Uniform circular motion (UCM) is the motion of an object ... · Uniform circular motion (UCM) is...

Uniform circular motion (UCM) is the motion of an object

in a perfect circle with a constant or uniform speed.

4. Is there an equation that relates the distance around a circle (circumference) to the radius of the circle?

1. Distance around a circle?

circumference

2. Distance from one side of circle to the opposite side which goes through the center of the circle

3. Distance from center of the circle to any point on the edge of the circle

diameter

radius

(C = 2πr)

5. radian: the angle made by taking the radius and wrapping it along the edge of a circle:

The radian is a pure measure based on the radius of the circle:

• There are π radians in a half circle • And also 180° in a half circle So π radians = 180° So 1 radian = 180°/π = 57.3°

Imagine you cut up pieces of string exactly the length from the center of a circle to its edge ... ... how many pieces do you need to go around the edge of the circle?

Answer: 2π (or about 6.283 pieces of string).

= 1 rev = 360°

Positive rotation is considered to be counterclockwise

# 6,7 & 8

# revolutions rads degrees

π/2

π

270°

1

9. Complete the following table

# revolutions rads degrees

1/4 π/2 90°

1/2 π 180°

3/4 3/2π 270°

1

2π

360°

9. Complete the following table

Linear Velocity

•  Objects moving in a circle still have a linear velocity = distance/time.

•  This is often called tangential velocity, since the direction of the linear velocity is tangent to the circle.

v

10a. A car travels in perfectly circular motion. If the radius of the circle is 25m meters how far does the car travel if…

It goes completely around the circle?

(C = 2πr)

C = 2 π 25 = 157 m

C = 2 π rads = 6.28 rads = 360°

10b. It goes halfway around the circle?

Halfway around is 78.5 meters or π rads = 3.14 rads = 180°

10c. It goes one fourth of the way around the circle?

1/4th of the way is 39.2 meters or π/2 rads = 1.57 rads = 90°

11. What is the equation to find the speed of an object with uniform circular motion ?

12. T is the time for one revolution or one cycle. This quantity T is called the period!

circumference equals ??

Average speed (v) = distance = circumference time time

V = 2 π r T

13. If the radius of the circle is 6 meters, what is the average speed of a minion on a motorcycle traveling completely around a circle in 4 sec?

V = 2 π 6m 4 sec

= 9.42 m/s V = 2 π r T

14. How far around the circle in meters and degrees would the minion move in 2.5 seconds?

(9.42 m/s) (2.5 sec) = ? d = vt

23.55 = ?° 2 π r 360°

= 23.55m

222° (a bit more than halfway)

ω = 2 π rads T

What if we do not know the radius of the circle but we know that the minion completes a revolution around a circle in 4 sec?

15. We can determine the angular velocity ω of the minion in rads/sec

ω = 2 π rads T


ω = 2 π rads 4 sec

= 1.57 rads/sec


ω = 2 π rads T



= 1.57 rads/sec

16. How far around the circle in rads would the minion move in 2.5 seconds?

θ= ωt where θ = angular displacement in rads (θ is theta)


ω = 2 π rads T



= 1.57 rads/sec


θ = 1.57 rads/sec) (2.5 sec) = ?


= 3.9 rads


17. How many degrees is this?

ω = 2 π rads T



= 1.57 rads/sec


θ = 1.57 rads/sec) (2.5 sec) = ?


(3.9 rads) (57.3°/rad) = ?

= 3.9 rads

223° (a bit more than halfway)


17. How many degrees is this?

18. A turn table completes one revolution in 5 seconds. If coin A is 4 meters from the center and coin B is 8 meters from the center, find the velocity of each coin.

V = 2 π r T

V = 2 π 4m 5 s

Velocity of coin A = 5.0 m/s

Velocity of coin B V = 2 π r T

V = 2 π 8m 5 s

= 10.0 m/s

The farther away an object is from the center of a circle the greater the velocity of that object

19..Conclusion?

as the radius is increased - greater r in equation = greater V Why?

After 5 seconds the turn table completes one revolution.

20. Each coin moves a total of 360° or 2 π rads while moving around the circle. So each coin moves the same “angular” distance in 5 seconds. That angular distance is 6.28 rads. If both coins move the same angular distance in the same time, they must have the same angular velocity (ω) – measured in rads/sec.

21. ω = _θ_ = 6.28 rads t 5 sec

= 1.25 rads/sec

A turn table completes one revolution in 5 seconds. If coin A is 4 meters from the center and coin B is 8 meters from the center, find the velocity of each coin.

V of coin A = 5.0 m/s and r = 4 m

V of coin B = 10.0 m/s and r = 8 m

22. Even though both coins have a different velocity in meters/sec they have the same angular velocity in rads/sec 23. v =ωr 24. Coin A v =ωr v = (1.25 rads/sec) (4 m) = 5 m/s

24. Coin B v =ωr v = (1.25 rads/sec) (8 m) = 10 m/s

25. All points on a rotating object rotate through the same angle in the same time

26. All points on a rotating object have the same angular velocity, ω, but different speeds, v, v =ωr.

•  Angular displacement = θ •  θ= ω t •  Angular Velocity = ω (Greek: Omega) •  ω = θ/t •  All points on a rotating object rotate through the same

angle in the same time, and have the same frequency. •  Angular velocity: all points on a rotating object have the

same angular velocity, ω, but different speeds, v, and v =ωr.

•  v =ωr

Angular & Linear Velocity

•  In symbols:

v = r ω

v

r ω

How would you describe the velocity in this situation where an object is moving in a circle?

•  Constantly changing! •  We say the velocity

vector is Tangent to the circular path of travel and call it the Tangential velocity.

What happens when an object is spun on a string and the string breaks?

The object moves off in a straight line tangent to the circle

Acceleration

•  As an object moves around a circle, its direction of motion is constantly changing.

•  Therefore its velocity is changing.

•  Therefore an object moving in a circle is constantly accelerating.

Centripetal Acceleration

•  The acceleration of an object moving in a circle points toward the center of the circle.

•  This is called a centripetal (center pointing) acceleration.

a

Centripetal Acceleration

•  The centripetal acceleration depends on: – The speed of the object. – The radius of the circle.

acent =

v2 r

Centripetal Force

•  Newton’s Second Law says that if an object is accelerating, there must be a net force on it.

•  For an object moving in a circle, this is called the centripetal force.

•  The centripetal force points toward the center of the circle.

Centripetal Force

•  Centripetal force on an object depends on: – The object’s mass - more mass means more

force. – The object’s speed - more speed means more

force. – And…

Centripetal Force

•  The centripetal force on an object also depends on: – The object’s distance from the axis (radius).

•  If linear velocity is held constant, more distance requires less force.

•  If angular velocity is held constant, more distance requires more force.

Centripetal Force

•  In symbols:

Fcent = mv2 r

“Centrifugal Force”

•  “Centrifugal force” is a fictitious force - it is not an interaction between 2 objects, and therefore not a real force.

• Nothing pulls an object away from the center of the circle.

“Centrifugal Force”

•  What is erroneously attributed to “centrifugal force” is actually the action of the object’s inertia - whatever velocity it has (speed + direction) it wants to keep.

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