NEWTON’S LAWSMOTION IN A CIRCLE PHY1012F CIRCULAR MOTION Gregor Leigh [email protected]

NEWTON’S LAWS MOTION IN A CIRCLE

PHY1012F

CIRCULAR

MOTION

Gregor [email protected]

NEWTON’S LAWS MOTION IN A CIRCLEPHY1012F

2

MOTION IN A CIRCLE

Learning outcomes:At the end of this chapter you should be able to…

Apply kinematics and dynamics knowledge, skills and techniques to circular motion.

Manipulate angular quantities and formulae against the background of an angular (rtz-) coordinate system.


3

In the particle model the centre of the circle lies outside the particle, and we speak of orbital motion.Later we shall apply the same principles to the rotation or spin of extended objects about axes within themselves.

Any particle travelling at constant speed around a circle is engaged in uniform circular motion.

UNIFORM CIRCULAR MOTION

v

v

v

r r

r

O

The magnitude of is constant, but since is everywhere tangent to the circle, its direction changes continuously.

v

v


4

Hence

The time taken for the particle to complete one revolution (rev) is called the period, T, of the motion.

PERIOD

v

r

O

E.g. Calculate the speed of a point on the rim of a CD in a 50x drive…

2 rvT

2 rvT 50 10000 rpm 6 msT

2 0.060.006

v 63 m/s


5

is positive when measured counterclockwise (ccw) from the positive x-axis;

is conveniently measured in radians (SI unit), where 1 rad is the angle subtended at the centre by an arc length s = r;

is the single time-dependent quantity of circular motion.

ANGULAR POSITION

It will be more convenient to describe the position of an orbiting particle in terms of polar coordinates rather than xy-coordinates.

y

xO

r s

, called the angular position of the particle, …


6

ANGULAR POSITION

Notes: and

s = r ( in rad).

The radian is a dimensionless unit (as is any unit of angle).

(rad) sr

3601 rad 57.3 602

y

x

O

r s

2 rad 2 radrr 1 rev360


7

ANGULAR VELOCITY

Change in angular position is called angular displacement, .

y

xiO

rti

f

tf = t i + t

Analogous to linear motion, the rate of change of angular position is called average angular velocity:

Allowing t0, we get (instantaneous) angular velocity:

average angular velocityt

0limt

dt dt

Units: [rad/s] (SI), but also[°/s, rev/s, and rev/min rpm]

r


8

ANGULAR VELOCITY

Notes: A particle moves with uniform circular motion if and only if its angular velocity is constant.

, sign by inspection…

Angular velocity is positive for counterclockwise motion….

…negative for clockwise motion.

The graphical relationships we developed for position s and velocity vs

in linear motion apply equally well to angular position and angular velocity …

> 0

< 0

2 radT


9

t

(rad/s)

–2

0

2 4 6 8 t (s)

For the first 3 s the

is

POSITION GRAPHS VELOCITY GRAPHS

Angular velocity is equivalent to the slope of a -vs-t graph. (rad)

–2

0

–4

2

t (s)2 4 6 8

4 2 2 rad/s3 0t

Eg: A particle moves around a circle…

velocityslope


10

(rad/s)

–2

2 4 60

8 t (s)

Between 3 s and 4 s the

is



t (s)–2

2 4 60

–4

2

8

4 4 0 rad/s1t


velocityslope


11

(rad/s)

–2

2 4 60

8 t (s)

Between 4 s and 8 s the

is



t (s)–2

2 4 60

–4

2

8

0 4 rad/s4t


velocityslope

t


12

FINDING POSITION FROM VELOCITY

A body’s angular position after a time interval t can be determined from its angular velocity using .

f i t

Graphically, the change in angular position ( = t) is given by the area “under” a -vs-t graph:

t

During the time interval 2 s to 8 s the body’s angular displacement is

2 4 60

8t (s)

2

(rad/s)

12 rad i.e. 6 revs ccw

2 rad/s 8 2 st


13

THE rtz-COORDINATE SYSTEM

To facilitate the resolution of angular quantities, we introduce the rtz-coordinate system (centred on the orbiting particle and travelling around with it) in which…

zt

O r

t

r

the r-axis (radial axis) points from the particle towards the centre of the circle;the t-axis (tangential axis) is tangent to the circle, pointing in the anticlockwise direction;the z-axis is perpendicular to the plane of motion.

z


14


Viewed from above (with the z-axis pointing out of the screen) the axes are shown travelling around with the particle…

r

zt


15

As in xyz-coordinate system, the r-, t-, and z-axes are mutually perpendicular.

The rtz-coordinate system is used only to resolve vector quantities associated with circular motion into radial and tangential components. The measurement of these quantities must necessarily take place in other reference frames.

Given some vector in the plane of motion, making an angle of with the r-axis,

Ar = A cos

At = A sin

tr

A


Notes:

A

A cos A sin


16

/s] m/smrad

VELOCITY and ANGULAR VELOCITY

The velocity vector has only a tangential component, vt .

rO

v

t

r

s = r

vr = 0 vt = r vz = 0

s

Differentiating with respect to time…

tds dv rdt dt

tvr

Hence vt = r and

vt vt

[


17

For uniform circular motion, since the lengths of successive ’s are all the same, the magnitude of is constant.

