Circular Motion, Centripital Forces and Rotation Spinning and Twisting Things

To this point, we've discussed things moving in circular motion, i.e., with centripital forces acting upon them. From our equation for centripital acceleration, we see that:

ac = v2/r or v =sqrt (rac)

As the distance away from the center is increased, the speed is increased, for any given centripital force. Imagine a spinning CD. Even though every point farther away from the center moves faster, every point makes the complete revolution in the same amount of time. We call this angular velocity. Each point on the CD moves through the same angle measure in a given time. Angular velocity, w (lower case omega), is given by:

w = q / t

The relationship between angular velocity and tangential velocity (what we've been calling velocity) is derived from the relation of the arclength to the radius of a circle. The length of an arc is given by:

s = r q

where s is the arclength, r is the radius of the circle, and q is the angle defining the arc, measured in radians. I repeat, measured in RADIANS! Tangential velocity (vT) is given by the displacement d traveled in a certain time t. As the angle gets smaller and smaller (i.e., the limit as theta goes to zero), d becomes closer and closer to s. Thus we have:

d = r q

q = d/r

w = d/rt

w = vT/r or vT = w r

Similarly, we can define an angular acceleration, a, for objects whose rate of spin is increasing or decreasing.

a = w / t = vT/rt = aT/r

Previously when talking about acceleration, we incorporated mass. Mass is a measure of the inertia, the resistance to a change in motion. You know that depending on shape, an object is harder or easier to spin. The axis about which the item is spun also affects the difficulty in getting it to spin. This concept, while related to mass, is obviously somewhat more complicated. We call this moment of inertia, represented by the letter I. In general, I has to do with the way that mass is distributed, or:

I = S miri2

For honors, we can use calculus to accomplish this for nearly any situation.

We can find the moment of inertia for some simple cases. For example, consider a lightweight bar (length 2R) with two heavy masses (mass M each) attached at the ends (i.e., a barbell). For the moment, we'll not consider the mass of the bar. If we turn the bar around its middle, each mass turns at a distance R from the center. So we have:

I = MR2 + MR2

I = 2MR2

The moment of inertia for the bar should also be included. For a thin rod about an axis through its center perpendicular to its length,

I = ML2/12

or in our case,

I = M4R2/12 = MR2/3

so the total moment of inertia for this system is:

I = 2MR2 + MR2/3 = 2.3 MR2

We've already seen several rotational quantities. It turns out that nearly every topic covered so far this year has a rotational equivalent. See the table below:

Straight Line Motion

Rotational Motion

displacement, x

angular displacement, q

velocity, v

angular velocity, w

acceleration, a

angular acceleration, a

mass, m

moment of inertia, I

KEtranslational, 1/2(mv2)

KErotational, 1/2(Iw2)

force, F = ma

torque, t = I a

momentum, p = mv

angular momentum, L = I w

The rotational kinetic energy can be used to figure out the speeds of things rolling down hills via energy conservation. The angular momentum can be used to explain things such as gyroscopes and phenomena such as why ice skaters move faster as they pull their arms inward.

Just as force is the most fundamental thing in straight line motion, torque is most fundamental in rotation. A torque is a twisting force, like that applied by a wrench to a nut, or your hand to a door. If you consider the case of trying to close a door, you can see that it is easier to do so by pushing farther away from the hinges. You can also see that if you push the door toward the hinges, you won't ever close the door.

The distance away from the axis the object is spinning about is called the moment arm. Torque, then, is given by:

t = F r

or more correctly,

t = F r cosq

where t is (tau) is the symbol for torque, F is the applied force, r is the length of the moment arm, and the angle is the angle between the force and the moment arm. We can easily relate torque to angular motion. We know that:

F = ma and a = a r

so F = m a r

and t = m a r r

t = m r2 a

t = I a

We can also discuss how torques can be conserved, making a situation of rotational equilibrium.