Applications of Trigonometric Functions in Physics HW

4. (5 points) To be sure this makes sense, draw a picture of the turntable from the top and draw x-y axes. Sketch your eye on the picture so that what you see from eye level is the x coordinate. Pick one location on the circle and indicate both and x for that location of the cup.

What is the x location of the cup at the following times (you are given that it starts at 15 cm)? 40s

x location of cup 15cm

Need for a more refined model

The mathematical model x=cos() is not quite correct or complete. First, we want to include in this description how changes in time, and second, the turntable is not a unit circle, but a circle of radius 15 cm. In this activity we will learn how to put these refinements in the mathematical model.

How  changes in time.
Recall that for an object moving at a constant rate in a straight line (e.g. a car moving down a straight

road) x=x0+vt, where x0 is the location at t=0.

Analogously, when the cup moves around a circle at a constant rate,  =0 + va t. (We will take 0=0 from now on). The notation va stands for angular velocity.

5. (5 points) Recalling linear motion for a moment, if an object moves 30m in 5 seconds, what is v? v=

Using the definition that va = change in angle/change in time, what is va for the cup on the turntable? Be sure to use radians (not degrees) for the angle. (We use radians to be consistent with the text.)

va =

t

0s

10 s

20s

30s

6

Using va, what angle has the cup traveled 20 seconds? Show your work for full credit and be sure to keep units. As a check, what fraction of a cycle is 20 seconds? Does that agree with your angle?

s= fraction of a cycle=
Using va, what angle has the cup traveled in 800 seconds? Show your work for full credit and be sure to

keep units. As a check, how many cycles is 800 seconds? Does that agree with your angle? 800s= number of cycles=

Do the last two equations have correct units for an angle? If not, look doe the mistake, or check with the instructor.

6. (5 points) Let’s do one more specific example, but this time assume it takes 12 seconds to go once around the turntable

What is va= in this example? va=

What angle has the cup traveled in 3 seconds? Show your work for full credit and be sure to keep units. What fraction of a cycle is 3 seconds?

= fraction of a cycle=

What angle has the cup traveled 48 seconds? Show your work for full credit and be sure to keep units. How many cycles is 48 seconds?

= number of cycles=