The drawing shows a baggage carousel at an airport. Your suitcase has not slid all the way down the slope and is going around at a constant speed on a circle of radius (625/8) cm as the carousel turns. The coefficient of static friction between the suitcase and the carousel is 0.8, and the angle θ in the drawing is 37°. The time (in seconds) required for your suitcase to go around once is 2t₁. Find t₁. Assume limiting friction on the suitcase and take π² = g.
Physics · Mixed Chapter
JEE Mains

Question

The drawing shows a baggage carousel at an airport. Your suitcase has not slid all the way down the slope and is going around at a constant speed on a circle of radius (625/8) cm as the carousel turns. The coefficient of static friction between the suitcase and the carousel is 0.8, and the angle θ in the drawing is 37°. The time (in seconds) required for your suitcase to go around once is 2t₁. Find t₁. Assume limiting friction on the suitcase and take π² = g.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The suitcase is at impending slip on the sloping carousel. At the limiting upward-slope static friction f=μN, resolve radial and vertical components using θ=37°. Step 2: The smallest centripetal acceleration corresponding to the geometry is a꜀/g=(μcosθ−sinθ)/(cosθ+μsinθ)=[0.8(0.8)−0.6]/[0.8+0.8(0.6)]=0.04/1.28=1/32. Step 3: With orbit radius r=(625/8) cm=0.78125 m and period 2t₁, ω=2π/(2t₁)=π/t₁. Therefore a꜀=ω²r=π²r/t₁². Step 4: π²r/t₁²=g/32. Given g=π², t₁²=32r=32×0.78125=25; t₁=5 s.