Worksheet: Acceleration for Uniform Circular Motion
Please don’t use the set of buttons at the bottom of the simulation, under the
“SHOW ACCELERATION – ANSWER THE QUESTIONS FIRST” label, and after you
have answered the questions on this worksheet.
In this exercise we’ll use the circular motion simulation and the basic definition of
r
r r
r ∆v v f − vi
acceleration, a =
=
, to determine whether the acceleration of an object
∆t
∆t
traveling in a circle at constant speed depends on the speed and the radius of the circle,
and if so how it depends on these parameters.
Using the simulation, we will focus on the red object’s acceleration at a point
270˚ (marked with a black circle) from its starting point.
r
Let’s call the initial velocity vi the velocity the red object has when it passes over
the light blue circle, just before it reaches the black circle.
r
Let’s call the final velocity v f the velocity the red object has when it passes over
the purple circle, just after it reaches the black circle.
The time interval ∆t can be the time it takes the red object to travel from the light
blue circle to the purple circle.
1.
Use the top left button to have the object move in a circle of radius R at
a speed v.
When you run the simulation a vector diagram will appear showing the vectors
r
r
vi and v f . Based on the definition of acceleration (see above) what is the
direction of the red object’s acceleration when it passes over the black circle? Rer
draw the vector diagram in the space below and draw in the vector ∆v ,
representing the change in the object’s velocity as it moves from the light blue
circle to the purple circle. The direction of the acceleration is the same as the
r
direction of ∆v .
What is ∆t in this case? In other words, how long does the red object take to move
from the light blue circle to the purple circle? Assume the time in the simulation
is in units of seconds.
2.
Now use the top middle button to have the object move in a circle of
radius R at a speed of 2v.
r
r
As before, sketch the vector diagram in the space below to show vi and v f , and
r
r
r
draw in the vector ∆v . How does this ∆v compare to the ∆v in step 1, when the
speed was half as large?
Again, find ∆t , the time it takes the red object take to move from the light blue
circle to the purple circle. How does this ∆t compare to the ∆t in step 1, when the
speed was half as large?
Has doubling the speed affected the acceleration? If so, how? What does this tell
you about how the acceleration in uniform circular motion is related to the speed?
3.
Now use the top right button to have the object move in a circle of
r
radius R at a speed of 3v. How do the values of ∆v and ∆t compare to
those of step 1, when the speed was one-third as large? How does the
acceleration compare to that in step 1? Is this consistent with your
conclusion in step 2, about how the acceleration in uniform circular
motion is related to the speed?
4.
Now let’s figure out whether the acceleration depends on the radius of
the circular path, and if so how it depends on the radius. Use the middle
button in the second row of buttons to have the object move in a circle
of radius R/2 at a speed of v.
r
r
As before, sketch the vector diagram in the space below to show vi and v f , and
r
r
r
draw in the vector ∆v . How does this ∆v compare to the ∆v in step 1, when the
radius was twice as large?
Again, find ∆t , the time it takes the red object take to move from the light blue
circle to the purple circle. How does this ∆t compare to the ∆t in step 1, when the
radius was twice as large?
Has reducing the radius by a factor of 2 affected the acceleration? If so, how?
What does this tell you about how the acceleration in uniform circular motion is
related to the radius?
5.
Now use the right button in the second row of buttons to have the object
move in a circle of radius R/4 at a speed of v. How do the values of
r
∆v and ∆t compare to those of step 1, when the radius was three times
larger? How does the acceleration compare to that in step 1? Is this
consistent with your conclusion in step 4, about how the acceleration in
uniform circular motion is related to the radius?
6.
Use the buttons in the “SHOW ACCELERATION” part of the
simulation to check your answers above. Were your conclusions
correct?