Question

Two blocks $A$ and $B$ of mass $1kg$ and $2kg$ respectively are placed over smooth horizontal surface,as shown in figure. The coefficient of friction between blocks $A$ and $B$ is $\mu =1/2$. An external fore, of magnitude $F$ is applied to the top block at an angle $\alpha ={30}^o$ below the horizontal.
a) If the two blocks move together. find their acceleration in terms of $F$.
b) Find the maximum values of $F$ no that both the blocks move together with same acceleration $(g=10m/s^2)$.

Answer 1

$f=uR$

$R=mg+F\sin \alpha$

If block will slide,

then $F\cos \alpha >f$

Resultant force= $F\cos \alpha -f$

$ma=F\cos \alpha -u(mg+F\sin \alpha )(m=combinedmass)$

$ma=F\cos \alpha =u(mg+F\sin \alpha )$

$ma=\frac{F\sqrt{3}}{2}-\frac{1}{2}(3\times 10+\frac{F}{2})$

$3a=\frac{F\sqrt{3}}{2}-\frac{1}{2}\left(\frac{60+F}{2}\right)$

$3a=\frac{F\sqrt{3}}{2}-\frac{60+F}{4}$

$3a=\frac{2F\sqrt{3}-60-F}{4}\Rightarrow a=\frac{F(2-\sqrt{3}-1)-60}{12}m/s$