In the adjacent figure a light and thin string is wound on a uniform disc of mass m and radius r very tightly. The disc is kept at rest in vertical position on a rough horizontal surface and string passes over a fixed pulley. A light pan is attached to the free end of the string. The maximum mass that can be placed on the pan so that disc will not slip is equal to mk/2. Find k if coefficient of friction between the surface and the disc is 0.1 and there is no friction between string and pulley.
Physics · Mixed Chapter
JEE Mains

Question

In the adjacent figure a light and thin string is wound on a uniform disc of mass m and radius r very tightly. The disc is kept at rest in vertical position on a rough horizontal surface and string passes over a fixed pulley. A light pan is attached to the free end of the string. The maximum mass that can be placed on the pan so that disc will not slip is equal to mk/2. Find k if coefficient of friction between the surface and the disc is 0.1 and there is no friction between string and pulley.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: If the disc rolls without slip under top horizontal tension T, its translation and rotation equations are ma=T+f and ½mr²(a/r)=Tr−fr. Step 2: Solving gives f=T/3 and a=4T/(3m). The pan/string speed at the disc's upper rim changes twice as fast, so pan acceleration=2a. Step 3: At the friction limit f=μmg=0.1mg, so T=0.3mg and a=0.4g. Step 4: For pan mass mₚ, mₚg−T=mₚ(2a)=0.8mₚg. Thus 0.2mₚg=0.3mg ⇒mₚ=1.5m. Step 5: mₚ=mk/2 ⇒k=3.