Question 419: Centre of Mass and Rotational Motion

Question

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is $$\frac{x}{5}$$. The value of $$x$$ is ___________.

Accepted Answer

For a rolling spherical shell, we must consider the fact that it has both translational and rotational kinetic energy. The total kinetic energy ($K_{total}$) can be expressed as the sum of the translational kinetic energy ($K_{trans}$) and the rotational kinetic energy ($K_{rot}$):
$$K_{total} = K_{trans} + K_{rot}$$
The translational kinetic energy of an object with mass (m) and linear velocity (v) is given by:
$$K_{trans} = \frac{1}{2}mv^2$$
The rotational kinetic energy of a rolling spherical shell with moment of inertia (I) and angular velocity (ÃÂ) is given by:
$$K_{rot} = \frac{1}{2}IÃÂ^2$$
For a rolling object without slipping, the relationship between linear velocity (v) and angular velocity (ÃÂ) is:
$$v = RÃÂ$$
Where R is the radius of the spherical shell.
The moment of inertia for a spherical shell is given by:
$$I = \frac{2}{3}mR^2$$
Now, we can substitute the moment of inertia and the relationship between linear and angular velocity into the equation for rotational kinetic energy:
$$K_{rot} = \frac{1}{2}\left(\frac{2}{3}mR^2\right)\left(\frac{v}{R}\right)^2$$
Simplifying the equation:
$$K_{rot} = \frac{1}{2}\left(\frac{2}{3}mR^2\right)\frac{v^2}{R^2}$$
$$K_{rot} = \frac{1}{3}mv^2$$
Now, we can find the ratio of rotational kinetic energy to total kinetic energy:
$$\frac{K_{rot}}{K_{total}} = \frac{\frac{1}{3}mv^2}{\frac{1}{2}mv^2 + \frac{1}{3}mv^2}$$
Simplifying the equation:
$$\frac{K_{rot}}{K_{total}} = \frac{\frac{1}{3}}{\frac{1}{2} + \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{5}{6}}$$
Multiplying both the numerator and the denominator by 6:
$$\frac{K_{rot}}{K_{total}} = \frac{2}{5}$$
Comparing this to the given ratio of $$\frac{x}{5}$$, we can determine that the value of $$x$$ is 2.

Centre of Mass and Rotational Motion

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is $\frac{x}{5}. The value of x$ is ___________.

Integer / Numeric Answer Type