4.18 Rotational Kinetic Energy

Introduction

Rotational kinetic energy is the energy an object possesses due to its rotation. Just as an object moving in a straight line has translational kinetic energy ($K.E{.}_{trans}=\frac{1}{2}m{v}^2$), an object spinning about an axis has rotational kinetic energy. This form of energy is crucial for analyzing the motion of any rotating system, from a spinning planet to a rolling ball. It is measured in Joules (J), which is a Derived Unit→.

Definition and Formula

The rotational kinetic energy ($K.E{.}_{rot}$) of a rigid body is defined by its moment of inertia ($I$) and its angular velocity ($\omega$).

Formula: $K.E{.}_{rot}=\frac{1}{2}I{\omega }^2$

This formula is the rotational analog of the translational kinetic energy formula, with:

• Moment of Inertia ($I$) being the rotational equivalent of mass ($m$)
• Angular Velocity ($\omega$) being the rotational equivalent of linear velocity ($v$)

Derivation for a Rigid Body

A rigid body can be considered as a collection of many small particles of mass $m_i$ at a distance $r_i$ from the axis of rotation. All particles in the rigid body rotate with the same angular velocity, $\omega$.

1. The translational kinetic energy of a single particle $i$ is $K.E{.}_i=\frac{1}{2}{m}_i{v}_i^2$.
2. The tangential velocity of this particle is related to the angular velocity by $v_i=r_i\omega$.
3. Substituting for $v_i$, the kinetic energy of the particle is $K.E{.}_i=\frac{1}{2}{m}_i(r_i\omega {)}^2=\frac{1}{2}{m}_i{r}_i^2{\omega }^2$.
4. The total rotational kinetic energy of the entire body is the sum of the kinetic energies of all its particles: $K.E{.}_{rot}=\sum K.E{.}_i=\sum \left(\frac{1}{2}{m}_i{r}_i^2{\omega }^2\right)$
5. Since $\frac{1}{2}$ and ${\omega }^2$ are constant for all particles, we can factor them out: $K.E{.}_{rot}=\frac{1}{2}\left(\sum m_i{r}_i^2\right){\omega }^2$
6. The term in the parentheses is the definition of the moment of inertia, $I=\sum m_i{r}_i^2$.
7. This gives us the final formula: $K.E{.}_{rot}=\frac{1}{2}I{\omega }^2$

Application: Rolling Objects on an Inclined Plane

When an object rolls without slipping down an incline, its initial potential energy ($P.E.=mgh$) is converted into both translational and rotational kinetic energy.

Total Kinetic Energy of a Rolling Object: $K.E{.}_{total}=K.E{.}_{trans}+K.E{.}_{rot}=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$

By the conservation of energy, the initial potential energy equals the final total kinetic energy: $mgh=\frac{1}{2}m{v}^2+\frac{1}{2}I{\omega }^2$

For a rolling object, we also know that $v=r\omega$, so $\omega =v/r$.

Case 1: A Rolling Solid Disc

• Moment of Inertia of a Disc: $I_{disc}=\frac{1}{2}m{r}^2$
• Energy Conservation: $mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\left(\frac{1}{2}m{r}^2\right){\left(\frac{v}{r}\right)}^2$ $mgh=\frac{1}{2}m{v}^2+\frac{1}{4}m{v}^2=\frac{3}{4}m{v}^2$
• Final Velocity of the Disc: $v_{disc}=\sqrt{\frac{4}{3}gh}\approx 1.15\sqrt{gh}$

Case 2: A Rolling Hoop (or Ring)

• Moment of Inertia of a Hoop: $I_{hoop}=m{r}^2$
• Energy Conservation: $mgh=\frac{1}{2}m{v}^2+\frac{1}{2}(m{r}^2){\left(\frac{v}{r}\right)}^2$ $mgh=\frac{1}{2}m{v}^2+\frac{1}{2}m{v}^2=m{v}^2$
• Final Velocity of the Hoop: $v_{hoop}=\sqrt{gh}$

Comparison

By comparing the final velocities, we see that $v_{disc}>v_{hoop}$.

• Conclusion: The solid disc will reach the bottom of the incline before the hoop.
• Reason: The hoop has a larger moment of inertia (its mass is distributed farther from the center), so more of the initial potential energy is converted into rotational kinetic energy and less is converted into translational kinetic energy. The disc, with a smaller moment of inertia, "invests" less energy into rotating and more into moving forward, so it travels faster.

Possible Questions and Answers

Q: What determines an object's moment of inertia?

A: The moment of inertia depends on two factors: the total mass of the object and the distribution of that mass relative to the axis of rotation. The farther the mass is from the axis, the larger the moment of inertia.

Q: Can an object have rotational kinetic energy but no translational kinetic energy?

A: Yes. An object spinning in place (like a stationary flywheel or a spinning top that is not moving across the floor) has rotational kinetic energy, but its center of mass is not moving, so its translational kinetic energy is zero.