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Two blocks of mass m and M, are connect ...

Two blocks of mass `m` and `M`, are connect to the ends of a string passing over a pulley. `M` lies on the plane inclined at an angle `theta` with the horizontal and `m` is hanging vertically as shown in Fig. The coefficient of static friction between `M` and the incline is `mu_(s)`. Find the minimum and maximum value of `m` so that the system is at rest.

Text Solution

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In the present problem motion will depend on the relative values of m and M . If m is heavier than M , then the block M will move upwards . If m is much lighter than M , then the block M will move downwards .

Consider following situations :
Case (i) : M is about to start sliding upwards (Motion impending upwards) . This will happen if m is much heavier than M .
As M is just at the point of sliding up , frictional force on M is `mu_(s)N` acting down the plane .
Balancing forces along x-axis and y-axis :
`N "Mg cos" theta , T = "Mg sin" theta + mu_(s)N , T = "mg" `
Hence , `"mg" = "Mg sin" theta + mu_(s) "Mg cos" theta`
Maximum value of m = `M ("sin" theta + mu_(s) "cos" theta)`
Case (ii) : M is about to start sliding downwards (motion impending downwards) . This will happen if m is much lighter than M .

As M is about to start sliding down , the frictional force is `mu_(s) N` acting upwards .
Balancing forces along x-axis and y-axis :
`N = "Mg cos" theta , T = "Mg sin" theta - mu_(s)N , T= "mg" `
Hence , mg = Mg sin `theta - mu_(s) "Mg cos" theta`

Minimum value of m = M (sin `theta - mu_(s) "cos" theta)`
Therefore the blocks are at rest if ,
M (sin `theta - mu_(s) "cos" theta ) lt m lt M ("sin" theta + mu_(s) "cos" theta )`
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