1. Unit 06 “Circular Motion, Gravitation and Black Holes”
Circular Motion Problem Solving

2. Centripetal Acceleration
Definition
Acceleration due to a change in direction.
Centripetal acceleration is always directed toward the center of the circle (inward).
Equation
Vt2
ac= r
ac  centripetal acceleration
Vt  tangential speed
r  radius

3. Fc= mVt2 r ----------- Centripetal Force Definition Equation
The force that causes circular motion.
The centripetal force is always directed toward the center of the circle, inward.
Equation
mVt2
Fc= r
Fc  centripetal force
m  mass
Vt  tangential speed
r  radius

4. “Easy” Example
1. A girl swings a bucket of water over her head so that it has a tangential speed of 6m/s. If the rope pulling the bucket is 0.75m long (radius), what is the bucket’s centripetal acceleration?

5. “Easy” Example ac = Vt2 r ac = __(6m/s)2 0.75m ac = (36m2/s2) 0.75m
1. A girl swings a bucket of water over her head so that it has a tangential speed of 6m/s. If the rope pulling the bucket is 0.75m long (radius), what is the bucket’s centripetal acceleration?
ac = Vt2 r
ac =
Vt =
r = ?
ac = __(6m/s)2
0.75m
6m/s
ac = (36m2/s2)
0.75m
0.75m
ac = 48m/s2

6. Easy 2. What is the centripetal force of a 400kg rollercoaster car being pushed around a track with a radius of 20m if its tangential speed is 15m/s?

7. Fc = mVt2 r Fc = (400kg)(15m/s)2 20m Fc = (400kg)(225m2/s2) 20m
Easy 2. What is the centripetal force of a 400kg rollercoaster car being pushed around a track with a radius of 20m if its tangential speed is 15m/s?
Fc = mVt2 r
Fc = ?
m = 400kg
Fc = (400kg)(15m/s)2
20m
Vt = 15m/s
r = 20m
Fc = (400kg)(225m2/s2)
20m
Fc = (90000kgm2/s2)
20m
Fc = N

8. Easy A girl on rollerblades turns a corner with a radius of 8m, if her centripetal acceleration is 10m/s2, what is her tangential speed around the corner?

9. ac = Vt2 r 10m/s2 =_Vt2_ 8m 80m2/s2 = Vt2 8.94m/s = Vt ac = Vt = r =
Easy A girl on rollerblades turns a corner with a radius of 8m, if her centripetal acceleration is 10m/s2, what is her tangential speed around the corner?
ac = Vt2 r
ac =
Vt =
r =
10m/s2
10m/s2 =_Vt2_ 8m
x 8m
8m x ?
80m2/s2 = Vt2 8m
8.94m/s = Vt

10. Easy + 4. A 200 kg bicycle turns a corner with a radius of 15m; a 2500N centripetal force is applied to it by the friction between the ground and the tires, what is the tangential speed of the bicycle?

11. 188m2/s2 = Vt2 Fc = mVt2 r 2500N =_(200kg)Vt2_ 15m
Easy + 4. A 200 kg bicycle turns a corner with a radius of 15m; a 2500N centripetal force is applied to it by the friction between the ground and the tires, what is the tangential speed of the bicycle?
Fc = mVt2 r
2500N =_(200kg)Vt2_
15m
m =
Fc =
Vt =
r =
200kg
2500N ?
2500N = (13.3kg/m)Vt2
15m
13.3kg/m
13.3kg/m
188m2/s2 = Vt2
13.7m/s = Vt

12. Medium 5. How long a rope attached to a 20kg bucket if it is pulled inward by a centripetal force of 300N with a tangential speed of 5m/s?

13. r = 1.67m 300N = (20kg)(5m/s)2 Fc = mVt2 r r 300N = (20kg)(25m2/s2) r
Medium 5. How long a rope attached to a 20kg bucket if it is pulled inward by a centripetal force of 300N with a tangential speed of 5m/s?
300N = (20kg)(5m/s)2 r
Fc = mVt2 r
300N = (20kg)(25m2/s2) r
300N = _(500kgm2/s2)_ r
r x
x r
r = ?
m = 20kg
300N(r) = (500kgm2/s2)
Fc = 300N
300N
300N
Vt = 5m/s
r = 1.67m

14. Answer the following questions about the horses and the people on the carousel.
What is the name of the force causing the horses and people to move in circular motion?
In what direction does the force on the horses and people pull?
If the carousel suddenly stopped, in which direction would the people move?
Why?
Which horse is moving faster?
The centripetal force
Inward
Tangent to the circle!
Their inertia!
They want to move in a straight line!
WHAT!? One moves faster!?
YES!

15. A Cartoon Guide to Tangential Speed and “HARD Problems”
B: “Sally, would you like to go on a date with me?”
S: “Bobby, we will have to see if you are smart enough to be in a relationship with me, where are we going?”
B: “I’m going to take you to the Roger Williams Park Carousel!”

16. S: “A carousel? What is the name of the force that
As they walk to Roger Williams Park, Sally begins to ask questions to see how smart Bobby is.
S: “A carousel? What is the name of the force that
causes the horses and the people to move in a
circle on the carousel?”
B: “ ___________________________________”
S: “In which direction does the force pull on
the people and the horses?”
B: “__________________________”
The centripetal force!
Inward!

17. The floor and the engine
S: “What provides the centripetal force?”
B: “_______________________________________”
S: “If the carousel suddenly stopped and the people weren’t holding on to the horses, which way would the people go?
B: “_______________________.”
S: “Why?” B:
The floor and the engine
Tangent!
“Because without a force to pull them in a circle, they would move in a straight line.”

18. S: “You sure are smart Bobby. You might be relationship material.”
B: “Great, we are here at the carousel, which horse would you like to go on?”
S: “The fastest horse, of course!”
B: “The fastest horse?!?!”

19. Oh no!
Which horse is the fastest?!?!

20. QUICK! I have to calculate which horse moves the fastest, the horse on the inside or the horse on the outside of the carousel.

21. Inside Horse T = 25s r = 3.5m VT = ? Outside Horse T = 25s r = 5.5m
I know…
The carousel takes 60seconds to complete one complete circle.
The inside horse is 1.5m away from the center of the carousel.
The outside horse is 2.9m away from the center of the carousel.
Inside Horse
T = 25s
r = 3.5m
VT = ?
Outside Horse
T = 25s
r = 5.5m
VT =?

22. Both horses take the SAME amount of time to complete one revolution.
But each horse goes a different distance!
The outside horse has to cover a bigger distance in the same time, greater speed!

23. Vt = 22.0m/25s Vt = 35.6m/25s Find the speed of the Inside Horse
Find the speed of the Outside Horse s
Vt = 22.0m/25s
Vt = 35.6m/25s
Vt = 0.88m/s
Vt = 1.42m/s
This makes sense!
The _______________horse travels more distance than the ______________horse.
They both take the same amount of time to move in a circle.
So, the __________________horse must be moving faster!
outside
inside
outside

24. B: “The faster horse? The faster horse is
the horse on the _________________!
S: “A person who knows their physics!
Perfect! <3”
THE END!
outside