Circular motion

Have you ever been to an amusement park?

Fig. 05.19

Why somebody needs to know about circular motion

• Otherwise we wouldn’t go on those rides?

• Space station , satellites, live from overseas

• Planets, Comets and Moons

Uniform Circular Motion

• Velocity is a vector, it has both direction and magnitude

• Simplest case: direction constant, magnitude changes. Linear Motion

• We will now study another “simple” case: magnitude is constant and direction continually changes.

• Uniform Circular Motion

7

Circular Motion

X

θ = 0

y

i

f

How do we locate something on a circle?Give its angular position θ

What is the location of

900 or π/2

8

Circular Motion

is the angular position.

Angular displacement:

if

Note: angles measured Clockwise (CW) are negative and angles measured (CCW) are positive. is measured in radians.

2 radians = 360 = 1 revolution

x

y

i

f

What is a radian?

• C =2πr = (total angle of one rotation) x r• Radian measure relates displacement

along circumference to angular displacement

10

Circular Motion

X

θ = 0

y

i

f

How do we locate something on a circle?Give its angular position θ

What is the location of

900 or π/2

From θ =0 to θ = 900 we have gone ¼ of way around ¼ of 2π =π/2

11

x

y

i

f

r

arclength = s = r

r

s

is a ratio of two lengths; it is a dimensionless ratio! This is a radian measure of angle

If we go all the way round s =2πr and Δθ =2 π

Question

• A child runs around the circular ledge of a fountain which has a diameter of 6.5 m. After the child has run 9.8 m, her angular displacement is

A) 0.48 rad. B) 6.0 rad. C) 1.5 rad. D) 3.0 rad.

13

If we move around a circle we change θ.

How fast does θ change?

The average and instantaneous angular velocities ω are:

tt t

0

av lim and

is measured in rads/sec.

v and ω

• For circular motion—How far do you travel during Δt if you are spinning at an angular velocity ω?

• How far? s = Δθ r = ω Δt r• How fast? (speed)• v = s/Δt =ωr

v =ωr

Question

• A child runs around the circular ledge of a fountain which has a diameter of 6.5 m. If it

takes her 72 seconds to run all the way around, her angular velocity is

• A) 0.087 rad/s• B) 0.57 m/s• C) 0.17 rad/s• D) 0.044 rad/s

Question

• A CD makes one complete revolution every tenth of a second. The angular velocity of point 4 is:

• A) the same as for pt 1.• B) twice that of pt 2.• C) half that of pt 2.• D) 1/4 that of pt 1.• E) four times that of pt 1.

Question

• A CD makes one complete revolution every tenth of a second. Which has the largest linear (tangential) velocity?

• A) point 1• B) point 2• C) point 3• D) point 4

18

The time it takes to go one time around a closed path is called the period (T).

T

rv

2

timetotal

distance totalav

Comparing to v = r:

ω = v/rf

T 2

2

f is called the frequency, the number of revolutions (or cycles) per second.

f = 1/T

19

Centripetal Acceleration

Here, v 0.

The direction of v is changing.

If v 0, then a 0. The net force cannot be zero.

Consider an object moving in a circular path of radius r at constant speed.

x

y

v

v

v

v

20

Conclusion: to move in a circular path, an object must have a nonzero net force acting on it.

It is always true that F = ma.

What acceleration do we use?

21

The velocity of a particle is tangent to its path.

For an object moving in uniform circular motion, the acceleration is radially inward.

22

All three angles =Δθ = Δr/r =vΔt/rΔv=vΔθ = v2 Δt/r

23

The magnitude of the radial acceleration is:

vrr

var 2

2

Δv=vΔθ = v2 Δt/r

ar = Δv/Δt = v2Δt/rΔt

What makes it go round?

A particle in circular motion is accelerating sothere must be a force providing this acceleration

• F =ma • = mv2/r

Question

A spider is sitting on a turntable that is rotating at 33rpm. The acceleration a of the spider is

• A) greater the closer the spider is to the central axis.

• B) greater the farther the spider is from the central axis.

• C) nonzero and independent of his location.• D) zero.

27

Example (text problem 5.12): The rotor is an amusement park ride where people stand against the inside of a cylinder. Once the cylinder is spinning fast enough, the floor drops out.

(a) What force keeps the people from falling out the bottom of the cylinder?

Draw an FBD for a person with their back to the wall:

x

y

w

N

fs

It is the force of static friction.

28

Example (text problem 5.12): The rotor is an amusement park ride where people stand against the inside of a cylinder. Once the cylinder is spinning fast enough, the floor drops out.

(a) What force keeps the people from falling out the bottom of the cylinder?

x

y

w

N

fs

(b) If s = 0.40 and the cylinder has r = 2.5 m, what is the minimum angular speed of the cylinder so that the people don’t fall out?

29

Example (text problem 5.79): A coin is placed on a record that is rotating at 33.3 rpm. If s = 0.1, how far from the center of the record can the coin be placed without having it slip off?

