Physics of Rolling Ball Coasters Cross Product (1) Cross Product (2) Digression Torque (1)
Rauch, Marc, Executive Vice President has reference to this Academic Journal, PHwiki organized this Journal Physics of Rolling Ball Coasters Cross Product Torque Inclined Plane Inclined Ramp Curved Path Examples Cross Product (1) The Cross Product of two three-dimensional vectors a =
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Digression Earlier, the definition of angular velocity was given, but details on the use of the cross product were not explained. Now that we have the cross product, we define the relation between tangential in addition to angular velocity in general: For circular motion, the velocity is always perpendicular to the position vector in addition to this reduces to v = r w. Similarly we define the relationship between tangential in addition to angular acceleration: Torque (1) Be as long as e dealing with a rolling ball, we must discuss how as long as ces act on a rotating object. Consider opening a door. Usually you grab the h in addition to le, which is on the side opposite the hinge, in addition to you pull it directly toward yourself (at a right angle to the plane of the door). This is easier than pulling a h in addition to le in the center of the door, in addition to than pulling at any other angle. Why When causing an object to rotate, it is important where in addition to how the as long as ce is applied, in addition to the magnitude. Torque is a turning or twisting as long as ce, in addition to it is a measure of a as long as ce’s tendency to produce rotation about an axis. Torque (2) There are two definitions of torque. First is in terms of the vectors F in addition to r, referring to the as long as ce in addition to position, respectively: Second is in terms of the moment of inertia in addition to the angular acceleration. (Angular acceleration is the time derivative of angular velocity). (Note the similarity to Newtons Second Law, F = m a. Here all the terms have an angular counterpart.)
Inclined Plane Consider a ball rolling down an inclined plane as pictured. Assume that it starts at rest, in addition to after rolling a distance d along the ramp, it has fallen a distance h in the y-direction. Inclined Plane (2) We will now consider the energy of the system. The system is closed, so energy must be conserved. Set the reference point as long as potential energy such that the ball starts at a height of h. Initially the ball is at rest, so at this instant it contains only potential energy. When it has traveled the distance d along the ramp, it has only kinetic energy (translational in addition to rotational). We can also express h in terms of d. This gives us the square velocity after the particle moves the distance d. Inclined Plane (3) If you know the square velocity of a particle after it travels a distance d, in addition to you know that the acceleration is constant, then that acceleration is unique. This derivation shows why, using definitions of average velocity in addition to average acceleration. Eliminating t in addition to vi=0, these expressions give Comparing this result to the previous slide, we can see that From the previous slide:
Inclined Track (1) When using physics to determine values like acceleration, there are often two perfectly correct approaches: one is using energy (like we just did), in addition to a second is by using as long as ces. While energy is often simpler computationally, it is not always as satisfying. For this next situation, the previous approach would also work, with the only difference being that However, to demonstrate the physics more explicitly, we will take an approach using as long as ces. When we build a track as long as a rolling ball coaster, there will actually be two contact points, one on each rail. Because the ball will now rest inside the track, we need to re-set the stage. The picture shows a sphere on top of a 2-rail track, with the radius R in addition to the height off the track b marked in. Inclined Track (2) These are all the as long as ces acting on the ball: friction, gravity, in addition to a normal as long as ce. The black square in the center represents the axis of rotation, which in this case is the axis connecting the two points where the ball contacts the track. The yellow arrow represents friction in addition to the blue arrow represents the normal as long as ce. Neither of these as long as ces torque the ball because they act at the axis of rotation. Thus the vector r is 0. The green arrow represents gravity. Convince yourself that the total torque is given by: Inclined Track (3) We also have a second definition of torque: Setting these equal in addition to solving as long as acceleration down the track: Notice that if b = R, then this reduces to the previous expression as long as acceleration
