1. Uniform Circular Motion
SPH4U – Grade 12 Physics
Unit 3

2. Circular Motion & Roller coasters
Roller coasters are exciting not just because they go fast, but because of the accelerations that are involved in the ride.
Roller coasters are able to produce feelings of weightlessness and weightiness which can be fun. (Or terrifying, depending on who you are…)

3. Circular Motion & Roller coasters
They are also able to accelerate us in one direction one moment, and then quickly change course to accelerate us in another direction the next.
Best of all, roller coasters seem dangerous, (even when they’re not), which is a thrilling experience for most people.

4. Circular Motion & Roller coasters
There are a lot of importance concepts related to circular motion employed in roller coasters. Roller coasters employ the physics of circular motion in their loops, banked turns, and in the small dips and hills found along otherwise straight sections of track.
Understanding circular motion is imperative if we are going to understand roller coasters.

5. Circular Motion & Roller coasters
The curvy sections of a rollercoaster can actually be thought of as parts of circles.

6. Uniform Circular Motion
Uniform motion is constant speed in a straight line.
Uniform circular motion is motion at a constant speed in a circular path.
Since the direction (velocity vector) if constantly changing during uniform circular motion, this type of motion is accelerated motion (even though the speed doesn’t change).

7. Centripetal Acceleration
Centripetal Acceleration is the instantaneous acceleration of an object directed toward the center of a circular path. All objects moving in a circle will experience centripetal (center-seeking) acceleration.

8. Centripetal Acceleration
Recall: Acceleration is defined as the change in velocity over the change in time.
For an object travelling in a circle with uniform circular motion, the tangential speed might not be changing, but the direction of the velocity is changing continuously with time.

9. Centripetal Acceleration
As you can see from the above diagrams, as v1 and v2 get closer together, Δv is directed more and more toward the exact center of the circular path. Since when we have the instantaneous acceleration, v1 and v2 should be about a fraction of a second apart, at that moment the direction is in fact directed directly toward the center of the circle.

10. Centripetal Acceleration
The direction of centripetal acceleration is always toward the center of the circular path.

11. Centripetal Acceleration
There are three equations that determine centripetal acceleration:
ac = centripetal acceleration f = frequency
r = radius of circular path T = period
v = tangential velocity

12. Centripetal Force
Since the acceleration is directed toward the center, we know that the net force on the object must be directed toward the center as well.

13. Centripetal Force
The net force that causes uniform circular motion is called the centripetal force.
The centripetal force might actually be made up of other forces, such as the force of gravity, tension, the normal force, or some other force.

17. Centripetal Force
From Newton’s 2nd law, we know that the following must be true:
If the centripetal force is the only force acting on the object, then by Newton’s second law:
Fc = mac
It can be shown that Fc is also equal to the mass of the object times it’s velocity squared over r:

18. Horizontal Circular Motion - a car going around a curve

19. Horizontal Circular Motion - a car going around a curve

20. Horizontal Circular Motion - a car going around a curve

21. Horizontal Circular Motion - a car going around a curve

25. Race car drivers will hug the inside of curves in order to increase the radius of curvature which in turn reduces the centripetal force they need from the friction of their tires.

26. For the car that produced these skid marks, the frictional force did not provide enough centripetal force to keep the car on the road. The car was going to fast for this curve and therefore required more centripetal force for the circular path than friction could provide.

27. Practice Problem 1

28. Vertical Circular Motion
At the top of the loop
Fc = Fnet = FN + Fg
Where FN = normal force
Fg = force of gravity FN
At the bottom of the loop
Fc = Fnet = FN - Fg
Where FN = normal force
Fg = force of gravity FN

29. Minimum Speed of a Loop
What is the minimum speed required by the object in order to complete the loop?
At the minimum speed, the Normal force will drop to zero. So, this formula will become….

30. Minimum Speed of a Loop
Thus the minimum speed required to complete the loop can be calculated with this equation.

31. Implications for Roller coasters
Your roller coasters will have curved horizontal turns in them. You need to consider what speed your marble is going when designing these turns. If your marble is going too fast it will not be able to make it around the curve.
Similarly, if your marble is going over top of a circular hill, if it has too much energy and speed it will fly off the track.

32. Implications for Roller coasters
How would you redesign your coaster to solve these problems?

33. Implications for Roller coasters
How would you redesign your coaster to solve these problems?
make the radius of curvature greater / add friction to slow the marble down

34. Implications for Roller coasters
Your roller coasters will have vertical loops in them.
We know that the loops ought to have an elliptical shape, rather than a circular one.

35. Implications for Roller coasters
You need to make sure the marble has enough kinetic energy to have the minimum speed required to make the loop.
(Granted.. Our equation is for a circular loop.. but for now you can use it as an estimation. In fact, if you make your loops like clothoid loops then you will need less speed).

36. Homework Read Sections 3.2-3.3.
Add in extra information from your textbook to supplement your notes
Answer the following questions:
Pg. 119 # 1, 6, 7, 9
Pg. 124 # 1, 2, 3, 4