Question

Two identical uniform discs roll without slipping on two different surfaces $AB$ and $CD$ (see figures) starting at $A$ and $C$ with linear speeds $v_1$ and $v_2$, respectively and always remain in contact with the surfaces. If they reach $B$ and $D$ with the same linear speed and $v_1=6m/s$, then $v_2$ in $m/s$ is _____. (Take $g=10m/s^2,\sqrt{19}=4.36$ )

Answer 1

Answer: 8.72

Final kinetic energy of both discs is same.
Applying conservation of energy,
$\frac{1}{2}m{v}_1^2\left(1+\frac{K^2}{R^2}\right)+mg{h}_1$
$=\frac{1}{2}m{v}_2^2\left(1+\frac{K^2}{R^2}\right)+mg{h}_2$
$\frac{v_1^2}{2}\left(1+\frac{K^2}{R^2}\right)+g{h}_1=\frac{v_2^2}{2}\left(1+\frac{K^2}{R^2}\right)+g{h}_2$
$\therefore \frac{6^2}{2}\times \frac{3}{2}+10\times 30$
$=\frac{v_2^2}{2}\times \frac{3}{2}+10\times 27$
$27+30=\frac{3}{4}{v}_2^2$
$\therefore 57=\frac{3}{4}{v}_2^2$
$v_2^2=\frac{57\times 4}{3}$
$\therefore v^2=76=2\sqrt{19}$
$\therefore v=8.72m/s$