Question

Two bodies of mass `m` and `4m` are attached with string of length `l` is excurting oscillations of angular amplitude `theta_(0)` while the other body is at rest . The minimum coefficient of friction between the mass `4m` and the horizontal surface should be:

A. `((2 - cos theta_(0))/(3))`
B. `2 cos^(2) ((theta_(0))/(2))`
C. `((1 - cos theta_(0))/(2))`
D. `((3 - cos theta_(0))/(4))`

Answer 1

Maximum tension in the string is in its lowest position.
`:.` Speed of mass `m` in its lowest position is

`v^(2) = 2gh = 2hl (1 - cos theta)`
`T_(max) - mg = (mv^(2))/(l)`
`T_(max) - mg + 2 mg (1 - cos theta)`
`= mg (3 - cos theta)`
Block of mass `4m` does not move.
So `mu (4 mg) ge T_(max)`
or `5 mu mg ge mg (3 - cos theta)or mu ge ((3 - cos theta_(0))/(4))`