Roller Coasters

Uniform or Nonuniform  Centripetal acceleration is constant for uniform circular motion.  It changes for nonuniform circular motion. The magnitude increases or decreases.The magnitude increases or decreases. There is a tangential acceleration.There is a tangential acceleration. Net vector is not antiparallel to radius.Net vector is not antiparallel to radius.

Angular Acceleration  In uniform circular motion there is a constant radial acceleration. a r = v 2 / r = r  2  If the angular velocity changes there is acceleration tangent to the circle as well as radially. a t = r  The angular acceleration is 

Rotational Motion  Kinematic equations with constant linear acceleration were defined. v av = ½ (v 0 + v)v av = ½ (v 0 + v) v = v 0 + atv = v 0 + at x = x 0 + v 0 t + ½at 2x = x 0 + v 0 t + ½at 2 v 2 = v a(x - x 0 )v 2 = v a(x - x 0 )  Kinematic equations with constant angular acceleration are similar.  av = ½ (  0 +  )  =  0 +  t  =  0 +  0 t + ½  t 2  2 =   (  -  0 )

Loop-the-Loop  A loop-the-loop is a popular rollercoaster feature.  There are only two forces acting on the moving car. GravityGravity Normal forceNormal force  There is a centripetal acceleration due to the loop. FgFg FNFN

Staying on Track  If the normal force becomes zero, the coaster will leave the track in a parabolic trajectory. Projectile motionProjectile motion  At any point there must be enough velocity to maintain pressure of the car on the track. FgFg

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. FgFg FNFN

Speed and Acceleration  Where is the acceleration greatest on the rollercoaster?  Use a = v 2 / r. High velocity causes large accelerationHigh velocity causes large acceleration Small radius causes large accelerationSmall radius causes large acceleration  Too much acceleration on a ride is fatal! next Try your own coaster