1. CIRCULAR MOTION

2. ROTATIONAL MOTION Objects that spin undergo rotational motion.
Any point on the object has circular motion around the axis.
The direction of motion is constantly changing.

3. ROTATIONAL QUANTITIES
 - angular displacement –(degrees, radians, or revolutions)
 = s/r
 - angular speed (rads/sec)
 = /t
 - angular acceleration (rads/sec2)
 = /t
1 radian = 57.3o
2 rads = 1 rev = 360o

4. Practice Problem
While riding on a carousel that is rotating clockwise, a child travels through an arc length of 11.5 m. If the child’s angular displacement is 165o, what is the radius of the carousel?

5. Practice Problem
A child at an ice cream parlor spins on a stool. The child turns counterclockwise with an average angular speed of 4.0 rads/sec. In what time interval will the child’s feet have an angular displacement of 8.0 rad?

6. Practice Problem
The wheel on an upside down bicycle moves through 11.0 rad in 2.0 s. What is the wheel’s angular acceleration of its initial angular speed is 2.0 rad/s?

7. T is period (time/# revolutions)
Tangential Speed
Instantaneous linear speed
Varies with position from axis of rotation
Speed along a line drawn tangent to the circular path
vt = r
vt = 2r/T
T is period (time/# revolutions)

8. Tangential Acceleration
Tangent to circular path
Occurs when rotating objects change speed
Example: A carousel speeds up
at = r

9. Practice Problem
What is the tangential speed of a child seated 1.2 m from the center of a rotating merry go round that makes one complete revolution in 4.0 s?
It takes 2.5 s for the merry go round to slow to a speed of .75 m/s. What is the tangential acceleration?

10. Centripetal Acceleration
Occurs as object moves in a circular path because it changes direction
Is constant or uniform
Directed toward the center
ac = vt2 ac = r2 r

11. Total Acceleration
When both centripetal and tangential acceleration exist, at is tangent to circular path
ac is toward the center
Components are perpendicular
atotal = square root of ac2 + at2
Direction = tan = ac/at ac at
atotal

12. Practice Problem
The Polar Express has an angular acceleration of .50 rad/s/s. A rider sits 6.6 m from the center and makes 10 revolutions in 13 seconds. Find the tangential, centripetal, and total accelerations.

13. Circular Motion object moves in a circular path
Continuous uniform acceleration
Ex: ball on the end of a string, Moon moving about the Earth (almost circular)
Can be vertical or horizontal
animation by Behrooz Mostafavi

14. Vertical Circular motion
Vmin occurs at the top
V = square root of rg
Ex: loop roller coaster Ft Fw Ft Fw

15. Practice Problem
A ball of mass .45 kg is swung in a vertical circle. If the centripetal force on the ball is 12.5 N, what is the tension in the string at the top and bottom of the circle?

16. How do you feel…
when sitting on the outside of the Polar Express ride at the State Fair
when you are in a car that turns sharply to the left.
What causes these feelings?

17. Centrifugal Force – not real

18. Centripetal Force – the real force

19. Explain how a bucket of water can be whirled in a a vertical circle without the water spilling out, even at the top of the circle when the bucket is upside down.

20. Horizontal Circular motion
Tension force acts horizontal and is constant.
If weight is small enough it can be ignored.
Ex: polar express

21. Centrifugal Force the fake force

22. What will happen…
when a car doesn’t have enough friction force to get around a curve?

23. If centripetal force is inward why will water not fall our of a cup that is swung in a vertical path?

24. What is required to cause the ball to have a curved path?

25. Centripetal Force
Required to maintain centripetal acceleration (Newton’s Laws)
Directed toward the center
Acts at right angles to motion
Ex: gravity, friction, strings…
Fc = mac = mr2 = mvt2/r

26. Practice Problem
What would be the centripetal force on a 1500 kg car rounding a 8.5 m curve at a speed of 10.0 m/s?

27. It is sometimes said that water is removed from clothes in the spin cycle by centrifugal force throwing the water outward. Is this correct?

29. The smaller the velocity of the object, the less centripetal force you will have to apply. The smaller the length of rope (radius), the more centripetal force you will have to apply to the rope.

30. The smaller the mass, the smaller the centripetal force (shown by the red vector labeled as the force of tension in the rope, FT) you will have to apply to the rope.

31. If you let go of the rope (or the rope breaks) the object will no longer be kept in that circular path and it will be free to fly off on a tangent.

32. Newton’s Law of Gravitation
Planets move in nearly circular orbits about the sun
Gravity acts as the centripetal force.
Any two masses are attracted
Inverse square law
Fg = Gm1m2 r2
G = 6.67 x Nm2/kg2

33. Practice Problem
What is the force of attraction between you and the Earth?

34. Pictures and animations from