓ)/5]. Step 2: Stress in 5-cm² part is F/5. Uniform 10-cm² bar under that stress has force 2F, so U_ref=(2F)²/(2Y)(100/10)=40F²/(2Y). Step 3: U=0.4U_ref ⇒ℓ/10+(100−ℓ)/5=16. Step 4: 20−ℓ/10=16⇒ℓ=40 cm=10x cm; x=4.