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Ramp Problem, Friction, Find Theta

  • Thread starter maryhem
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Challenging Ramp and Pulley Problem

Homework Statement



Two blocks of masses 2m and m are connected by a weightless string over a frictionless, massless pulley, as shown in the figure. The coefficient of kinetic friction between the block and the incline is [itex]\mu[/itex]. The system is in a uniform gravitational field directed downward of strength [itex]g[/itex]. Find the incline angle [itex]\theta[/itex] such that the blocks move at a constant speed. Distinguish between the cases of upward and downward motion. Rationalize your solutions using a simple physical picture.

Homework Equations


[tex]\mathbf{F} = m\mathbf{a} [/tex]

The Attempt at a Solution



So we start by looking at the forces acting on each block. In this case, we will be looking at downward motion. For the [itex]2m[/itex] mass:
[tex] 2mg\sin\theta - \mu mg\cos\theta - T = 0 [/tex]
And for the second block:
[tex] T - mg = 0 \implies T = mg [/tex]
Using the second equation and plugging into the first equation, we find:
[tex] 2\sin\theta - 2\mu\cos\theta = 1 [/tex]
I can't figure out how to solve for [itex]\theta[/itex]. Wolfram's answer is pretty ugly.
 

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Answers and Replies

  • #2
Simon Bridge
Science Advisor
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You need trig identities and a bit of manipulation:

Divide through by ##\cos(\theta)##
##1/\cos(\theta)=\sec(\theta)##

Square both sides and expand the RHS
##\sec^2(\theta)=1+\tan^2(\theta)##

Change variables: ##x=\tan(\theta)##
Look familiar?

Note - you have misplaced a minus sign in the first equation.
gravity and friction both point in the opposite direction to tension.
 

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