Curved Paths Until now we have considered only straight paths. While these are much simpler, they would make a very boring rollercoaster. Now we need to put together all the theory discussed to this point. Given a parameterized path r(s), define , , in addition to as the principal unit vectors in the tangential, normal, in addition to binormal directions, respectively. At any instant along the path there are two vectors acting on the particle, gravity in addition to a as long as ce exerted by the track which we will call the normal as long as ce. Note: Do not confuse the normal as long as ce with the normal direction . While they coincide in 2D systems, in a 3D system the normal as long as ce may point in any direction along the plane defined by the unit normal in addition to unit binormal vectors. Finding the normal as long as ce will tell us how much as long as ce the track must be able to withst in addition to at a given point. Also important are the total (resultant) as long as ces on the system. They will be discussed after the normal as long as ce. Curved Paths (2) To apply Newtons second law, consider all as long as ces in the normal direction. Acting in the positive direction is gravity, in addition to in the negative direction is the normal as long as ce, N. The sum of these as long as ces must result in curved motion around the instantaneous radius R. NN refers to the component of the normal as long as ce in the normal direction Curved Paths (3) This as long as mula has three parts. First is finding an expression as long as . Second is finding an expression as long as k. And third is getting an expression as long as v2. We will take these one at a time. The most convenient expression as long as is given by where r is the second derivative of the path, rN is the acceleration in the normal direction, rT is the acceleration in the tangential direction, in addition to is the unit tangent vector. with
Curved Paths (4) Next is the curvature, which is most useful expressed as As as long as the v2 term, we will get this from energy. We assume that friction is negligible, in addition to since the system is closed, energy is conserved. In general, the initial types of energy include potential as well as both kinetic energies, in addition to at any position s along the track there are the same types. Using the definition of w, we can relate it to v by . Then Curved Paths (5) Thus the general expression as long as the magnitude of the normal as long as ce in the normal direction is where r(s) is the path of the center of mass, m is the mass of the object, I is the moment of inertia of the object, ry(s) is the height of the center of mass at position s in addition to b is the height of the center of mass from the axis connecting the points of contact. If the track is banked such that there are no as long as ces acting in the binormal direction (so no lateral as long as ces), then the normal as long as ce is in the direction of the unit normal vector. Curved Paths (6) Now that the expression as long as the normal vector is found, we can focus on as long as ces important to the rider. For these we consider only as long as ces that push on your skin. To prove this to yourself, consider an astronaut in orbit around the earth. They are in constant free-fall, so gravity is acting on them. However, they feel weightless. What this means to us is that only the normal as long as ce of the track can be felt by the ball ( in addition to only the normal as long as ce of the seat on a bobsled coaster is felt by the rider). So in this case, the only as long as ce felt by the ball is the normal as long as ce.
Example – Parabola Set r(s)=
. Consider a ball starting at rest, with b= R. Then we have the expression as long as the magnitude of the normal as long as ce. The graph looks like this: (vertical axis in gs, horizontal axis has arbitrary units) Some may wonder why this curve has the shape it does. The reason as long as this is that the resultant (or net) as long as ce is predefined by the track in addition to the initial conditions. The graph of N is the difference of the component of gravity in addition to the resultant as long as ce. Graphically that means the normal as long as ce is the difference between the two black curves, which explains the shape of the graph as long as the magnitude of the normal as long as ce (blue). Force of gravity Resultant as long as ce N
Example – Cosine where A is the amplitude of the curve, in addition to l is the wavelength. Example Unit Normal Direction, The unit normal always points toward the center of curvature. Thus when making calculations involving the normal direction, take special care around points of inflection! (In this diagram there are arrows immediately be as long as e in addition to after the inflection point, but at the inflection point there is no normal direction defined) Normal as long as ce pushing up out of the track A ball, starting at rest, b=2R/3, amplitude is 1, wavelength is 2p. This shows the normal as long as ce in addition to its direction at different points on the curve.
Force of gravity Resultant as long as ce N As be as long as e, here is a breakdown of the different components of the as long as ces on the the system. This shows the normal as long as ce at different initial speeds. Where are they coincident in addition to why Speeds increase 1 m/s with each line. Blue is initially at rest, in addition to orange is initially at 5 m/s. Loop-de-Loop Un as long as tunately, we were unable to find the actual parameterization used in the design of coasters. However, we think its something like this:
This shows the local coordinate system as long as certain points along the path. This shows the normal vector along the track. One last comment on as long as ces in the normal in addition to binormal directions. We have calculated the necessary resultant as long as ce as long as the normal direction, but there has been no discussion on the binormal direction. Since there is no curvature in this direction, the only as long as ce that will act in the binormal direction is gravity. To prevent the ball from having any net lateral as long as ce, the binormal component of gravity should be balanced by a component of the normal as long as ce. Consider the as long as mula in addition to diagrams below:
Rauch, Marc Executive Vice President
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