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 = <a1,a2,a3> and b = <b1,b2,b3> is defined as follows: If q is the angle between the vectors, then

Cross Product (2) Important facts about the cross product: The cross product is always perpendicular to the vectors a and b. The direction of the cross product is given by the right hand rule (see diagram, where ). The cross product is greatest when While the dot product produces a scalar, the cross product produces a vector. Therefore it is sometimes called a vector product.

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 and angular velocity in general: For circular motion, the velocity is always perpendicular to the position vector and this reduces to v = r w. Similarly we define the relationship between tangential and angular acceleration:

Torque (1) Before dealing with a rolling ball, we must discuss how forces act on a rotating object. Consider opening a door. Usually you grab the handle, which is on the side opposite the hinge, and you pull it directly toward yourself (at a right angle to the plane of the door). This is easier than pulling a handle in the center of the door, and than pulling at any other angle. Why? When causing an object to rotate, it is important where and how the force is applied, in addition to the magnitude. Torque is a turning or twisting force, and it is a measure of a force's tendency to produce rotation about an axis.

Torque (2) There are two definitions of torque. First is in terms of the vectors F and r, referring to the force and position, respectively: Second is in terms of the moment of inertia and the angular acceleration. (Angular acceleration is the time derivative of angular velocity). (Note the similarity to Newton’s 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, and 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 for 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 and 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) From the previous slide: If you know the square velocity of a particle after it travels a distance d, and you know that the acceleration is constant, then that acceleration is unique. This derivation shows why, using definitions of average velocity and average acceleration. Eliminating t and vi=0, these expressions give Comparing this result to the previous slide, we can see that

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), and a second is by using forces. 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 forces. When we build a track for 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 and the height off the track b marked in.

Inclined Track (2) These are all the forces acting on the ball: friction, gravity, and a normal force. 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 and the blue arrow represents the normal force. Neither of these forces 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 and solving for acceleration down the track: Notice that if b = R, then this reduces to the previous expression for acceleration The “+mb^2” term in this second definition of torque comes from the Parallel Axis Theorem. In the “a_{CM}”, CM stands for Center of Mass.