1. Circular Motion

2. Circular Motion SPEED remains CONSTANT Velocity CHANGES
An object that moves in a circle at constant speed is said to experience uniform circular motion.
During Circular Motion:
SPEED remains CONSTANT
Velocity CHANGES
Therefore, there is an acceleration due to the change in velocity. We call this Centripetal Acceleration

3. We study Uniform Circular Motion to understand how things that move in a circle work.
We can determine:
A planet’s orbiting velocity or acceleration
How fast you need to go over a hill to feel “weightless”
How fast a coaster has to go in order to stay on the loop
Tensions in cables that keep things in rotation

4. Circular Motion Equations
Rotational Period A term describing the length of time it takes for an object to make a complete rotation. Can be given in: Rotations, cycles, laps, turns, revolutions, etc.
Time (s)
Period (s)
T = t .
cycles
Revolutions

5. Circular Motion Equations
Centripetal Acceleration
Centripetal Velocity
Vc = (2πr) T
ac = V2 R
Velocity Radius
Radius
Period
Tangential Velocity can also be interchanged with centripetal velocity: These terms are used to describe an object moving about a circle at a constant speed while changing direction.

6. Examples using Velocity and Acceleration

7. Circular Motion Equations
Centripetal Force
( )
Fc = m V2 R
Fc = mac
Mass
Mass
Centripetal
Acceleration
Radius Velocity

8. 1.5 s T = t . cycles Vc = (2πr) T ac = V2 R Fc = mac 5.23 m/s
Example: A 5 kg object is swung in a circle from a chain that is 1.25 m long. It is swung around 7 times in 10.5 seconds. Find T, Vc, Ac, Fc.
1.5 s
T =
Vc =
Ac =
Fc =
T = t .
cycles
Vc = (2πr) T
ac = V2 R
Fc = mac
5.23 m/s
T = 7
Vc = (2 π 1.25)
1.5
ac = 5.232
1.25
Fc = (5)(21.88)
21.88 m/s2
109.4 N

9. Circular Motion Equations
Centripetal Force – A force that causes an object to move in a circle.
Types of Centripetal Forces:
Gravitational Force
Tension
Friction
Friction (for a car on the road)
Fg = Gm1m2 r2
Ff = FN μ
Ff = m g μ

10. We don’t really look into displacement……
In UMC, we study force, acceleration, and velocity of things going in circles!
We don’t really look into displacement……
Hmm..??!?!
What would the displacement be of this blue dot when it completes a cycle?

11. Directions FORCE always points TOWARD THE CENTER V
ACCELERATION always points TOWARD THE CENTER
VELOCITY is always TANGENT TO THE CIRCLE V F A

12. Velocity is TANGENT
A tangent line is a line that touches a circle at one point but does not intersect it.

13. Centripetal Force keeps things in ORBIT
We call this the Gravitational Force (Fg)

14. We call this the Frictional Force (Ff)
Centripetal force keeps the car on the road while turning
We call this the Frictional Force (Ff)

15. We call this the Tension Force (T)
You can swing an object around. There will be a centripetal force
We call this the Tension Force (T)

17. Circular Motion Review
Force and Acceleration are TOWARD the CENTER
Velocity is TANGENT to the circle

18. Welcome back ΣF equations!
Vertical Circles
Welcome back ΣF equations!

19. What is the Tension at the top and at the bottom?
Consider an object of mass “m” being swung around in a vertical circle of radius “R” with a velocity of “v”
What is the Tension at the top and at the bottom?
ΣF: -T - Fg = m(-ac)
ΣF: T + mg = mv2 R T Fg
ΣF: T = mv mg R v R
@ the bottom:
ΣF: T - Fg = m(ac) T T
ΣF: T - mg = mv2 R T
ΣF: T = mv mg R Fg v

20. Consider a rollercoaster loop
@ the top:
Remember, there are other forces that are considered centripetal forces…
Consider a rollercoaster loop
ΣF: -Fn - Fg = m(-ac)
ΣF: Fn+ mg = mv2 R Fn Fg
ΣF: Fn = mv mg R v R
@ the bottom:
ΣF: Fn - Fg = m(ac) T Fn
ΣF: Fn - mg = mv2 R
ΣF: Fn = mv mg R Fg v

21. Fg = Fc mg = mv2 r Fg = Fc Gm1m2 = mv2 Ff = Fc m g μ = mv2 r2 r
Circular Motion
Remember the Centripetal force can be equal to many other things.
Used if the Fg is the only thing acting on it. (Weightless)
Fg = Fc mg = mv2 r
Fg = Fc
Ff = Fc m g μ = mv2
Ff = Fc Fn μ = mv2
Gm1m2 = mv2
r r
Used when talking about orbiting planets or satellites
Used when an object on a flat surface is going in a circle (car on the road )
On a flat surface Fn=Fg
Used when an object going in a circle (car on the road on a slopped road )