A block with mass `m_(1)` is placed on an inclined plane with slope angle `alpha` and is connected to a second hunging block with mass `m_(2)` by a cord passing over a small . Friction less pulley as shown in fig 7.247 . The coefficient of static friction is `mu_(2)` and the coefficient of kinetic friction is `mu_(s)`

a. Find the mass `m_(2)` for which block `m_(1)` moves up plane at constant speed once it is set in motion
b. Find the mass `m_(2)` for which block `m_(1)` moves down the plane at constant speed once it is set in motion
c.For what range of `m_(2)` will the blocks remain at rest if they are released from rest?

Answer

`m_(1)(sin alpha + mu_(k)cos alpha)`
b.`m_(1)(sin alpha - mu_(k)cos alpha)`
c.`m_(1)(sin alpha - mu_(k)cos alpha)< m_(2) < m_(1)(sin alpha + mu_(s)cos alpha)`

Explanation

a. The tension in the cort must be `m_(2) g` in order that the banging block moves at constant speed .This tension must overcome friction and the component of the gravitational force the incline , so
`m_(2) g = (m_(2) g sin alpha + mu_(k) m_(2) g cos alpha)`
and `m_(2)= m_(1)(sin alpha + mu_(k)cos alpha)`
b. In this case the friction in the same direction as the tension on the block of mass `m_(1)` so ,
`m_(2) g = (m_(1) g sin alpha - mu_(k) m_(2) g cos alpha)`
or `m_(2)= m_(1)(sin alpha - mu_(k)cos alpha)`
c. similar to the analysis of parts (a) and (b), the largest could be is `m_(1)(sin alpha + mu_(k)cos alpha)` and the similar `m_(2)` could `m_(1)(sin alpha - mu_(k)cos alpha)`