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## High school physics

### Course: High school physics>Unit 7

Lesson 5: Rotational kinetic energy

# Rotational kinetic energy

David explains what rotational kinetic energy is and how to calculate it. Created by David SantoPietro.

## Want to join the conversation?

• How come the units aren't equal on both sides of the formula?
If you write out the units of K = 1/2 I 𝝎^2 in SI-units you get:
kg m^2 s^-2 = kg m^2 rad^2 s^-2
This would mean the rad unit just appears and dissapears. Is this correct? How can this be explained?
• Since no one solved your question...

Radians can disappear or reappear in an equation because the unit itself is a ratio (if I remember correctly it should be the ratio of the radius to the arc length where it is the same length as the radius [1 rad is like the 2π in 2πr=C where this time it's just 1r=part of C])].

Due to it being an equal (1:1) ratio, it effectively cancels itself out if you want it to. You can google dimensionless units for more information.

• At , why we got different masses for different points? Aren't they the same? because they are all on the same ball.
• With calculus, to get the real rotational kinectic energy, you would use all points of the baseball, each with the same infinitely small mass (considering the density is constant for the ball).
• David says that the moment of inertia is mr^2, but isn't the moment of inertia of a sphere 2/5mr^2? Does the summation result in that quantity? Or is 2/5mr^2 only used in special circumstances?
• the moment of inertia of a single point mass a distance r from the turning point is I = mr^2

for all other shapes is more complicated; the half circle is just one example
• In the baseball problem shown in this video, why did the relationship v=ωr fail?
v=40m/s but ωr=50*0.07 m/s=3.5 m/s ! Did I make any mistake here?
• In the formula v=wr, v is the tangential speed of a particular point on the baseball.
What you have calculated using the formula is the tangential speed of a point on the outermost edge of the ball for which the w is 50s^(-2) and r=0.07m.
What v=40m/s here in the question given is the translational velocity of the center of mass of the baseball
• If the total kinetic energy is the sum of both kinds of energies, does that mean that the energy I've used to throw the ball is going to be divided between translational and rotational kinetic energy? In an ideal world, if I could minimize the rotation when throwing, would the ball go faster/land further away from me?
• If you could minimize rotation without otherwise reducing the energy you put into the throw, then sure, you would be able to throw the ball harder.
Whether it would go further is a different questions. The spin of a ball causes interesting effects with air resistance and in some cases can cause a spinning ball to pass more easily through the air than a non spinning ball. Golf balls have dimples to maximize the effect of spin on the distance of the golf shot.
• How would you find the moment of inertia of a sphere using calculus? Would you use integration or derivation? And how?
• You would use integration because you are adding up a bunch of little inertias to get the total moment of inertia.
• How could be the angular velocities the same for every point ?
• I think we need to review the basic concepts of rotation here. Each point on a rotating rigid object possesses the same angular velocity but different linear velocities. If the angular velocities weren't the same at every point, the object would break apart.
• So what is the kinectic energy of the speed of the center of a wheel? is it still 1/2 mv^2 ?
(1 vote)
• How does changing the radius of an object change its rotational kinetic energy? Logically, it seems like it would increase it because KEr= (1/2)(mr^2)(angular velocity)^2 but wouldn't the radius affect the angular velocity?
(1 vote)
• Looking at the equation:
Kr = (1/2) * m * r^2 * ω^2

Without the summation portion this is the rotational kinetic energy of a small piece of the object.

The term r is not the radius of the whole object. It is the distance of the small part of the object we are looking at is away from the axis of rotation.

It is the same with the term m, it is not the mass of the whole object, it is just the mass of the small part we are looking at.

The term ω while being for the small piece we are looking at the whole object has the same angular velocity since it is a rigid object, a point 1/2 the way to the center of the object completes a rotation in the same time that a point on the surface does.

Lets assume that the size and mass of the pieces of the object we are looking at are all the same so mass is a constant.

What happens to the value of Kr for parts of the object at different distances from the axis of rotation (different values of r). For a given ω as the value of r increases the overall value of Kr increases since as r increases the part has to travel around a larger and larger circles.

Now what happens to the value of Kr if we look at different values of ω for the same value of r. As ω increases Kr increases because the piece at distance r is moving around the circle faster.

The angular velocity ω is not tied to the distance from the axis of rotation r unless you want to keep Kr constant.