Physics of Rolling Ball Coasters

1
Physics of Rolling Ball Coasters

• Cross Product
• Torque
• Inclined Plane
• Inclined Ramp
• Curved Path
• Examples

2
Cross Product (1)

• The Cross Product of two three-dimensional
  vectors a lta1,a2,a3gt and b ltb1,b2,b3gt is
  defined as follows
• If q is the angle between the vectors, then

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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.

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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 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

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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.

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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 Newtons Second Law, F
  m a. Here all the terms have an angular
  counterpart.)

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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.

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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.

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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 vi0, these expressions give
Comparing this result to the previous slide, we
can see that
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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.

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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

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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