An ant of mass m is standing peacefully on top of a horizontal stretched rope. The rope has mass per unit length μ and is under tension F. Without warning, a student starts a sinusoidal transverse wave of wavelength λ propagating along the rope. The wave is in vertical plane. What minimum wave amplitude (in mm) can make that ant feel weightless momentarily? Assume m is so small that presence of the ant has no effect on wave propagation. [Given λ=0.5 m, μ=0.1 kg/m, F=3.125 N, take g=π².]
Physics · Mixed Chapter
JEE Mains

Question

An ant of mass m is standing peacefully on top of a horizontal stretched rope. The rope has mass per unit length μ and is under tension F. Without warning, a student starts a sinusoidal transverse wave of wavelength λ propagating along the rope. The wave is in vertical plane. What minimum wave amplitude (in mm) can make that ant feel weightless momentarily? Assume m is so small that presence of the ant has no effect on wave propagation. [Given λ=0.5 m, μ=0.1 kg/m, F=3.125 N, take g=π².]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Transverse wave speed on the rope is v=√(F/μ)=√31.25 m/s. Wave angular frequency ω=2πv/λ. Step 2: At the top of a sinusoidal oscillation, the ant just loses contact (normal reaction=0) when its downward acceleration equals g: Aω²=g. Step 3: A=g/ω²=gλ²μ/(4π²F). Substitute g=π², λ=.5, μ=.1, F=3.125. Step 4: A=(.25×.1)/(4×3.125)=0.002 m=2 mm.