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Challenging Ramp and Pulley Problem
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.
[tex]\mathbf{F} = m\mathbf{a} [/tex]
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.
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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