A truck pulling a trailer is going down a hill. The truck has brakes but the tra

Question

A truck pulling a trailer is going down a hill. The truck has brakes but the trailer does not, meaning that the truck and the trailer have very different 1 coefficients of friction. The truck is braking and the truck and trailer move together. Assume that the hill makes an angle below the horizontal, the truck has mass m, the trailer has mass M, and that the coeficents of friction on the truck and trailer are given by µm and µM, respectively. Find an expression for the force of the trailer on the truck in terms of g, m, M, µm, and µM. Explain why that force is independent of the accelerations of the truck and trailer.

Explanation / Answer

given a truck pulling a trailer is going down the hill, truck has brakes, trailer does not
angle of hill below the horizontal = theta
mass of truck = m, coefficient of friciton = um
mass of trailer = M, coefficient of friction = mM

let force between truck and trailer be F
then
from force balance on the trucck
mgsin(theta) + F - um*mgcos(theta) = ma ( where a is acceleration of both truck and the trailer)
similiarly using force balance on the trailer
Mgsin(theta) - F - uM*Mgcos(theta) = Ma

a = (Mgsin(theta) - F - mU*Mgcos(theta))/M
hence
F = ma - mgsin(theta) + um*mgcos(theta)
F = m* (Mgsin(theta) - F - mU*Mgcos(theta))/M - mgsin(theta) + um*mgcos(theta)

as we can see this force is independent of accelerations of the trailer and the truck, because both accelerations are the same and will cancel out in the simultaneous equations
leaving rest of the forces to be taken into considerations
moreover, this is an internal foirce of the system which dfoesx not effect external acceleration of the ssytem, similiar to tyhe fact that external forces onthe system do not effect the internal forces on the system

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