1. Roller Coaster Design Project
Lab 3: Coaster Physics Part 2

2. Introduction
The focus of today's lab is on the understanding how various features influence the movement and energy loss of the ball.
Loops - Understanding centripetal acceleration and how to calculate the frictional losses in a loop
Hills - Understanding the effects of momentum and projectile motion and how to ensure the ball remains on the track
Banks - Modeling G-forces in banked turns and estimating energy loss

3. Forces Involved in the Coaster

4. Centripetal Force
To move along a curved path requires centripetal force
This force is pointed inward toward the center point of the circle (or arc) along which the object moves
The ball applies an equal and opposite force to the track, called the reactive centrifugal force
Centripetal force (and thus centrifugal force) for a given velocity, v and radius R, can be found with the equation:

5. Gravitational Force At all times, gravity acts upon the ball
This force is pointed directly downward at all times
Can be found using the following equation:
Fg = ma = mg

6. Normal Forces
Normal forces are the result of two objects pressing against one another.
Normal forces are always perpendicular to the plane surfaces that are pressing together
Frictional losses in the coaster will scale with the normal forces

7. Free Body Diagrams

8. Free Body Diagrams (FBDs)
Simple illustration of the scenario to be analysed showing all forces on the body and where they are acting
For the coaster, we will include gravitational, centripetal, centrifugal and normal forces
We will use vector mathematics & trigonometry to find the relevant components of each force
Goal: calculate the total force normal to the track at a given point.

9. FBD Example 1

10. FBD Example 2
The green vector contributes to the normal force.
How do we find it?
W*cos(Θ)

11. FBD Example 3 - Loop
In this case, the ball is traveling through the loop. There are 3 key scenarios:
The bottom of the loop
The side of the loop
The top of the loop
How do we the normal force at each point?

12. FBD Example 3 - Loop Bottom
Bottom of the loop
As the track curves up, the ball is pressed into the track in the same direction as gravitational force
Gravitational force (green arrow) and reactive centrifugal force (blue arrow) are modeled as additive when finding the normal force
Frictional loss scales with total force perpendicular to the track
Motion
Systematic view of the design review phases.
Gravitational Force
Reactive Centrifugal Force

13. FBD Example 3 - Loop Sides
Sides of the loop
Gravity and centrifugal force are perpendicular.
We can model the normal force as being equal to FC only
Systematic view of the design review phases.

14. FBD Example 3 - Loop Top Top of the loop
Gravity and centrifugal force oppose one another.
In this case, the normal force is modeled as (FC - W)
Additionally, to keep the ball on the track, the balls velocity must be great enough for the centrifugal force to overcome gravity.
Systematic view of the design review phases.

15. FBD Example 3 - Loop Consider:
The (very rough) average force felt by the track from the ball is:
or:
So the average force felt by the track from the ball through the loop is equal to FC!
Note: This is a VERY rough approximation!
Systematic view of the design review phases.

16. FBD Example 4 - Bank Model
When the ball travels around a banked turn, centripetal forces again play a roll.
In this example, we will look at a cross section of track
Note that both weight and centrifugal force contribute to the counter force to the normal force
Use the bank angle and trig to find these components and add them
Turn Center
Systematic view of the design review phases.
Normal Force

17. FBD Example 5 - Hill
We can model a hill or bump on the coaster as the exterior of a circular loop. As with the loop top, gravity and centrifugal force oppose one another.
Now, to keep the ball on the track, the balls velocity must be small enough for the centrifugal force to NOT overcome gravity.
Systematic view of the design review phases.

18. Questions?