Chapter 5: Circular Motion

• Uniform circular motion

• Radial acceleration

• Unbanked turns (banked)

• Circular orbits: Kepler’s laws

• Non-uniform circular motion

• Tangential & Angular acceleration

• (apparent weight, artificial gravity)

• Hk: CQ 1, 2. Prob: 5, 11, 15, 19, 39, 49.1

2

angular measurement

• degrees (arbitrary numbering system, e.g. some systems use 400)

• radians (ratio of distances)

• e.g. distance traveled by object is product of angle and radius.

3

Radians

r

s

s = arc length

r = radius

rs rad/s

t

s

rad/s t

4

motion tangent to circle

rs

r

t

r

t

sv

r

t

r

t

va

5

Angular Motion

• radian/second (radian/second)/second

tavg

tavg

6

angular conversions

Convert 30° to radians:

radradrad

52.06360

2

1

30

Convert 15 rpm to radians/s

sradsrev

radrev/57.1

60

min12

min

15

7

Angular Equations of Motion

to

22

1 tto

to 21

222o

Valid for constant- only

8

Centripetal Acceleration

• Turning is an acceleration toward center of turn-radius and is called Centripetal Acceleration

• Centripetal is left/right direction

• a(centripetal) = v2/r

• (v = speed, r = radius of turn)

• Ex. V = 6m/s, r = 4m. a(centripetal) = 6^2/4 = 9 m/s/s

Top View Back View

f f

NF

mg

Centripetal Force

Acceleration with Non-Uniform Circular Motion

• Total acceleration = tangential + centripetal

• = forward/backward + left/right

• a(total) = r (F/B) + v2/r (L/R)

• Ex. Accelerating out of a turn; 4.0 m/s/s (F) + 3.0 m/s/s (L)

• a(total) = 5.0 m/s/s

11

Centripetal Force

• required for circular motion

• Fc = mac = mv2/r

• Example:

• 1.5kg moves in r = 2m circle v = 8m/s.

• ac = v2/r = 64/2 = 32m/s/s

• Fc = mac = (1.5kg)(32m/s/s) = 48N

12

Rounding a Corner

• How much horizontal force is required for a 2000kg car to round a corner, radius = 100m, at a speed of 25m/s?

• Answer: F = mv2/r = (2000)(25)(25)/(100) = 12,500N

• What percent is this force of the weight of the car?

• Answer: % = 12,500/19,600 = 64%

13

Mass on Spring 1

• A 1kg mass attached to spring does r = 0.15m circular motion at a speed of 2m/s. What is the tension in the spring?

• Answer: T = mv2/r = (1)(2)(2)/(.15) = 26.7N

14

Mass on Spring 2

• A 1kg mass attached to spring does r = 0.15m circular motion with a tension in the spring equal to 9.8N. What is the speed of the mass?

• Answer: T = mv2/r, v2 = Tr/m

• v = sqrt{(9.8)(0.15)/(1)} = 1.21m/s

15

Kepler’s Laws

Kepler’s Laws of Orbits

1. Elliptical orbits

2. Equal areas in equal times (ang. Mom.)

3. Square of year ~ cube of radius

Elliptical Orbits

• One side slowing, one side speeding

• Conservation of Mech. Energy

• ellipse shape

• simulated orbits

18

Summary

• s = r v = r

• a(tangential) = r.

• a(centripetal) = v2/r

• F(grav) = GMm/r2

• Kepler’s Laws, Energy, Angular Momentum

19

Centrifugal Force

• The “apparent” force on an object, due to a net force, which is opposite in direction to the net force.

• Ex. A moving car makes a sudden turn to the left. You feel forced to the right of the car.

• Similarly, if a car accelerates forward, you feel pressed backward into the seat.

20

rotational speeds

• rpm = rev/min

• frequency “f” = cycles/sec

• period “T” = sec/cycle = 1/f

• degrees/sec

• rad/sec = 2f

7-43

• Merry go round: 24 rev in 3.0min.

• W-avg: 0.83 rad/s

• V = rw = (4m)(0.83rad/s) = 3.3m/s

• V = rw = (5m)(0.83rad/s) = 4.2m/s

22

Rolling Motion

v = vcm = R

23

Example: Rolling

A wheel with radius 0.25m is rolling at 18m/s. What is its rotational rate?

RvRv /: sradsmsm mm /72/7225.0//18

srads

radsrad /4127

2

36072/72

24

Example

to

A car wheel angularly accelerates uniformly from 1.5rad/s with rate 3.0rad/s2 for 5.0s. What is the final angular velocity?

sradssradsrad /5.16)0.5)(2/0.3(/5.1

What angle is subtended during this time?

to 21 rad45)5(5.165.12

1

22

1 tto rad455355.1 22

1

222o 25.27245325.1 2

srad /5.1625.272

25

Ex: Changing Units

srevrad

rev

s

radsrad /1592.0

2

11/1

srevsrad

srevsradsrad /5.11

/1

/1592.0/72/72

rpms

rad

rev

s

radsrad 688

min1

60

2

172/72

srev

sT 0869.

5.11

1

26

Rotational Motion

rvt

ttct vvvvv 0222

rat

22

rr

vac

vt at

ac

ac

vt

r

0cv

Convert 50 rpm into rad/s.

• (50rev/min)(6.28rad/rev)(1min/60s)

• 5.23rad/s