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 $ν_1$ and $ν_2$, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and $ν_1 = 3m/s$, then $ν_2$ in m/s is $(g = 10 m/s^2)$

Answer 1

For $1^{st}$ body (Applying conservation of energy)
$ \frac{1}{2}mν^{2}_{1} \left(1+\frac{k^{2}}{R^{2}}\right) = mg \left(30\right) + \frac{1}{2}mν^{2} \left(1+\frac{k^{2}}{R^{2}}\right)\quad\quad\dots\dots\left(i\right)$
For $2^{nd}$ body (Applying conservation of energy)
$\frac{1}{2}mν^{2}_{2} \left(1+\frac{k^{2}}{R^{2}}\right) = mg\left(27\right)+\frac{1}{2}mν^{2} \left(1+\frac{k^{2}}{R^{2}}\right)\quad\quad\dots\dots\left(ii\right)$
equation $\left(i\right) \& \left(ii\right)$
$\frac{1}{2} m \left(ν^{2}_{1}-ν^{2}_{2}\right) \left(1+\frac{k^{2}}{R^{2}}\right) = mg \left(3\right)$
$\frac{1}{2}\left(9-ν^{2}_{2}\right) \left(1+\frac{1}{2}\right) = 30$
$\left(9-ν^{2}_{2}\right)\times\frac{3}{4} = 30$
$9-ν^{2}_{2} = 40$
$ν_{2} = 7\,m/s$