Two small pulses are created on the rope at points A and B respectively as shown. The rope is rotating in a horizontal plane with constant angular velocity about end A whereas end B is free. Sₐ and Sᵦ are the distances covered by pulses A and B respectively in the rope frame before they cross each other. If k=Sₐ/Sᵦ, find the value of (k−1)².
Physics · Mixed Chapter
JEE Mains

Question

Two small pulses are created on the rope at points A and B respectively as shown. The rope is rotating in a horizontal plane with constant angular velocity about end A whereas end B is free. Sₐ and Sᵦ are the distances covered by pulses A and B respectively in the rope frame before they cross each other. If k=Sₐ/Sᵦ, find the value of (k−1)².

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Let rope length be L and mass per unit length μ. Tension at distance x from fixed end A equals centrifugal load beyond x: T(x)=½μω²(L²−x²). Step 2: Local wave speed is v(x)=√[T(x)/μ]=(ω/√2)√(L²−x²). Step 3: Times for pulses to reach x are t_A=∫₀ˣds/v(s)=(√2/ω)sin⁻¹(x/L) and t_B=∫ₓᴸds/v(s)=(√2/ω)[π/2−sin⁻¹(x/L)]. Equality gives x=L/√2. Step 4: k=x/(L−x)=(1/√2)/(1−1/√2)=√2+1; thus (k−1)²=2.