On a pulley of mass M hangs a rope with two masses m ₁ and m ₂ ( ~m ₁> m ₂ ) tied at the ends as shown in the figure. The pulley rotates without any friction, whereas the friction between the rope and the pulley is large enough to prevent any slipping. Which of the following plots best represents the difference between the tensions in the rope on the two sides of the pulley as a function of the mass of the pulley?

Step-by-step solution

mg - T ₁= m ₁ a ...... (1) T ₂- m ₂ ~g = m ₂ a ...... (2) array l ( T ₁- T ₂ ) R = I [ a = r ] = Ia R (1)+(2) ( m ₁- m ₂ ) g + ( T ₂- T ₁ )= ( m ₁+ m ₂ ) a ( T ₁- T ₂ )= I R ² [ ( m ₁- m ₂ ) g + ( T ₂- T ₁ ) m ₁+ m ₂ ] ( T ₁- T ₂ ) [1+ I k ² ( ~m ₁+ m ₂ ) ] = I k ² ( m ₁- m ₂ ) m ₁+ m ₂ ~g T ₁- T ₂= [ I ( m ₁- m ₂ ) g I + ( m ₁+ m ₂ ) R ² ] = 1 2 MR ² ( ~m ₁- m ₂ ) g ( 1 2 MR ²+ ( m ₁+ m ₂ ) R ² ) T ₁- T ₂= M ( m ₁- m ₂ ) g M +2 ( ~m ₁+ m ₂ ) array