Ramp Problem, Friction, Find Theta

[maryhem]

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 $\mu$. The system is in a uniform gravitational field directed downward of strength $g$. Find the incline angle $\theta$ 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

$$\mathbf{F} = m\mathbf{a}$$

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 $2m$ mass:
$$2mg\sin\theta - \mu mg\cos\theta - T = 0$$
And for the second block:
$$T - mg = 0 \implies T = mg$$
Using the second equation and plugging into the first equation, we find:
$$2\sin\theta - 2\mu\cos\theta = 1$$
I can't figure out how to solve for $\theta$. Wolfram's answer is pretty ugly.

 

[Simon Bridge]

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.