Answer to Question #180901 in Physics for ansaynathaniel22

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

Draw the figure:

Sliding will start when the inclination exceeds angle $\theta$.

Forces acting on each block (the lower one is 1 and the upper one is 2):

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

We also know that $f=\mu N$ and see that $W\cos\theta=N$, so:

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

To exclude T, add these equations:

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