Two masses are connected by the rope The rope is passed thro

Two masses are connected by the rope. The rope is passed through the pulley. The masses have the magnitude of the acceleration a_1 if the first mass is on the horizontal surface and the second mass is vertically hanged. The masses have the magnitude of the acceleration a_2 if the second mass is on the horizontal surface and the first mass is vertically hanged. Find the magnitude of the acceleration if both masses are hanged on the pulley.

Solution

Here,

let the larger mass is m1

smaller mass is m2

for the acceleration a1

Using second law of motion

a1 = (m1 * g)/(m1 + m2)

m1 = (m1 + m2) * a1/g ----(1)

using second law of motion for case 2

a2 = (m2 * g)/(m1 + m2)

m2 = (m1 + m2) * a2/g ----(2)

--- Now , using magnitude of acceleration for third case

a = (m1 - m2) * g/(m1 + m2)

a = ((m1 + m2) * a1/g - (m1 + m2) * a2/g)/(m1 + m2)

a = (a1/g - a2/g) = (a1 - a2)/g

the magnitude of acceleration in the third case is (a1 - a2)/g