Question

a disc and a ring of same mass are rolling down fritionless inclined plane with same kinetic energy .the ratio of velocity of disc to velocity of ring is ??????????????????????

Answer 1

a disc and a ring of same mass are rolling down frittionless inclined plane with same kinetic energy. the ratio of velocity of disc to velocity of ring is ?

Solution:

Kinetic energies of the ring and disc are equal:

$$E_{\text{disc}} = E_{\text{ring}}$$

The kinetic energy of rotational motion:

$$E = \frac{J \cdot \omega^2}{2} + \frac{m \cdot \vartheta^2}{2}$$

, where $J$ - moment of inertia, $\omega = \frac{\vartheta}{r}$ - angular velocity, $r$ - radius, $m$ - mass.

Moment of inertia of the disc and the ring:

$$\begin{aligned} J_{\text{disk}} &= \frac{m r_{\text{disc}}^2}{2} \Rightarrow \\ E_{\text{disc}} &= \frac{m r_{\text{disc}}^2}{2 \cdot 2} \cdot \omega_{\text{disc}}^2 + \frac{m \cdot \vartheta_{\text{disc}}^2}{2} = \frac{m r_{\text{disc}}^2}{4} \left(\frac{\vartheta_{\text{disc}}}{r_{\text{disc}}}\right)^2 + \frac{m \cdot \vartheta_{\text{disc}}^2}{2} = \frac{m \vartheta_{\text{disc}}^2}{4} + \frac{m \cdot \vartheta_{\text{disc}}^2}{2} \\ &= \frac{3m \cdot \vartheta_{\text{disc}}^2}{2} \end{aligned}$$ $$\begin{aligned} J_{\text{ring}} &= m r^2 \Rightarrow \\ E_{\text{ring}} &= \frac{m r_{\text{ring}}^2}{2} \cdot \omega_{\text{ring}}^2 + \frac{m \cdot \vartheta_{\text{ring}}^2}{2} = \frac{m r_{\text{ring}}^2}{4} \left(\frac{\vartheta_{\text{ring}}}{r_{\text{ring}}}\right)^2 + \frac{m \cdot \vartheta_{\text{ring}}^2}{2} = \\ &= \frac{m \vartheta_{\text{ring}}^2}{2} + \frac{m \cdot \vartheta_{\text{ring}}^2}{2} = m \cdot \vartheta_{\text{ring}}^2 \end{aligned}$$

(3) and (2) in (1):

$$\begin{aligned} \frac{3m \cdot \vartheta_{\text{disc}}^2}{2} &= m \cdot \vartheta_{\text{ring}}^2 \\ \frac{\vartheta_{\text{disc}}^2}{\vartheta_{\text{ring}}^2} &= \frac{2}{3} \\ \frac{\vartheta_{\text{disc}}}{\vartheta_{\text{ring}}} &= \sqrt{\frac{2}{3}} = 0.8 \end{aligned}$$

Answer: the ratio of velocity of disc to velocity of ring is 0.8.