In fig. a ball of mass `m_(1)` and a block of mass `m_(2)` are joined together with an inextensible string. The ball can slide on a smooth horizontal surface. If `v_(1)` and `v_(2)` are the respective speeds of the ball and the block, then determine the constraint relation between the two.

Explanation

Method 1: Distances are assumed from the center of the pulley as shown in fig.

Consraint: Length of the string remains constant.
`sqrt(x_(1)^(2)+h_(1)^(2))+x_(2)` = constant
Differentiating both the side w.r.t. time, we get
`(2x_(1))/(2sqrt(x_(1)^(2)+h_(1)^(2)))(dx_(1))/(dt)+(dx_(2))/(dt)=0`
Since the ball moves so as to increases `x_(1)` with time and block moves so as to decrease `x_(2)` with time,
`(dx_(1))/(dt) +v_(1)` and `(dx_(2))/(dt)=-v_(2)`
also, `(x_1)/(sqrt(x_(1)^(2)+h_(1)^(2))) = cos theta` or `v_(2)=v_(1) cos theta`
Method 2: The problem can be solved very easily if we look at the problem from a different viewpoint and identify a difference constraint. i.e., the velocity of any two points along the string is same. Obviously, from fig., we have `v_(1) cos theta=v_(2)`

Method 3: Change in length of segment I,
`Delta l_(1)=0+-(x_(2))=-x_(2)`

Change in length of segment II,
`Delta l_(1)=(x_(2)cos theta)+0=x_(1) cos theta`
Total change in the length of all segments should be zero, as the length of string is constant.
`Delta l=Delta l_(1)+Delta l_(2)`
`=-x_(2)+x_(1) cos theta=0`
`implies x_(2)=x_(1) cos theta` or `v_(2)=v_(1) cos theta`.