Apply Newton’s 2nd Law:

0 2

1 2

wNF

rmmafF

y

rsx

x

y

w

N

fs

Draw an FBD for the coin:

Centrifuge

• Salad spinners, spin drying , Uranium centrifuges• Need force to keep spinning. If object overcomes

that force it moves out of circle.(spinners, drying)• More massive particles are harder to rotate• Fr = Mv2/r (uranium)

• If there is no normal force or friction (which depend on M) more massive particles will be “thrown” out of circle.

Question

• An object is in uniform circular motion. Which of the following statements is true?

• A) Velocity = constant• B) Speed = constant• C) Acceleration = constant• D) Both B and C are true

Fig. 05.12

Fig. 05.13

35

Unbanked and Banked Curves

Example (text problem 5.20): A highway curve has a radius of 825 m. At what angle should the road be banked so that a car traveling at 26.8 m/s has no tendency to skid sideways on the road? (Hint: No tendency to skid means the frictional force is zero.)

Take the car’s motion to be into the page.

36

Nonuniform Circular MotionHere, the speed is not constant.

There is now an acceleration tangent to the path of the particle.

The net acceleration of the body is 22tr aaa

at

v

ar

a

Recall v is tangential =ωr

37

at

ar

a

at changes the magnitude of v.

ar changes the direction of v.

tt

rr

maF

maFCan write:

Fig. 05.19

39

Example: What is the minimum speed for the car so that it maintains contact with the loop when it is in the pictured position?

FBD for the car at the top of the loop:

N w

y

x

Apply Newton’s 2nd Law:

r

vmwN

r

vmmawNF ry

2

2

r

40

The apparent weight at the top of loop is:

gr

vmN

r

vmwN

2

2

N = 0 when

grv

gr

vmN

0

2

This is the minimum speed needed to make it around the loop.

Example continued:

• A boy swings on a tire swing. When is the tension in the rope the greatest?

• A) At the highest point of the swing.

• B) Midway between his highest • and lowest points.• C) At the lowest point in the

swing.• D) It is constant.

42

Tangential and Angular Acceleration

The average and instantaneous angular acceleration are:

tt t

0

av lim and

is measured in rads/sec2.

43

Recall that the tangential velocity is vt = r. This means the tangential acceleration is

rrtt

tt

v

44

The kinematic equations:

xavv

tatvxx

tavv

2

2

1

20

2

200

0

2

2

1

20

2

200

0

tt

t

Linear Angular

With rarv tt and

• r=67.5m• T=30 min• tto cruise =20s

• α = ?• Δθ =?

Gravity and Circular Motion

• What keeps the Moon going around the Earth?

Why do planets go around the Sun? What determines the orbits of

satellites?• How can we “weigh” the Earth? the Sun?

• What is the Universe made of?

Circular Orbits

r

vmam

r

MGmFF srs

esg

2

2

Consider an object of mass m in a circular orbit about the Earth.

Earth

rThe only force on the satellite is the force of gravity:

r

GMv

r

vm

r

MGm

e

ses

2

2

The speed of the satellite:

How much does the Earth weigh?Solve for Me

G

rvM

r

vm

r

MGm

e

ses

2

2

2

We measure v of satellite , we measure how high it is (r) , we can measure G in the lab, hence we calculate Me .

We have weighed the Earth!

Weigh the Sun

• What about the Earth going around the sun?• Gravity keeps the Earth moving around Sun.

Same equations as for the satellite. Gravity is universal

• r R (distance from Sun to Earth)

• ms Ms

G

RvM

R

vM

R

MGM

s

eeS

2

2

2

Weighing the Sun

• How fast are we traveling around the Sun? (v)

• v=2πR/T T = 1 year

• From astronomy we can measure R

• We have weighed the SunG

RvM s

2

Gravitational Orbits

• We can do this for any objects in gravitational orbit around each other. Independent of the mass of the orbiting object. Depends only on the mass of the central object.

• Planets around Sun, Moons around Planets …..• General formula

R

GMv

What is the Universe made of ?

What is the Universe made of ?

• We see mostly galaxies. • We can measure the velocity of stars in a

galaxy• Most of the bright stuff (luminous matter,

stars) is in the center. We are made of the same stuff as stars.

• We expect

R

GMv

Rv

1

Rv

1

R

GMv

56

Example: How high above the surface of the Earth does a satellite need to be so that it has an orbit period of 24 hours?

r

GMv eFrom previous problem: Also need,

T

rv

2

Combine and solve for r:3

1

224

TGM

r e

km 000,35 ee RrhhRr

57

Example: How high above the surface of the Earth does a satellite need to be so that it has an orbit period of 24 hours?

r

GMv eFrom previous slide: Also need,

T

rv

2

Combine these expressions and solve for r:

31

224

TGM

r e

m 10225.4

s 864004

kg 1098.5/kgNm1067.6

7

31

2

2

242211

r

km 000,35 ee RrhhRr