Roller Coaster Design Project Lab 3 Coaster Physics

Roller Coaster Design Project Lab 3: Coaster Physics Part 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

Forces Involved in the Coaster

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:

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

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

Free Body Diagrams

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.

FBD Example 1

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

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?

FBD Example 3 - Loop 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 Gravitational Force Reactive Centrifugal Force

FBD Example 3 - Loop Sides of the loop Gravity and centrifugal force are perpendicular. ▪ We can model the normal force as being equal to FC only

FBD Example 3 - Loop 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.

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!

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 Turn Center ▪ Use the bank angle and trig to find these components and add them Normal Force

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.

Questions?