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 ${\nu }_1$ and ${\nu }_2$, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and ${\nu }_1=3m/s$, then ${\nu }_2$ in m/s is $(g=10m/s^2)$

Answer 1

Answer: 7

For $1^{st}$ body (Applying conservation of energy)
$\frac{1}{2}m{\nu }_1^2\left(1+\frac{k^2}{R^2}\right)=mg\left(30\right)+\frac{1}{2}m{\nu }^2\left(1+\frac{k^2}{R^2}\right)\dots \dots \left(i\right)$
For $2^{nd}$ body (Applying conservation of energy)
$\frac{1}{2}m{\nu }_2^2\left(1+\frac{k^2}{R^2}\right)=mg\left(27\right)+\frac{1}{2}m{\nu }^2\left(1+\frac{k^2}{R^2}\right)\dots \dots \left(ii\right)$
equation $\left(i\right)\&\left(ii\right)$
$\frac{1}{2}m\left({\nu }_1^2-{\nu }_2^2\right)\left(1+\frac{k^2}{R^2}\right)=mg\left(3\right)$
$\frac{1}{2}\left(9-{\nu }_2^2\right)\left(1+\frac{1}{2}\right)=30$
$\left(9-{\nu }_2^2\right)\times \frac{3}{4}=30$
$9-{\nu }_2^2=40$
${\nu }_2=7m/s$