Hey, guys. In this video, we're going to talk about rotational kinetic energy, which is the energy associated with the motion of spinning. Let's check it out. Alright. So if you remember, if you had linear speed, which is v, you had kinetic energy. Now there's going to be 2 types of kinetic energy. So we're going to specify that this is linear kinetic energy and you're used to this equation k=12mv2. I put a little l there to indicate that this is the linear type of kinetic energy. And that's because now we have a new one, which is if you have rotational speed instead of v, it's ω or omega, you have rotational kinetic energy. And instead of k_{l}, we call it k_{r}. Now the equation is very similar. It's 12. Now instead of using m, we're going to use the rotation equivalent of m, which is I moment of inertia. And instead of V, we're going to use the rotation equivalent of V, which is Omega. So I get this. Right? So if you remember the first equation, it should be easy to remember the second one. Now on a special case, there's a special situation when you're moving and rotating. So you have a v and a ω. This is called rolling motion. And one example of this is if you have a toilet paper roll that is sort of moving this way while rolling around itself. So it's a toilet paper that's rolling on the floor, has both kinds of motion. Therefore, it has both kinds of kinetic energy. So I'm going to say that the k_{total} is k_{l}+k_{r}. Cool? And the last thing I want to remind you, we'll do a quick example, is that for point masses, point masses are tiny objects that don't have a shape, that have negligible size and radius. They have no volume. The moment of inertia I is mr2, where r is a distance between the objects and the axis. Okay? Remember also that if you have a shape or a rigid body, an object with non-negligible radius and volume, we're going to get the moment of inertia from a table lookup. For example, if you have a solid cylinder or a solid disk, same thing, the equation for that is 12mr2. So point mass is always this and some sort of shape will have a different equation each time. Cool? Awesome. So let's do a very quick example here. I have a basketball player that spins a basketball around itself on top of his finger. K. So I'm going to try to draw this. It's going to come out terrible. So here's a basketball player. Here's his finger, exaggerating some stuff and here's a basketball. And he's rotating the basketball around itself, so it looks kind of like this. Basketball spinning around itself on top of your finger. Right? And it says here the ball has a mass of 0.62, a diameter of 24 centimeters, so 0.24 meters, and it spins at 15 radians per second. Radians per second is angular speed, angular velocity, omega. K. 15. And we want to know the ball's linear, rotational, and total kinetic energy. In other words, we want to know what is k_{l}, what is k_{r}, and what is k_{total}. Alright. So first things first, you may already have caught this. In physics, we never use diameter. We always use radius. So when you see diameter, you immediately convert it to radius. Radius is half, so it's 0.12. Now we're going to plug into the equation here. Kinetic energy is 12mv2 and this ball has no kinetic energy. No linear kinetic energy I should say and that's because it spins in place. It's rotating, but it's not actually moving. Right? It doesn't have it has rotational motion, but it doesn't have linear motion. It doesn't have translational motion. It just stays in place spinning around itself. So we're going to say that it has no linear kinetic energy. It does have rotational kinetic energy because it's spinning around itself and that's given by 12Iω2. Okay? Now a basketball a basketball has moments of inertia. The moment of inertia of a hollow sphere. Okay? I didn't give you the equation for that. I didn't explicitly say it was a hollow sphere, but you should know that a basketball is a shell and then there's air inside. So it is a hollow sphere. So I for a hollow sphere, you would look it up or it would be given to you, is 23mr2. So what I'm going to do is I'm going to plug that in here. 23mr2 and then omega squared, which I have. Okay. So now we can just plug in numbers. The 2 cancels with the 2 and I'm left with 13. The mass is 0.62. The radius is 0.12 squared and omega, we have it right here, 15^{2}. And if you multiply all of this, I got it here, you get 0.67 joules. 0.67 joules. And so that's it. For the last part, we want to do the total kinetic energy. Remember the total kinetic energy is just an addition of the 2 types kinetic linear plus kinetic rotational. There is no kinetic linear. So the total kinetic energy is just the 0.67 that's coming from the rotational kinetic energy. Cool? So that's how this stuff works. Hopefully this made sense. Let me know if you have any questions and let's keep going.

