1. CIRCULAR MOTION

2. CIRCULAR MOTION
Rotation along a circle, circular path, or circular orbit
Can be uniform or non-uniform

3. UNIFORM CIRCULAR MOTION
When an object is moving in a circle and its speed is constant
Recall that speed is the magnitude of velocity

4. VELOCITY and ACCELERATION
The direction of the object’s velocity is always tangent to the circle
The direction of the motion is always changing
The object is always ACCELERATING

5. VELOCITY and ACCELERATION
As an object moves from point A to point B, its velocity changes from v1 to v2
The direction of the acceleration is the same as the direction of the change in velocity (∆v=v2- v1) v2 v1 B A v2
-v1 ∆v

6. CENTRIPETAL ACCELERATION
The acceleration vector points directly towards the centre of the circle
“Centre-seeking” or CENTRIPETAL ACCELERATION (ac)
ac = v2/r

7. DERIVATION
r1=r2 because they are radii of the same circle v1=v2 because speed is constant r1 is perpendicular to v1 because the velocities are tangent to the circle The angle between corresponding members of sets of perpendicular lines are equal Since the angles between the equal sides of 2 isosceles triangles are equal, the triangles are similar The two triangles can be used to find the magnitude of acceleration.

8. DERIVATION
1. ∆r = ∆v r v 2. The object travels from point A to point B in time interval ∆t. ∆d= v∆t 3. As the angle between A and B becomes very small the length of ∆r becomes more nearly identical to the arc from A to B ∆r = ∆d

9. DERIVATION
4. Substitute ∆r for ∆d ∆r = v∆t 5. Substitute #4 into #1 v∆t = ∆v r v 6. Divide both sides by ∆t v= ∆v r v∆t

10. DERIVATION
7. Recall the definition for acceleration a = ∆v ∆t 8. Substitute #7 into #6 v = a r v 9. Rearrange to solve for a a = v2 r

11. CENTRIPETAL FORCE
The force causing centripetal acceleration always points towards the centre of the circular path
It is not a “type of force” like friction or gravity
It is a force that is required for an object to travel in a circular path

12. CENTRIPETAL FORCE
Centripetal force can be supplied by any type of force.
Gravity provides the centripetal force that keeps the moon in roughly a circular orbit
Friction provides the centripetal force that causes a car to move in a circular path on a flat road
Tension in a string tied to a ball will cause the ball to move in a circular path when you twirl it.
Fc= mv2 r

13. DERIVATION Recall Newton’s Second Law F = ma
2. Recall the equation describing centripetal acceleration
ac = v2 r
Substitute #2 into #1, omit the vector notation because the Force and Acceleration always point towards the centre of the circular path.
Fc = mv2

14. Practice Problems
1. A car with a mass of 2135kg is rounding a curve on a level road. If the radius of the curvature is 52m and the coefficient of friction between the tires and the road is 0.70, what is the maximum speed at which the car can make the curve without skidding off the road?

15. Practice Problems
2. A yo-yo has a mass of 225g. The full length of the string is 1.2m. You decide to see how slowly you can swing it in a vertical circle while keeping the string fully extended, even when the yo-yo is at the top.
Calculate the minimum speed
Find the tension on the string at the side and bottom of the swing.

16. References