An early arrangement for measuring the acceleration of gravity, called Atwood\'s

Question

An early arrangement for measuring the acceleration of gravity, called
Atwood's Machine, is shown in the gure. The pulley P and cord C have
negligible mass and friction. The system is balanced with equal masses M
on each side, and then a small rider of mass m is added to one side. After
accelerating through a certain distance h, the rider is caught on a ring and
the two equal masses then move on with constant speed v. Show that the
value of g that results is given by

g = (2M+m)v2 / 2mh .

An early arrangement for measuring the acceleration of gravity, called Atwood's Machine, is shown in the gure. The pulley P and cord C have negligible mass and friction. The system is balanced with equal masses M on each side, and then a small rider of mass m is added to one side. After accelerating through a certain distance h, the rider is caught on a ring and the two equal masses then move on with constant speed v. Show that thevalue of g that results is given by g = (2M+m)v2 / 2mh .

Explanation / Answer

Initial :

(M+m)g - T = (M+m)a

T - Mg = Ma

adding both these two

mg = (2M+m)*a

a = mg/(2M+m) .....................(1)

v ^2= u^2 + 2*a*h

v = final speed

u = initial speed = 0

v^2 = 2ah

a = v^2/2h.........................(2)

From (1) and (2)

v^2/2h =   mg/(2M+m)

g = (2M+m)v^2 / 2mh ....(hence proved)

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