- 0. Math Review31m
- 1. Intro to Physics Units1h 23m
- 2. 1D Motion / Kinematics3h 56m
- Vectors, Scalars, & Displacement13m
- Average Velocity32m
- Intro to Acceleration7m
- Position-Time Graphs & Velocity26m
- Conceptual Problems with Position-Time Graphs22m
- Velocity-Time Graphs & Acceleration5m
- Calculating Displacement from Velocity-Time Graphs15m
- Conceptual Problems with Velocity-Time Graphs10m
- Calculating Change in Velocity from Acceleration-Time Graphs10m
- Graphing Position, Velocity, and Acceleration Graphs11m
- Kinematics Equations37m
- Vertical Motion and Free Fall19m
- Catch/Overtake Problems23m

- 3. Vectors2h 43m
- Review of Vectors vs. Scalars1m
- Introduction to Vectors7m
- Adding Vectors Graphically22m
- Vector Composition & Decomposition11m
- Adding Vectors by Components13m
- Trig Review24m
- Unit Vectors15m
- Introduction to Dot Product (Scalar Product)12m
- Calculating Dot Product Using Components12m
- Intro to Cross Product (Vector Product)23m
- Calculating Cross Product Using Components17m

- 4. 2D Kinematics1h 42m
- 5. Projectile Motion3h 6m
- 6. Intro to Forces (Dynamics)3h 22m
- 7. Friction, Inclines, Systems2h 44m
- 8. Centripetal Forces & Gravitation7h 26m
- Uniform Circular Motion7m
- Period and Frequency in Uniform Circular Motion20m
- Centripetal Forces15m
- Vertical Centripetal Forces10m
- Flat Curves9m
- Banked Curves10m
- Newton's Law of Gravity30m
- Gravitational Forces in 2D25m
- Acceleration Due to Gravity13m
- Satellite Motion: Intro5m
- Satellite Motion: Speed & Period35m
- Geosynchronous Orbits15m
- Overview of Kepler's Laws5m
- Kepler's First Law11m
- Kepler's Third Law16m
- Kepler's Third Law for Elliptical Orbits15m
- Gravitational Potential Energy21m
- Gravitational Potential Energy for Systems of Masses17m
- Escape Velocity21m
- Energy of Circular Orbits23m
- Energy of Elliptical Orbits36m
- Black Holes16m
- Gravitational Force Inside the Earth13m
- Mass Distribution with Calculus45m

- 9. Work & Energy1h 59m
- 10. Conservation of Energy2h 51m
- Intro to Energy Types3m
- Gravitational Potential Energy10m
- Intro to Conservation of Energy29m
- Energy with Non-Conservative Forces20m
- Springs & Elastic Potential Energy19m
- Solving Projectile Motion Using Energy13m
- Motion Along Curved Paths4m
- Rollercoaster Problems13m
- Pendulum Problems13m
- Energy in Connected Objects (Systems)24m
- Force & Potential Energy18m

- 11. Momentum & Impulse3h 40m
- Intro to Momentum11m
- Intro to Impulse14m
- Impulse with Variable Forces12m
- Intro to Conservation of Momentum17m
- Push-Away Problems19m
- Types of Collisions4m
- Completely Inelastic Collisions28m
- Adding Mass to a Moving System8m
- Collisions & Motion (Momentum & Energy)26m
- Ballistic Pendulum14m
- Collisions with Springs13m
- Elastic Collisions24m
- How to Identify the Type of Collision9m
- Intro to Center of Mass15m

- 12. Rotational Kinematics2h 59m
- 13. Rotational Inertia & Energy7h 4m
- More Conservation of Energy Problems54m
- Conservation of Energy in Rolling Motion45m
- Parallel Axis Theorem13m
- Intro to Moment of Inertia28m
- Moment of Inertia via Integration18m
- Moment of Inertia of Systems23m
- Moment of Inertia & Mass Distribution10m
- Intro to Rotational Kinetic Energy16m
- Energy of Rolling Motion18m
- Types of Motion & Energy24m
- Conservation of Energy with Rotation35m
- Torque with Kinematic Equations56m
- Rotational Dynamics with Two Motions50m
- Rotational Dynamics of Rolling Motion27m

- 14. Torque & Rotational Dynamics2h 5m
- 15. Rotational Equilibrium3h 39m
- 16. Angular Momentum3h 6m
- Opening/Closing Arms on Rotating Stool18m
- Conservation of Angular Momentum46m
- Angular Momentum & Newton's Second Law10m
- Intro to Angular Collisions15m
- Jumping Into/Out of Moving Disc23m
- Spinning on String of Variable Length20m
- Angular Collisions with Linear Motion8m
- Intro to Angular Momentum15m
- Angular Momentum of a Point Mass21m
- Angular Momentum of Objects in Linear Motion7m

- 17. Periodic Motion2h 9m
- 18. Waves & Sound3h 40m
- Intro to Waves11m
- Velocity of Transverse Waves21m
- Velocity of Longitudinal Waves11m
- Wave Functions31m
- Phase Constant14m
- Average Power of Waves on Strings10m
- Wave Intensity19m
- Sound Intensity13m
- Wave Interference8m
- Superposition of Wave Functions3m
- Standing Waves30m
- Standing Wave Functions14m
- Standing Sound Waves12m
- Beats8m
- The Doppler Effect7m

