Answer to Question #181308 in Physics for Mark Laurence

Question

Two blocks of weights W1 = W2 = 10KN rest on a rough inclined plane and are connected by a short piece of string as shown. If the coefficient of friction is μ1 = 0.20 and μ2 = 0.30 respectively: Determine a) The inclination of the plane for which sliding will impend.

Answer 1

Sliding will start when the inclination will exceed some angle $\theta$.

Write forces acting on each block:

$$1.\space x:f_1+T-W\sin\theta=0,\\ y: W\cos\theta=N.\\\space\\ 2.\space x:f_2-T-W\sin\theta=0,\\ y: W\cos\theta=N.$$

Since $f=\mu N$,

$$1.\space \mu_1W\cos\theta+T-W\sin\theta=0,\\ 2.\space \mu_2W\cos\theta-T-W\sin\theta=0,\\$$

add these equations:

$$\cos\theta ·W(\mu_1+\mu_2)-2W\sin\theta=0,\\ \cos\theta ·(\mu_1+\mu_2)=2\sin\theta,\\\space\\ \tan\theta=\frac{\mu_1+\mu_2}{2},\\\space\\ \theta=\arctan\frac{\mu_1+\mu_2}{2}=14°.$$