Friday, October 16, 1998 Chapter 7: periods and frequencies Centripetal acceleration & force Newton’s Law of Gravity

Hint: Be able to do the homework (both the problems to turn in AND the recommended ones) you’ll do fine on the exam! Friday, October 23, 1998 in class Chapters inclusive You may bring one 3”X5” index card (hand-written on both sides), a pencil or pen, and a scientific calculator with you. I will put any constants, math, and Ch formulas which you might need on a single page attached to the back of the exam.

Angular Velocity Angular Acceleration

For an object moving in a circle with a constant angular velocity (  ), we can define a frequency and period associated with the motion. The period of the rotation is simply the time required for the object to go around the circle exactly one time.

The frequency is the number of revolutions completed each second. Notice that frequency and period are inversely related to one another… That is...

Period frequency

Equations for Systems Involving Rotational Motion with Constant Angular Acceleration Again, these are completely analogous to what we derived for the kinetic equations of a linear system with constant linear acceleration!

A wheel rotates with constant angular acceleration of  0 = 2 rad/s 2. If the wheel starts from rest, how many revolutions does it make in 10 s?

We’ve seen how arc length relates to an angle swept out: Let’s look at how our angular velocity and acceleration relate to linear quantities. First, a question... In which direction does the instantaneous velocity of an object moving in a circle point?

For an object moving in a circle with a constant linear speed (a constant angular velocity), the instantaneous velocity vectors are always tangent to the circle of motion! The magnitude of the tangential velocity can be found from our relationship of arc length to angle...

For an object moving in a circle with a constant linear speed (a constant angular velocity), the instantaneous tangential acceleration is ZERO! The tangential acceleration tells us how the tangential velocity changes. It will point either parallel or anti-parallel to the tangential velocity vector. Under what circumstances will the tangential acceleration be NON-ZERO?

For an object moving around a circle with a changing angular velocity, and hence a changing tangential velocity, the instantaneous tangential acceleration is NON-ZERO! Let’s look at how the tangential velocity changes with time in such a case:

A wheel of radius 0.1 m rotates with constant angular acceleration of  0 = 2 rad/s 2. If the wheel starts from rest, what is the tangential acceleration at t = 10 s?

The tangential acceleration, however, is not the only acceleration we need to consider in problems of circular motion... The blue arrows represent the direction of the CENTRIPETAL ACCELERATION, which always points towards the center of the circle.

Although our objects moves in a circle at constant speed, it still accelerates. WHY? Recall that acceleration and velocity are both vector quantities. Since acceleration is a change in velocity over some change in time, an object whose velocity changes direction is accelerating!

Recall our definition of acceleration: You can demonstrate to yourself (and the book provides a nice proof on p. 186)

So, tangential acceleration occurs when the angular velocity of an object changes (i.e., the angular acceleration is not equal to 0). Centripetal acceleration exists for any object moving in a circle. The instantaneous centripetal acceleration is related to the instantaneous tangential velocity as defined before. Centripetal acceleration always points towards the center of the circle.

The total acceleration will be the vector sum of the tangential acceleration and the centripetal acceleration. a tan acac a tot Centripetal acceleration vector Tangential acceleration

When an object experiences a centripetal acceleration, there must be a force acting which results in such an acceleration. F c = m a c For objects moving in a circle, Newton’s 2nd Law takes the form

What kind of forces might result in a centripetal force? Force of Tension What makes this car turn to the right? Frictional Force of Tires on Road

Component of Normal Force of Road on Car What if the car now goes around an inclined bend? Frictional Force of Tires on Road Could point up or down the road. On what will the direction of the frictional force depend? Angle of the incline Speed of the car