10.5 Angular Momentum and Its Conservation

The result is valid for any solid cylinder, implying that all solid cylinders will roll down an incline at the same rate.

If the cylinder slid down the incline without rolling, the entire potential energy would go into translation.

The cylinder would go quicker at the bottom.

An example of each type of energy is given.

Yes, the two main types of energy are rotational and translational.

The energy of motion is involved with the coordinated movement of mass relative to a reference frame.
Straight line motion is the only difference between the two.
There is a bike tire with both a bike tire and a bike tire on it.
The movement of the bike along the path and the rotation of the tire mean they both have the same energy.
If you lift the front wheel of the bike and spin it while the bike is stationary, the wheel would have only a small amount of energy relative to the Earth.

You can build your own heavenly bodies and watch the ballet.

You can set initial positions, velocities, and mass of 2, 3, or 4 bodies, and then see them in motion.

The answers to questions like these are based on the movement of the body.

The pattern is clear by now.

The definition of linear momentum is an analog to this equation.

An object with a large moment of inertia, such as Earth, has a very large angular momentum.
An object that has a large angular velocity, such as a centrifuge, also has a large angular momentum.

Linear momentum was first presented in.

It has the same implications in terms of carrying rotation forward when the net external Torque is zero.
The atoms and particles have a similar property to linear momentum.

We have to look up data before we can calculate the problem.

Earth's mass is and its radius is.

The Earth has one revolution per day, but we have to covert to radians per second to do the calculation.

The number is large and shows that Earth has a strong momentum.

We assumed a constant density for Earth in order to estimate its moment of inertia.

You exert a Torque when you push a merry-go-round, spin a bike wheel, or open a door.

If the Torque you exert is greater than the opposing Torque, the rotation will accelerate.

The relationship between force and linear momentum is similar to this expression.

The equation is applicable in many ways.
The rotational form of the second law is what it is.

A person is looking at a food tray while trying to get sustenance.

A partygoer exerts a Torque on a lazy Susan.

The equation shows the relationship between the two.

The given information can be used to calculate the Torque.

The lazy Susan starts from rest because the final angular momentum is the same as the change in angular momentum.

The final velocity is determined from the definition of in.

The property of the object doesn't matter when it comes to the imparted angular momentum.

The determination of the time period is left as an exercise for the reader, which is about right for a lazy Susan.

The lever arm is 2.20 cm.

The moment of inertia of the lower leg is a good place to start.

The muscles in the upper leg give the lower leg a force that can be felt in the knee.

The situation is examined in this example.

The second law can be used to find the angular acceleration.

The moment of inertia can be found easily from the given force and lever arm.
The final angular velocity and rotational kinetic energy can be calculated once the acceleration is known.

The values are reasonable for a person kicking his leg.

When the pivot in the knee is directly beneath the center of gravity of the lower leg, the weight of the leg can be neglected.
The force that the upper leg exerts is larger than that created by the weight of the lower leg as it rotates.
The lower leg has enough energy to give a ball a significant velocity by transferring some of it in a kick.

It is similar to energy and linear momentum.

Another sign of underlying unity is this universally applicable law.
When the net external force is zero, linear momentum is the same as it is when the net external force is zero.

We now know why Earth keeps spinning.

In the previous example,.
The equation says that a Torque must act over a period of time.
A large Torque is needed to change the Earth's rate of spin.
It will take tens of millions of years before the change in Earth's rotation is significant.
The length of the day is thought to have been 18 h some 900 million years ago.
The tides retard the spin on Earth, and so it will continue to spin for billions of years.

There is another law here.

If the net Torque is zero, the momentum is constant or conserved.
We can see this by looking at the situation in which the net Torque is zero.

There are as many conservativism laws as there are important.

The net Torque on her is very close to zero, because there is relatively little friction between her skates and the ice and because of the close proximity to the pivot point.

She can spin for a long time.
She can do more than that.
She can increase her spin rate by pulling her arms and legs in.

The moment of inertia equation is expressed in terms of the primed quantities, where they refer to conditions after she has pulled in her arms.

To keep the momentum constant, the angular velocity must increase.
The following example shows how dramatic the change can be.

The net Torque on her is small.

Her moment of inertia decreases when she pulls in her arms.
The work she does increases the amount of energy in her body.

She has a moment of inertia with her arms extended and close to her body.

We are looking for the skater's speed after she has pulled in her arms.

The moments of inertia and initial angular velocity are given to find this quantity.

The conserved of angular momentum is applicable because of the negligible amount of Torque.

There is an increase in both parts.

Most world-class skaters can achieve spin rates about this great, but the final angular velocity is large.
The final energy is more than the initial energy.
The skater pulling in her arms increases the amount of energy in the air.
This work depletes the skater's food energy.

Some objects increase their rate of spin because of something.

One example is tornadoes.

There are storm systems that create tornadoes.
Another example is Earth.
Our planet was born from a huge cloud of gas and dust, the rotation of which came from turbulence in an even larger cloud.
The rotation rate increased as a result of the contracting cloud.

The solar system coalesced from a rotating cloud of gas and dust.

The motions and spins of the planets are the same as the original spins of the parent cloud.

When a body interacts with the environment as it pushes its foot off the ground, one wouldn't expect the movement of the body's axis of motion to be conserved.

If a person is motionless, the International Space Station has no angular momentum relative to the inside of the ship.
If they don't push themselves off the side of the vessel, their bodies will continue to have zero value.

Both linear and angular momentums are very similar.