# Relationship between angular momentum and energy

### Angular momentum - Wikipedia

A rolling object has both translational and rotational kinetic energy. Angular momentum is also a vector, pointing in the direction of the. In Newtonian particle dynamics the covariance of the kinetic energy equation to transformation between different inertial frames may he demonstrated only if the. In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a.

One needs to make this distinction because without it, the sign of the cross product would be meaningless. From the above, it is possible to reformulate the definition to either of the following: For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector: Conservation of angular momentum The torque caused by the two opposing forces Fg and -Fg causes a change in the angular momentum L in the direction of that torque since torque is the time derivative of angular momentum.

Relation b/w kinetic energy and momentum

This causes the top to move back upright. In a closed system, angular momentum is constant. This conservation law follows mathematically from what is called the "continuous directional symmetry" of space—that is, no direction in space is any different from any other direction. The change of angular momentum over time is called torque.

Mathematically, it is expressed as the time derivative of angular momentum, as follows: When the angular momentum is a constant for a closed systemthe mathematical expression for that is equivalent to the mathematical equation showing that the external torque acting on the system is zero. This equivalence may be written as follows: Angular momentum of planetary orbits For a planet in orbit, the angular momentum is distributed between the spin of the planet itself and the angular momentum of its orbit: Usefulness The conservation of angular momentum is used extensively in analyzing what is called central force motion.

If the net force on some body is always directed toward a fixed point, the center, then there is no torque on the body with respect to the center, and the angular momentum of the body about the center is constant.

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.

## Angular momentum

When the arms are pulled in close to the body, the skater spins faster because of conservation of 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.

### Angular momentum - New World Encyclopedia

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: 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.

## Angular momentum and energy

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.

The angular momentum of an object is changed by giving it an angular impulse. You usually give your closet door a gentle push and it swings closed gently in 5 seconds. But today you are in a rush and exert 3 times the normal torque on it. If you increase the force by a factor of 3 and push for the same amount of time, the angular momentum increases by a factor of three.