7:3 Periodic and Circular Motion

This is motion of objects as they travel in a circle; twirling a rock on the end of a string, a car going around a race track, riding a Ferris Wheel, the earth orbiting the sun......

There are two types of uniform circular motion:

• (1) A horizontal circle; the net acceleration is constant.
• (2) A vertical circle; the net acceleration is not constant

Lets use the example of twirling a rubber stopper in a horizontal (level) circle on the end of a string.

• The stopper will have a centripetal (centripetal = "center-seeking" )acceleration of a_c , where  .
• v is the velocity of the stopper; if the string breaks this value becomes V_xi .
• The distance the object travels in one revolution is  , and since v = d/t, the velocity of the stopper is  , where T is the period of revolution, the time it takes to complete one revolution.
• Then substituting the previous value for velocity,  a_c can be expressed as  .
• The stopper will experience a force called the centripetal force, F_c , or F_c = ma_c .
• If you substitute  , you get  ;
• and if you substitute  for a in the F = ma formula, you get  .
   

 Note then, that a centripetal force is necessary to keep an object accelerating centripetally. Can you think of any possible uses for this in space?

Lets take a look at how it applies to keeping a car going around a flat, circular course.

This section deals with motion in which the force varies directly with displacement; such as Hook's Law
F = kx, where F is the applied force, k is a constant of proportionality (dependent on the configuration of the system), and x is the distance the object moves. Other examples are a pendulum, a child on a swing, a guitar string, a spring....etc.

The period, T, of a pendulum is  , where l is the length of the string (or more precisely, as measured from the center of mass of the system) , and g is the acceleration due to gravity. A pendulum is used to measure the gravity at
different locations; this is an important tool in the search for minerals.
Check this website out: A Pendulum Clock 

Rotational Motion

Rotary motion is the motion of a body about an internal axis. Examples are a spinning bicycle wheel, the spinning crankshaft of a car, and a cheerleader doing a cartwheel.

• In Chapter 3, we studied the motion of an object in a straight line (moving back and forth).
• Next we studied the motion of this object in two-dimensions (moving back and forth and also up and down).
• Then we studied the motion of this object as it moved in a circle (around a track).
• In this section we will study the motion of this object as it spins around.

In the previous chapters we expressed linear displacement in terms of mm, cm, m, and km.
In this section we define angular displacement , (theta), the measure of the angle in which the object moves through. The units of angular displacement are revolutions, degrees, and radians. Angular velocity, (omega), is the time rate of angular displacement or  .

Angular acceleration , (alpha), is the rate of change of angular velocity or  .

Rotational Inertia:

• Rotational inertia is the resistance of a rotating object to changes in its angular velocity. The angular acceleration is directly proportional to the torque, T , but inversely proportional to the rotational inertia, I .

, or, which for rotary motion is analogous to F = ma for linear motion.

For linear motion, inertia is equal to the mass of the object. For rotary motion, inertia must take into account both shape and mass; therefore different shapes have different rotational inertia.

Homework: set #4 Ch 7 questions 10-14 page 170

Set #5 Ch 7 problems 49-55, page 172-173