Static Friction Of Masses Connected By A Rod

[inkliing]

This isn't homework...I'm reviewing physics after many years of neglect.

 

Given 2 masses, [math]m_1, m_2[/math], connected by a rigid, massless rod, stationary with respect to a ramp which makes an angle of [math]\theta[/math] with the horizontal, with coefficients of static friction between the masses and the ramp = [math]\mu_{s1}, \mu_{s2}[/math] respectively, what is the magnitude of the tension or compression in the rod, and what are the magnitudes of the static friction, [math]f_{s1}, f_{s2}[/math], acting on each mass?

 

This assumes [math]\theta[/math] is small enough that the masses do not lose traction, i.e.,

 

[math]\theta \leq \arctan \frac{\mu_{s1} m_1 + \mu_{s2} m_2}{m_1 + m_2}[/math]

 

Note that if the masses are assumed to already be moving, then the problem is straightforward:

 

[math]T = (\mu_{k1} - \mu_{k2})\frac{m_1 m_2}{m_1 + m_2}g\cos\theta[/math]

[math]f_{k1} = \mu_{k1} m_1 g\cos\theta[/math]

[math]f_{k2} = \mu_{k2} m_2 g\cos\theta[/math]

 

Where T is the tension in the rod (T<0 implies compression). Such problems are common in basic physics, e.g., Halliday, Resnick, & Krane, 4th Ed., chap.6, problem 29.

 

So I assumed it would be straightforward to do the same problem, but with the masses stationary. But I get 2 equations (sum of the forces for each mass) and 3 unknowns (T, [math]f_{s1}, f_{s2}[/math]).

 

I find it surprising that such a simple situation is undetermined, and assume I've missed something simple.

 

Note: I've looked through several physics texts and can find only problems in which the masses are already moving.

 

Also, it's driving me crazy!