Ch+8+-+Rotational+Dynamics

This chapter contains the second half of the information on rotation. The topics it deals specifically with are:
- the new quantity, torque - equilibrium - center of gravity - rotational inertia - moment of inertia & rotational inertia - rotational kinetic energy - angular momentum

Important Equations:

Torque=Force*Distance

Center of Gravity- cg=m* x (or y)/sum of the masses

Rotational Inertia- I= mr^2

Rotational KE= 1/2*I*w^2

Torque is the tendency of a force to rotate an object about some axis. Rotational inertia depends on mass and the location of the mass. The angular momentum of an object will change when a net torque is applied to the object. Torque is measured in Nm and inertia is measured in kg*m^2.

Equilibrium is a state in which an object either remains motionless or remains in motion at a constant speed. It is described as a condition in which both the sum of the torques and the sum of the forces in a system are equal to zero.

In a system of fluctuating or unbalanced mass distribution, the center of gravity for the system (on an x-y plane) can be described as the sum of the products of each mass and its corresponding x-position. (This equation is mirrored for the y-dimension if it is involved, as well as (theoretically) the z-dimension): Here is the formula:

Returning to our original lessons on Torque, we can relate this to the next topics discussed in the chapter, which are known as Rotational Inertia and the Moment of Inertia. Normally, the force of Rotational Inertia acts on each particle in a system individually, based on each ones' mass and distance from the center of the system. However, in, say, a rotating disk, each particle has the same rotational acceleration. Therefore, unlike in the Torque equation, Angular Acceleration is not taken into account, as it cancels. We're then left with the sum of each mass and it's radius squared, which is the formula for the Moment of Inertia, or in layman's terms, mass distribution. Rotational Inertia Moments also vary between types of objects.

Moving on, we will now discuss the Energy and Momentum found in Rotational Systems.

In Linear Systems, Kinetic Energy is simply the product of the mass and it's squared velocity.

In Rotational Systems, the Kinetic Energy is shown as the sum of these values throughout the system. As we've covered, linear velocity can be displayed as the product of the rotational velocity and the system's radius. Applying this to the system, we are left with 1/2 of the sum of the masses and their radii squared, times rotational velocity squared. Because the Moment of Inertia is the sum of the masses times their radii squared, the formula simplifies to 1/2*I*w^2, or 1/2 the Moment of Inertia times Rotational Velocity squared.

As a final topic, we will now discuss Angular Momentum. Like each of our other mirrored subtopics, Angular Momentum is simply momentum in a rotational sense. Where Linear Momentum is described as the product of mass and velocity, Angular Momentum is described as being the product of the previously described Inertia and Angular Velocity. We can relate Angular Momentum to Torque in that an object, when put under a constant Torque, will increase in angular speed over a period of time. In this, we gain another formula.

Momentum, in its Angular form, is conserved throughout a system's actions. The Conservation of Momentum in a Rotational System applies equal amounts of Angular Momentum ONLY in the absence of external Torque. Therefore, we must consider that for Conservation of Angular Momentum to work in any system, the Sum of the Torques must be zero.