1. Dynamics of Uniform Circular Motion
Chapter 5

2. Uniform circular motion
Uniform motion is motion at a uniform (constant) speed.
Circular motion is motion along the arc of a circle.
Uniform circular motion is motion at a constant speed along a circular arc.

3. For motion along a circular arc, use the distance along the arc.
Period and velocity
Units for period T: seconds
For motion along a circular arc, use the distance along the arc.

4. Example 1: Tire-Balancing Machine
A tire is rotating at 830 revolutions per minute on a tire balancing machine. The tire's radius is 0.29m. How fast is the outer edge of the tire moving?
The tire rotates 830 revolutions in 1 minute or 60 seconds.

5. Velocity direction is always changing
The velocity magnitude stays constant, but the velocity direction is always changing toward the center of the circle.
toward the center
Uniform circular motion is always accelerated motion because the velocity vector direction is always changing.

6. Centripetal acceleration
Since acceleration is the velocity change each one second, the acceleration is in the same direction as the velocity change direction and that direction is toward the center of the circle.
Centripetal means “toward the center”. (Centrifugal means "away from the center".)
Centripetal acceleration is "toward-the-center acceleration ".
The centripetal acceleration magnitude ac is
For uniform circular motion, the acceleration vector is always

7. Newton's second law
For uniform circular motion is always accelerated motion because the velocity vector direction is always changing, .
According to Newton’s 2nd law, there must be a net force in the direction of the acceleration.
This net force is called the net centripetal force because its direction is always toward the center.
The direction of the net centripetal force is always toward the center of the circle.

8. Example 5: Effect of Speed on Centripetal Force
A 0.9 kg model airplane flies around in a circle at a constant speed of 19 m/s. Find the plane's acceleration and the tension in the string. The string is 17 m long.
17 m
Acceleration can be expressed in multiples of g.
The acceleration magnitude is 2.17 times more than 9.8 m/s2.
[see maximums in Wikipedia g-force]

9. in the vertical direction lift and gravity are balanced
Example 5: Effect of Speed on Centripetal Force
force diagram
in the vertical direction lift and gravity are balanced
17 m
Newton's 3rd law
String exerts a centripetal force on the plane. Plane exerts a centrifugal force on the string.

10. Unbanked curves (also called flat turns)
Horizontal friction force provides the net centripetal force needed for uniform circular motion along a curved road.
Vertical forces of gravity and support are balanced because the car has zero acceleration in the vertical direction.

11. Banked curves
Horizontal component of the support force FN provides the net centripetal force needed for uniform circular motion along a banked road. No friction force is needed when the car is driven at the proper velocity.
To keep the vertical velocity from changing, the vertical component of the support force balances the gravity force .

12. Find the acceleration, forces, and banking angle. θ
Banked turn example
Find the acceleration, forces, and banking angle. θ
force diagram θ
vector diagram

13. Satellites in circular orbits
Gravity provides the net centripetal force needed to keep communications satellites orbiting around the Earth.

14. Acceleration due to gravity "g"

15. Communications satellites
Satellite stays above the same Earth location. Earth rotates once in 24 h and satellite orbits once in 24 h.
These satellites have an orbital radius of about 42 Mm (M = mega, 106) which is about 22,000 miles above the Earth.
With a velocity of about 3,000 m/s, the satellite travels about 3000 m in one second.
The gravity acceleration at that altitude is about 0.2 m/s2.
In one second the satellite "falls" about 0.1 m toward Earth.
Instead of traveling in a straight line, the satellite "falls" just enough, about 0.1 m, to keep on its circular orbit.

16. Communications satellites
Earth

17. Orbital motion
Objects that fall just right can orbit in a circle or an ellipse.

18. Orbital motion

19. Apparent weightlessness
apparent weight example from chapter 4
When the person and the scale are both falling with the same acceleration, the person is not being supported by the scale. The scale measures zero and the person is "weightless".

20. Artificial gravity
rotation
The support force of the floor provides the net centripetal force needed to keep the people and things moving in uniform circular motion.
A person standing on a scale would have the scale provide the centripetal support force and the scale would indicate that the person had "weight" just as if the person were experiencing weight due to gravity.
Orbiting Space Station

21. Artificial gravity
What period of rotation would produce a centripetal acceleration of 9.8 m/s2 in a 200 m diameter space station?

22. Vertical circular motion
At the position shown, the skier is moving along a circular arc and therefore, must have a net centripetal force
FN must be greater than Fg by the amount of the net centripetal force required for uniform circular motion.

23. Vertical circular motion
Find FN for a 100 kg skier moving at 15 m/s with a arc radius of 20 m.

24. Vertical circular motion
The net centripetal force is the vector sum of the normal support force and the gravity force acting on the moving object.
Gravity force always acts downward and the support force always acts toward the center because the track is a circle.
At the bottom, Fg and FN are in opposite directions.
At the top, Fg and FN are in the same direction.
FN is smaller at the top and larger at the bottom.
is the same magnitude at each location

25. Vertical circular motion
At the top, the smallest support force that the track can have is zero.
This occurs when the velocity is so slow that the cycle barely touches the track.
is the same magnitude at each location
The velocity in this equation is the slowest velocity the cycle can have and still touch the track at the top.

26. Vertical circular motion
How fast must the cycle move and still be in contact with the track at the top if the radius is 10 m?

27. Clothes dryer
Compare the motion of the clothes in a dryer to the motion of the motorcycle on the circular track.
What about the motions is the same?
Explain.
What about the motions is different?

28. The End