Roller Coasters

1. Roller Coasters

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

3. Angular Acceleration • In uniform circular motion there is a constant radial acceleration. • ar = v2 / r = rw2 • If the angular velocity changes there is acceleration tangent to the circle as well as radially. • at = ra The angular acceleration is a

4. Kinematic equations with constant linear acceleration were defined. vav = ½ (v0 + v) v = v0 + at x = x0 + v0t + ½at2 v2 = v02 + 2a(x - x0 ) Kinematic equations with constant angular acceleration are similar. wav = ½ (w0 + w) w = w0 + at q = q0 + w0t + ½at2 w2 = w02 + 2a(q - q0 ) Rotational Motion

5. 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 • There is a centripetal acceleration due to the loop. FN Fg

6. 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

7. 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

8. 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 next