Question

The upper portion of an inclined plane of inclination $\alpha$ is smooth and the lower portion is rough. A partlcles slldes down from rest from the to and just comes to rest at the foot If the ratio of smooth length to rough length is m : n, the coefficient of functlon is:

• A. $\left[\frac{m+n}{n}\right]tan\alpha$
• B. $\left(\frac{m+n}{n}\right)cot\alpha$
• C. $\left(\frac{m-n}{n}\right)cot\alpha$
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\left[\frac{m+n}{n}\right]tan\alpha$

Let the mass of the particle be m, let the length of the incline be l and let the coefficient of friction be $\mu$
We'll use the equation ${KE}_1+{PE}_1+W_{ext}={KE}_2+{PE}_2$
${KE}_1=0$
${PE}_1=mgh=mgl\sin \alpha$
${PE}_2=0$
${KE}_2=0$
The only external force acting on the particle is friction, $f$.
$f=\mu N=\mu mg\cos \alpha$
$W_{ext}=f\times s=\mu mg\cos \alpha \left(\frac{n}{m+n}\right)l$
Substituting the values into the equation, we get
$\mu mg\cos \alpha \left(\frac{n}{m+n}\right)l=mgl\sin \alpha$
$\mu =\frac{m+n}{n}\tan \alpha$