# Relationship between angular momentum and moment of inertia

### Rotational inertia (article) | Khan Academy

There is a close relationship between the result for rotational energy and the energy held Due to conservation of angular momentum this process transfers angular A general relationship among the torque, moment of inertia, and angular. The Relationship Between Angular Momentum & Torque . So you're already on your way to calculating the moment of inertia for an object. Relation between Angular Momentum and Moment of inertia. When a body moves along a straight line, then the product of the, mass m and the.

Energy and angular momentum If we push on an object in the forward direction while the object is moving forward, we do positive work on the object.

The object accelerates, because we are pushing on it. It gains kinetic energy. When the speed of a car doubles, its energy increases by a factor of four.

A rotating object also has kinetic energy. When the angular velocity of a spinning wheel doubles, its kinetic energy increases by a factor of four.

### Rotational kinetic energy and angular momentum

When an object has translational as well as rotational motion, we can look at the motion of the center of mass and the motion about the center of mass separately. The total kinetic energy is the sum of the translational kinetic energy of the center of mass CM and the rotational kinetic energy about the CM. Rolling Consider a wheel of radius r and mass m rolling on a flat surface in the x-direction.

Suppose you are designing a race bicycle and it comes time to work on the wheels.

You are told that the wheels need to be of a certain mass but you may design them either as wheels with spokes like traditional bike wheels or you may make them as having solid rims all the way through. Which design would you pick given that the racing aspect of the machine is the most important?

By clicking the button below, you can play or to step through a video clip frame-by-frame. The torque is the product of a weight and a small lever arm. The moment of inertia of the ruler-like object changes because masses are added at larger distances away from the center. Compare the angles through which the ruler turns per step without attached masses and with masses attached at different locations.

Three particles are connected by rigid rods of negligible mass lying along the y-axis as shown. Here ri is the perpendicular distance of particle i from the x-axis. The sum of the kinetic energies of the three particles is J.

Angular momentum about an axis is a measure of an objects rotational motion about this axis. A light rod 1 m in length rotates in the xy plane about a pivot through the rod's center.

Two particles of mass 4 kg and 3 kg are connected to its ends.

### MyRank: Relation between Angular Momentum and Moment of inertia

We assume that the mass and moment of inertia of the rod can be neglected. Let the z-axis pass through the center of the rod and point out of the page. Here k is a unit vector or direction indicator pointing in the z-direction out of the page.

Angular momentum is a vector. For a single particle its direction is the direction of the angular velocity given by the right hand rule. In the same way that linear momentum is always conserved when there is no net force acting, angular momentum is conserved when there is no net torque.

If there is a net force, the momentum changes according to the impulse equation, and if there is a net torque the angular momentum changes according to a corresponding rotational impulse equation. Angular momentum is proportional to the moment of inertia, which depends on not just the mass of a spinning object, but also on how that mass is distributed relative to the axis of rotation. This leads to some interesting effects, in terms of the conservation of angular momentum.

A good example is a spinning figure skater.

Consider a figure skater who starts to spin with their arms extended. When the arms are pulled in close to the body, the skater spins faster because of conservation of angular momentum.

## Angular momentum

Pulling the arms in close to the body lowers the moment of inertia of the skater, so the angular velocity must increase to keep the angular momentum constant.

Parallels between straight-line motion and rotational motion Let's take a minute to summarize what we've learned about the parallels between straight-line motion and rotational motion. Essentially, any straight-line motion equation has a rotational equivalent that can be found by making the appropriate substitutions I for m, torque for force, etc.

Example - Falling down You've climbed up to the top of a 7. Just as you reach the top, the pole breaks at the base. Are you better off letting go of the pole and falling straight down, or sitting on top of the pole and falling down to the ground on a circular path? Or does it make no difference? The answer depends on the speed you have when you hit the ground. The speed in the first case, letting go of the pole and falling straight down, is easy to calculate using conservation of energy: In the second case, also apply conservation of energy.

If you have negligible mass compared to the telephone pole, just work out the angular velocity of the telephone pole when it hits the ground. In this case we use rotational kinetic energy, and the height involved in the potential energy is half the length of the pole which we can call hbecause that's how much the center of gravity of the pole drops. So, for the second case: Solving for the angular velocity when the pole hits the ground gives: For you, at the end of the pole, the velocity is h times the angular velocity, so: So, if you hang on to the pole you end up falling faster than if you'd fallen under the influence of gravity alone.