Question

A solid sphere with a moment of inertia of 2.20 kgm2 is rolling along the ground (without slipping) with an angular velocity of 15.1 rad/s . Find (a) its rotational kinetic energy and (b) its translational kinetic energy.

Answer 1

Explanation:

It is given that,

Moment of inertia of a solid sphere, $I=2.2\ kg-m^2$

Angular velocity of the solid sphere, $\omega=15.1\ rad/s$

(a) The rotational kinetic energy of the solid sphere is given by :

$K_r=\dfrac{1}{2}I\omega^2$

$K_r=\dfrac{1}{2}\times 2.2\times (15.1)^2$

$K_r=250.811\ J$

(b) The translational kinetic energy of the solid sphere is given by :

$K_t=\dfrac{1}{2}mv^2$

Moment of inertia of solid sphere is :

$I=\dfrac{2}{5}mr^2$

$K_t=\dfrac{1}{2}m(r \omega)^2$

$K_t=\dfrac{5 I\omega^2}{4}$

$K_t=\dfrac{5 \times 2.2\times (15.1)^2}{4}$

$K_t=627.02\ J$

Hence, this is required solution.

Answer 2

a) The rotational kinetic energy of the solid sphere is 250.811 joules.

b) The translational kinetic energy of the solid sphere is 627.028 joules.

The principle of energy conservation, definitions of kinetic energy and moment of inertia and the D'Alembert's principle we can derive the mass (m), in kilograms, and the rotational and translational kinetic energy ($K_{R}$, $K_{T}$) of the solid sphere as well.

The rotational and translational kinetic energy are determined by the following formulae:

Rotational kinetic energy

$K_{R} = \frac{1}{2}\cdot I\cdot \omega^{2}$   (1)

Translational kinetic energy

$K_{T} = \frac{1}{2}\cdot \left(\frac{5\cdot I}{2\cdot R^{2}} \right)\cdot R^{2}\cdot \omega^{2}$

$K_{T} = \frac{5}{4}\cdot I\cdot \omega^{2}$   (2)

Where:

• I - Moment of inertia of the solid sphere, in kilogram-square meters.
• ω - Angular speed of the solid sphere, in radians per second.

a) If we know that $I = 2.20\,kg\cdot m^{2}$ and $\omega = 15.1\,\frac{rad}{s}$, then the rotational kinetic energy is:

$K_{R} = \frac{1}{2}\cdot (2.20\,kg\cdot m^{2})\cdot \left(15.1\,\frac{rad}{s} \right)^{2}$

$K_{R} = 250.811\,J$

The rotational kinetic energy of the solid sphere is 250.811 joules. $\blacksquare$

b) If we know that $I = 2.20\,kg\cdot m^{2}$ and $\omega = 15.1\,\frac{rad}{s}$, then the rotational kinetic energy is:

$K_{T} = \frac{5}{4}\cdot (2.20\,kg\cdot m^{2})\cdot \left(15.1\,\frac{rad}{s} \right)^{2}$

$K_{T} = 627.028\,J$

The translational kinetic energy of the solid sphere is 627.028 joules. $\blacksquare$

To learn more on kinetic energy, we kindly invite to check this verified question: https://brainly.com/question/999862