1

Uniform Circular Motion

http://www.physicsclassroom.com/mmedia/circmot/circmotTOC.html

2

Uniform circular motion

• motion of an object in a circle with a constant or uniform speed

• constant change in direction

3

Uniform Circular Motion: Period

Object repeatedly finds itself back where it started.

The time it takes to travel one “cycle” is the “period”.

distance = rate time

time =distance

rate v

T =2 r

v

2

r

4

Quantifying Acceleration: Magnitude

v1

v2

Similar Triangles: v

v

x

x

5

v

v

v t

r

vv t

r

av

t

v

r

2

2

Quantifying Acceleration: Magnitude

Centripetal Acceleration

6

Applying Newton’s 2nd Law:

F ma

Fmv

r

2

Centripetal Force

Always points toward center of circle. (Always changing direction!)

Centripetal force is the magnitude of the force required to maintain uniform circular motion.

7

Direction of Centripetal Force, Acceleration and Velocity

Without a centripetal force, an object in

motion continues along a straight-line path.

Without a centripetal force, an object in

motion continues along a straight-line path.

8

Direction of Centripetal Force, Acceleration and Velocity

9

What if velocity decreases?

10

What if mass decreases?

11

What if radius decreases?

12

What provides the centripetal force?

• Tension• Gravity• Friction• Normal Force

Centripetal force is NOT a new “force”. It is simply a way of quantifying the magnitude of the force required to maintain a certain speed around a circular path of a certain radius.

13

Relationship Between Variables of Uniform Circular Motion

Suppose two identical objects go around in horizontal circles of identical diameter but one object goes around the circle twice as fast as the other. The force required to keep the faster object on the circular path is

A. the same as

B. one fourth of

C. half of

D. twice

E. four times

the force required to keep the slower object on the path.

The answer is E. As the velocity increases the centripetal force required to maintain the circle increases as the square of the speed.

14

Relationship Between Variables of Uniform Circular Motion

Suppose two identical objects go around in horizontal circles with the same speed. The diameter of one circle is half of the diameter of the other. The force required to keep the object on the smaller circular path is

A. the same as B. one fourth of C. half of D. twice E. four times the force required to keep the object on the larger path.

The answer is D. The centripetal force needed to maintain the circular motion of an object is inversely proportional to the radius of the circle. Everybody knows that it is harder to navigate a sharp turn than a wide turn.

15

Relationship Between Variables of Uniform Circular Motion

Suppose two identical objects go around in horizontal circles of identical diameter and speed but one object has twice the mass of the other. The force required to keep the more massive object on the circular path is

A. the same as

B. one fourth of

C. half of

D. twice

E. four times

Answer: D.The mass is directly proportional to centripetal force.

16

Tension Can Yield a Centripetal Acceleration:

If the person doubles the speed of the airplane, what happens to the tension in the cable?

F = mamv

r

2

Doubling the speed, quadruples the force (i.e. tension) required to keep the plane in uniform circular motion.

17

Friction Can Yield a Centripetal Acceleration:

18

Friction provides the centripetal acceleration

Car Traveling Around a Circular Track

19

Friction Can Yield a Centripetal Acceleration

W

FN

fs

Force X Y

W 0 -mg

FN 0 FN

fs -sFN 0

Sum ma 0

What is the maximum speed that a car can use around a curve of radius “r”?

20

Force X Y

W 0 -mg

FN 0 FN

FC -sFN 0

Sum ma 0

F mg F

F mg

F ma mg

mv

rmg

v g r

v g r

y N

N

x s

s

s

s

0

2

2

max

max

max

max

Maximum Velocity

21

F = mamv

r

2

Centripetal Force: Question

Smaller radius: larger force required to keep it in uniform circular motion.

A car travels at a constant speed around two curves. Where is the car most likely to skid? Why?

22

Gravity Can Yield a Centripetal Acceleration:

Hubble Space Telescopeorbits at an altitude of 598 km(height above Earth’s surface).What is its orbital speed?

23

Gravity and Centripetal Acceleration:

Centripetal acceleration provided by gravitational force

G m M

R

m v

RE

2

2

24

Gravity and Centripetal Acceleration:

G m M

R

m v

RE

2

2

Solve for the velocity….

vG m M R

m R

vG M

R

vG M

R

E

E

E

22

2

25

Hubble Space Telescope:

vGM

R km

v

v

E

E

598

6 67 10 974 10

7 600

11 24( . ) (5.

,

m kg s kg)

6,976,000 m

m / s

3 -1 -2

26

Banked Curves Why exit ramps in highways are banked?

27

Banked CurvesQ: Why exit ramps in highways are banked?

28

Banked CurvesQ: Why exit ramps in highways are banked?

A: To increase the centripetal force for the higher exit speed.

29

The Normal Force Can Yield a Centripetal Acceleration:

How many forces areacting on the car (assumingno friction)?

Engineers have learned to “bank” curves so that cars can safely travel around the curve without relying on friction at all to supply the centripetal acceleration.

30

Banked CurvesWhy exit ramps in highways are banked?

FN cos = mg

31

Banked CurvesWhy exit ramps in highways are banked?

FN cos = mg

32

The Normal Force as a Centripetal Force:

Two: Gravity and Normal

Force X Y

W 0 -mg

FN FNsin FNcos

Sum ma 0

33

The Normal Force as a Centripetal Force:

F mg F

mg

F F mamv

r

mg mv

r

v

gr

y N

x N

cos

cos

sin

cossin

tan

0

2

2

2

FN

34

The Normal Force and Centripetal Acceleration:

tan v

gr

2

How to bank a curve…

…so that you don’t rely on friction at all!!

35

Artifical Gravity

36

Vertical Circular Motion

37

 Vertical Circular Motion

38

The End!