A block with mass m_1 is placed on an inclined plane with slope angle a and is c

Question

A block with mass m_1 is placed on an inclined plane with slope angle a and is connected to a second hanging block with mass m_2 by a cord passing over a small, frictionless pulley (Fig. P5.68). The coefficient of static friction is jus and the coefficient of kinetic friction is mu_k. Find the mass m_2 for which block m_1 moves up the plane at constant speed once it is set in motion, Find the mass m_2 for which block m_1 moves down the plane at constant speed once it is set in motion. For what range of values of m_2 will the blocks remain at rest if they are released from rest?

Explanation / Answer

for simplicity purpose let slope angle alpha be a

a) forces on m1 are: friction: F1= m1*g*cos(a) * uK , gravity: F2 = m1*g*sin(a) , cord tension: F3 = F1 + F2 = m1*g * [sin(a) + uK*cos(a)]
also F3 = m2*g
Hence equating two equations
m2 = m1*[sin(a) + uK*cos(a)]

b) when block m1 moves downward:
F3 + F1 = F2
F3 = m1*g * [sin(a) - uK*cos(a)]
also F3 = m2*g

Hence
m2 = m1 * [sin(a) - uK*cos(a)]

c) The minimum mass m2 that allows system to be at rest
m2 min = m1*[ sin(a) - uS*cos(a) ]

and maximum mass m2 will be
m2 max = m1*[ sin(a) + uS*cos(a)]

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