A uniform rod rotates about one end at angular speed ω in gravity-free space. If the ratio of mean strains of the nearer and farther halves is k (>1), find 20k/11.
Physics · Mixed Chapter
JEE Main Practice

Question

A uniform rod rotates about one end at angular speed ω in gravity-free space. If the ratio of mean strains of the nearer and farther halves is k (>1), find 20k/11.

G1 Physics · Accepted Answer

Step 1: At distance x from the pivot, required centripetal pull from material beyond x gives tension T(x)=∫ₓᴸλω²s ds=½λω²(L²−x²). Step 2: Strain at x is T(x)/(YA), hence proportional to L²−x². Step 3: Average near-half strain ∝ (2/L)∫₀ᴸ⁄²(L²−x²)dx =11L²/12. Step 4: Average far-half strain ∝(2/L)∫ᴸ⁄²ᴸ(L²−x²)dx=5L²/12, so k=11/5. Step 5: 20k/11 =20/5 =4.