Transcript Roller Coasters

Roller Coasters
Uniform or Nonuniform

Centripetal acceleration is constant for uniform
circular motion.
v2
2
a

r
 r
It changes for nonuniform circular motion.
• The magnitude increases or decreases.
• There is a tangential acceleration.
• Net vector is not antiparallel to radius.
Angular Acceleration


ar
at 
 
at v
v

r
 ra
t
t
The angular acceleration is a
In uniform circular motion
there is a constant radial
acceleration.
• ar = v2 / r = r2

If the angular velocity
changes there is
acceleration tangent to the
circle as well as radially.
• at = ra
Rotational Motion

Kinematic equations with
constant linear acceleration
were defined.
•
•
•
•
vav = ½ (v0 + v)
v = v0 + at
x = x0 + v0 t + ½at2
v2 = v0 2 + 2a(x - x0 )

Kinematic equations with
constant angular
acceleration are similar.
•
•
•
•
av = ½ (0 + )
 = 0 + at
q = q0 + 0 t + ½at2
2 = 0 2 + 2a (q - q0 )
Loop-the-Loop

A loop-the-loop is a popular
rollercoaster feature.

There are only two forces
acting on the moving car.
• Gravity
• Normal force
FN

There is a centripetal
acceleration due to the loop.
Fg
Staying on Track

If the normal force becomes
zero, the coaster will leave
the track in a parabolic
trajectory.
• Projectile motion

At any point there must be
enough velocity to maintain
pressure of the car on the
track.
Fg
Force at the Top

The forces of gravity and the
normal force are both
directed down.

Together these must match
the centripetal force.

The minimum occurs with
almost no normal force.
Fg
FN
Fc  FN  Fg
m v2 / r  FN  m g
v
FN r
 gr
m
vmin  gr
Speed and Acceleration

Where is the acceleration greatest on the
rollercoaster?

Use a = v2 / r.
• High velocity causes large acceleration
• Small radius causes large acceleration

Too much acceleration on a ride is fatal!
Try your own coaster
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