- 19. Fluid Mechanics2h 27m
- 20. Heat and Temperature3h 7m
- Temperature16m
- Linear Thermal Expansion14m
- Volume Thermal Expansion14m
- Moles and Avogadro's Number14m
- Specific Heat & Temperature Changes12m
- Latent Heat & Phase Changes16m
- Intro to Calorimetry21m
- Calorimetry with Temperature and Phase Changes15m
- Advanced Calorimetry: Equilibrium Temperature with Phase Changes9m
- Phase Diagrams, Triple Points and Critical Points6m
- Heat Transfer44m

- 21. Kinetic Theory of Ideal Gases1h 50m
- 22. The First Law of Thermodynamics1h 26m
- 23. The Second Law of Thermodynamics3h 11m
- 24. Electric Force & Field; Gauss' Law3h 42m
- 25. Electric Potential1h 51m
- 26. Capacitors & Dielectrics2h 2m
- 27. Resistors & DC Circuits2h 7m
- 28. Magnetic Fields and Forces2h 23m
- 29. Sources of Magnetic Field2h 30m
- Magnetic Field Produced by Moving Charges10m
- Magnetic Field Produced by Straight Currents27m
- Magnetic Force Between Parallel Currents12m
- Magnetic Force Between Two Moving Charges9m
- Magnetic Field Produced by Loops and Solenoids42m
- Toroidal Solenoids aka Toroids12m
- Biot-Savart Law (Calculus)18m
- Ampere's Law (Calculus)17m

- 30. Induction and Inductance3h 37m
- 31. Alternating Current2h 37m
- Alternating Voltages and Currents18m
- RMS Current and Voltage9m
- Phasors20m
- Resistors in AC Circuits9m
- Phasors for Resistors7m
- Capacitors in AC Circuits16m
- Phasors for Capacitors8m
- Inductors in AC Circuits13m
- Phasors for Inductors7m
- Impedance in AC Circuits18m
- Series LRC Circuits11m
- Resonance in Series LRC Circuits10m
- Power in AC Circuits5m

- 32. Electromagnetic Waves2h 14m
- 33. Geometric Optics2h 57m
- 34. Wave Optics1h 15m
- 35. Special Relativity2h 10m

# Intro to Rotational Kinetic Energy - Online Tutor, Practice Problems & Exam Prep

Rotational kinetic energy (K_{r}) is the energy associated with spinning motion, represented by the equation ${\frac{1}{2}}^{I}\omega \omega $. In rolling motion, both linear (K_{l}) and rotational kinetic energy contribute to total kinetic energy (K_{total} = K_{l} + K_{r}). The moment of inertia (I) varies based on the object's shape, with point masses calculated as I = m r^{2} and hollow spheres using I = (2/3) m r^{2}. Understanding these concepts is crucial for analyzing motion in physics.

### Intro to Rotational Kinetic Energy

#### Video transcript

A flywheel is a rotating disc used to store energy. What is the maximum energy you can store on a flywheel built as a solid disc with mass 8 × 10^{4} kg and diameter 5.0 m, if it can spin at a max of 120 RPM?

^{6}J

^{6}J

^{7}J

^{7}J

### Mass of re-designed flywheel

#### Video transcript

Hey, guys. So here we have a rotational kinetic energy problem of the proportional reasoning type. And what that means, it's one of those questions where I ask you, how does changing one variable affect another variable? It's one of those. Okay. So let's check it out. I'm gonna show you what I think is the easiest way to solve these. So it says you're tasked with redesigning a solid disc flywheel, and you want to decrease the radius by half. So first things first, solid disc means that the moment of inertia is half *m*r². That's the equation for a solid disc or solid cylinder. And you want the new radius, I'm going to call this *r₂*, to be half of *r₁*. And I want to know by how much mass or how much mass must the new flywheel have, so what's the new mass relative to the original mass so that you can store the same amount of energy. You want the amount of energy that you stored to be the same. The amount of energy you stored is given by *k*r, that's energy stored, right, which is given by half *I*ω². This is energy stored as rotational kinetic energy in a flywheel. You want this number not to change. You want this number to be constant, constant. Okay? So how do you do this? Well, if *r* changes if *r* changes right here, then *I* is going to change. And if *I* changes, *k* is going to change and that's bad news. So how do we change something else so that the *k* doesn't change? Well, for the *k* not to change, for the *k* not to change, you have to make sure that the *I* doesn't change. And for the *I* not to change, you have to cancel out changing *r* with changing *m*. Okay? So what I'm gonna do here is I'm gonna expand this equation, half, *I* is half *m*r²ω². So now I see all the variables that affect my *k*. And again, the *k* has to remain constant. So if my radius is becoming half as large, it means that it is decreasing by a factor of 2. Okay. So but the the *r* is squared, which means that when I reduce *r* by a factor of 2, I also have to square this. And *r* is becoming half as large, but then the whole thing, *r* squared, is becoming 4 times smaller. Okay? 4 times smaller. What that means is that if you wanna keep everything constant, my mass has to grow by a factor of 4×. Okay? So my new mass has to be 4 times my old mass, and that's the answer. Cool? So again, *r* decreases by the factor of 2, but then you have to square because there's a square here, you get a 4. If one variable decreases by 4, the other one has to increase by 4. Notice there are no squares in the *m*. So it's just a 4, not a 2. Nothing crazy like that. Cool? That's it for this one. Let me know if you have any questions.

