Consider the scenario shown in the diagram below. A box (mass m1) is on a slope,

Question

Consider the scenario shown in the diagram below. A box (mass m1) is on a slope, which is inclined at an angle theta with respect to the horizontal. The box is connected via a string and pulley to a hanging weight (mass m2). The string and the pulley are assumed to be massless and frictionless, but there is friction between the box and the incline (with coefficient of kinetic friction mu k). Find an expression for the tension T in the string as the box slides down the incline. Your answer should be written in terms

Explanation / Answer

let tension in the string is T and acceleration of the system is a m/s^2

then for m2:

m2*g-T=m2*a
==>T+m2*a=m2*g...(1)
for m1:
normal force=m1*g*cos(theta)
friction force=mu*m1*g*cos(theta)

writing force balance equation:
T+m1*g*sin(theta)-mu*m1*g*cos(theta)=m1*a

==>T-m1*a=m1*g*(mu*cos(theta)-sin(theta))..(2)

equation 1* m1 + equation 2*m2:

m1*T+m2*T=m1*m2*g+m1*m2*g*(mu*cos(theta)-sin(theta))

==>T*(m1+m2)=m1*m2*g*(1+mu*cos(theta)-sin(theta))
==>T=(m1*m2*g*(1+mu*cos(theta)-sin(theta)))/(m1+m2)

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