1. Curved Tracks

2. Force on a Curve
A vehicle on a curved track has a centripetal acceleration associated with the changing direction.
The curve doesn’t have to be a complete circle.
There is still a radius (r) associated with the curve
The force is still Fc = mv2/r directed inward r Fc

3. Friction on a Wheel A rolling wheel does not slip.
It exhibits static friction.
As a car accelerates the tire pushes at the point of contact.
The ground pushes back, accelerating the car.
Acceleration of the contact point is upward
FGW
FWG
Point in contact doesn’t slip

4. Curves and Friction
On a turn the force of static friction provides the centripetal acceleration.
In the force diagram there is no other force acting in the centripetal direction. r Fc

5. Skidding
The limit of steering in a curve occurs when the centripetal acceleration equals the maximum static friction.
A curve on a dry road (ms = 1.0) is safe at a speed of 90 km/h.
What is the safe speed on the same curve with ice (ms = 0.2)?
90 km/h = 25 m/s
rdry = v2/ ms g = 64 m
v2icy = ms g r = 120 m2/s2
vicy = 11 m/s = 40 km/h

6. Banking Curves intended for higher speeds are banked.
Without friction a curve banked at an angle q can supply a centripetal force Fc = mg tan q.
The car can turn without any friction.

7. Hill Top A car going over a hill follows a vertical curve.
There are only two forces acting on the moving car.
Gravity
Normal force
If the normal force is zero the car has lost contact with the ground. FN Fc Fg

8. Speed Bump A neighborhood speed bump has a radius of 1 ft.
What is the maximum speed (mph) a car can travel without flying off the bump?
The normal force is zero is the car goes airborne.
1 ft = 12*2.54 cm = 30 cm
30 cm = 0.3 m
The maximum speed is 1.7 m/s.
1.7 m/s * 3600 s/h / 1000 m/km = 6.2 km/h
6.2 km/h / 1.6 km/mi = 3.9 mi/h
This is 4 mph.

9. 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 from the curve. FN Fg

10. 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

11. 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