# Calculating the angle $\theta$ of an incline which allows two masses to move at a constant speed? [closed]

Two blocks of unequal mass are connected by a string over a smooth pulley. If the coefficient of kinetic friction is $\mu_k$, what angle $\theta$ of the incline allows the masses to move at a constant speed?

So here is my attempt:

Essentially we have two 1D problems for each respective mass. So, for $v_x, v_y$ to be constant, that requires that $\ddot{x} = 0,$ and $\ddot{y} = 0$.

First, I'll consider the block of mass $m$ which moves vertically: $m \ddot{y} = T - mg = 0 \implies$ $T = mg$.

Since both masses are connected by the same string we can assume that the tensile force acting on each mass is identical.

Now, for the block of mass $2m$ which moves only horizontally:

$(2m) \ddot{x} = 2mg \sin \theta - T - F_f = 0$.

Given that $F_f := \mu F_N$, and this block does not move vertically, $F_N = 2mg \cos \theta$. This gives us that:

$2mg \sin \theta - mg - \mu_k (2mg \cos \theta) = 0$, or $\, 2\sin \theta - 1 = \mu_k (2 \cos \theta)$.

I'm not really sure how to proceed from here. The only thing I can think of doing is dividing both sides by $2 \cos \theta$, which simplifies to:

$\tan \theta - \frac{1}{2 \cos \theta} = \mu_k$, but I am not sure if this is really any better.

Any help is very greatly appreciated. Thanks!

EDIT: I just want to know if I have done the physics correctly...

• If your solution is correct, then you've actually already solved the problem... The only thing missing is algebraic. I didn't check your calculations, but the equation you've gotten to is not easy to solve. Aug 30, 2016 at 19:22
• Right... the issue is I am not sure if what I've done is exactly 100% correct. I'm actually a math major taking a theoretical/classical mechanics class and the last time I had intro. mechanics was 3 years ago.. So I'm quite rusty on the fundamentals. Aug 30, 2016 at 19:30
• What is your question? What 'hint' are you asking for? As QB says, you've got an equation; simplifying it further is math, not physics. Aug 30, 2016 at 19:32
• Did I analyze the mechanics of the system correctly? Aug 30, 2016 at 19:35