When solid sphere 4 m in diameter spins around its central axis at 120 RPM, it has 10,000 J in kinetic energy. Calculate the sphere's mass.

^{6}kg

## Do you want more practice?

More sets### Here’s what students ask on this topic:

What is the formula for rotational kinetic energy?

The formula for rotational kinetic energy (K_{r}) is given by:

${K}_{r}=\frac{1}{2}I{\omega}^{2}$

Here, $I$ is the moment of inertia, and $\omega $ (omega) is the angular velocity. This equation is analogous to the linear kinetic energy formula ${K}_{l}=\frac{1}{2}m{v}^{2}$, where mass (m) and linear velocity (v) are replaced by their rotational counterparts.

How do you calculate the moment of inertia for different shapes?

The moment of inertia (I) depends on the shape and mass distribution of the object. For point masses, it is calculated as:

$I=m{r}^{2}$

where m is the mass and r is the distance from the axis of rotation. For rigid bodies, the moment of inertia varies:

- Solid cylinder or disk: $I=\frac{1}{2}m{r}^{2}$
- Hollow sphere: $I=\frac{2}{3}m{r}^{2}$

These values can be found in tables or derived based on the geometry of the object.

What is the difference between linear and rotational kinetic energy?

Linear kinetic energy (K_{l}) is associated with the motion of an object moving in a straight line and is given by:

${K}_{l}=\frac{1}{2}m{v}^{2}$

where m is the mass and v is the linear velocity. Rotational kinetic energy (K_{r}) is associated with the spinning motion of an object and is given by:

${K}_{r}=\frac{1}{2}I{\omega}^{2}$

where I is the moment of inertia and ω (omega) is the angular velocity. In rolling motion, both types of kinetic energy contribute to the total kinetic energy.

How do you calculate the total kinetic energy in rolling motion?

In rolling motion, an object has both linear and rotational kinetic energy. The total kinetic energy (K_{total}) is the sum of these two components:

${K}_{\mathrm{total}}={K}_{l}+{K}_{r}$

where:

${K}_{l}=\frac{1}{2}m{v}^{2}$

and

${K}_{r}=\frac{1}{2}I{\omega}^{2}$

Here, m is the mass, v is the linear velocity, I is the moment of inertia, and ω (omega) is the angular velocity. By calculating both components and adding them, you get the total kinetic energy.

What is the moment of inertia for a hollow sphere?

The moment of inertia (I) for a hollow sphere is given by:

$I=\frac{2}{3}m{r}^{2}$

where m is the mass of the sphere and r is the radius. This formula accounts for the mass being distributed on the surface of the sphere, with the interior being hollow. This is different from a solid sphere, which has a different moment of inertia due to its mass distribution.

### Your Physics tutor

- A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at...
- An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm (rev/...
- If we multiply all the design dimensions of an object by a scaling factor f, its volume and mass will be multi...
- A uniform sphere with mass 28.0 kg and radius 0.380 m is rotating at constant angular velocity about a station...
- The flywheel of a gasoline engine is required to give up 500 J of kinetic energy while its angular velocity de...
- A 140-g baseball, with a diameter of 7.5 cm, is pitched at 145 km/h. It spins at 1200 rpm. If the baseball is ...
- An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25 incline. It rolls,...
- A thin, 100 g disk with a diameter of 8.0 cm rotates about an axis through its center with 0.15 J of kinetic e...
- What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth c...
- A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. b. What is the rotat...
- Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy ...
- (III) The 1100-kg mass of a car includes four tires, each of mass 35 kg (including wheels) and diameter 0.80 m...
- (III) The 1100-kg mass of a car includes four tires, each of mass 35 kg (including wheels) and diameter 0.80 m...
- (II) A merry-go-round has a mass of 1240 kg and a radius of 7.50 m. How much net work is required to accelerat...
- (II) Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s w...
- (II) Hurricanes can involve winds in excess of 120km/h at the outer edge. Make a rough estimate of(a) the ener...
- (II) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (a) that due to...
- (II) Estimate the kinetic energy of the Earth with respect to the Sun as the sum of two terms, (b) that due to...
- (III) The 1100-kg mass of a car includes four tires, each of mass 35 kg (including wheels) and diameter 0.80 m...