These are all average velocity vectors…

ACCELERATION and ANGULAR VELOCITY

Although the magnitude of remains constant in uniform circular motion, its direction changes continuously, so the particle must be accelerating.

v

Motion diagram analysis reveals that the acceleration is centripetal.

fv

iv

a

a

a

Notes:

v

a iv

v

fv

a


18

O

The instantaneous velocity and acceleration vectors are everywhere at right angles to each other.


v

v

v

a

a

a

P

P'During time interval t …

r

the particle travels an arc length vt between P and P' (PP' vt);

both the angular position and turn through angles of ;

v

v

v

r

Q

Q'

…so OPP' ||| P'QQ'

v v tv r

2v vt r

2

0limt

v vat r


19

In vector notation:

arO

t

r

at = 0 az = 0

2var

, towards centre of circle

And since v = r… ar = 2r

vt


Centripetal acceleration has only a radial component, ar …

22

rva rr


20

DYNAMICS OF UNIFORM CIRCULAR MOTION

From Newton II…

2

netmvF ma

r

, towards centre of circle

z t

Or v

netF

Note! As always, is simply the result of any number of forces being applied by identifiable agents.(It is NOT some new, disembodied force!)

netF


21

DYNAMICS OF UNIFORM CIRCULAR MOTION

In terms of r-, t-, and z-components:

2

2net r rr

mvF F ma m rr

z t

Or v

netF

net 0t ttF F ma

net 0z zzF F ma

Necessarily so!


22

s maxmax

rfv

m

Determine the maximum speed at which a car can corner on an unbanked, dry tar road without skidding.

O

tr

z

vr

n

w

2

srmvF f

r

sf

srfv

m

v will be a maximum when fs

reaches its maximum value: fs =

fs max = snFz = n – w = 0 n = w = mg

max sv rg

sr mgm


23

r

z

A highway curve is banked at an angle to the horizontal. Determine the maximum speed at which a car can take this corner without the assistance of friction.

O

tz

r vr

2

r rmvF n

r

rrnv

m

nr = n sin

Fz = nz – w = 0 nz = n cos = w = mg

sincos

rmgv

m

tanv rg

n

w

nr

nz

cosmg

n


24

vorbit

CIRCULAR ORBITS

The force which keeps satellites (including the Moon) moving in circular orbits around the Earth is nothing other than the gravitational force of the Earth on them.A near-Earth satellite will maintain its circular orbit only if its centripetal acceleration ar is equal to

g.I.e. if

2orbit

r

va g

r

orbitv rg

w

w

wr


25

In 1957 Earth’s first artificial satellite, Sputnik I, was put into orbit 300 km above the Earth’s surface by the USSR. How long did observers have to wait between sightings?

6 5orbit 6.37 10 3 10 9.8 8085 m/sv rg

2 rvT

62 6.67 10 5184 s 86 min8085

T

Whereas the period of Earth’s natural satellite, the Moon (384 000 km away) is…

83.84 1029.8

T

The problem lies in the fact that g is only a local constant which can be used only near the surface of the Earth…

??!39330 s 11 hours2 rTv 2 r

rg 2 r

g


26

NEWTON’S LAW OF UNIVERSAL GRAVITATION

Any two particles in the universe exert a mutually attractive force on each other which is proportional to the product of their masses and inversely proportional to the square of the distance of their separation.

1 21 on 2 2 on 1 2

m mF F G

r

G is the universal gravitation constant.

G = 6.67 10–11 N m2/kg2.

The equation holds for extended spherical masses (e.g. planets) provided r, the distance between their centres, is large compared to their sizes.

Notes:


e2

e

Mg G

R

PHY1012F

27

NEWTON’S LAW OF UNIVERSAL GRAVITATION

eEarth on 2

em

M mF G

R

For a body of mass m at the surface of the Earth:

But this is the body’s weight, w = mg…

Hence, for Earth,

The value of the gravitational constant on the surface of any planet, gplanet, is thus a direct consequence of the

size and mass of that planet.

(Why not 9.80 m/s2?)

11 2 2 24

26

6.67 10 Nm /kg 5.98 10 kg

6.37 10 m

29.83 m/s


28

e2

e( )

GMg

R h

VARIATION OF g WITH HEIGHT ABOVE GROUND

For a satellite a distance h above the earth’s surface:

e2 2e e(1 / )

GM

R h R

earth

2e(1 / )

g

h R

Height, h Example g (m/s2)9.83Sea level0 m

9.81Kilimanjaro5 900 m

9.80Jet airliner10 000 m

8.85International Space Station350 000 m

0.22Geosynchronous satellite35 900 000 m


on 2M mMmF Gr

PHY1012F

29

vorbit

CIRCULAR ORBITS

We can now derive more universal formulae for any satellite:

And, since ,

orbitGMv

r

r

FM on mm

orbit2 r GMvT r

22 34T r

GM

rma2

orbitmvr

M


30

CIRCULAR ORBITS

So the correct period of the Moon is…

22 34T r

GM

2 32 811 244 3.84 10

6.67 10 5.98 10T

T = 2.37 106 s = 27.4 days


31

To date, however, the most precise experiments have been unable to determine any measurable difference between the two.

INERTIAL and GRAVITATIONAL MASS

The connection between inertial mass (found by meas-uring a body’s acceleration a in response to a force F) and the gravitational mass which causes two bodies to attract each other is not immediately apparent…

Einstein’s general theory of relativity explains this principle of equivalence (minert = mgrav) as a fundamental

property of space-time.


32

rt

t

at

NON -UNIFORM CIRCULAR MOTION

If the speed of an orbiting body varies, the body is exhibiting non-uniform circular motion.

at

In such cases, in addition to centripetal acceleration, the body also has non-zero tangential acceleration: t

tdv

adt

The net acceleration vector, , is given by ,

where and .

2 2r ta a a 1tan t

r

aa

neta

ar

neta

ar

net r ta a a

neta


33

r

t

DYNAMICS OF NON-UNIFORM CIRCULAR MOTION

The resultant force (the sum of any number of individual forces) acting on an orbiting particle can always be resolved into tangential and radial components if required…

(Fnet)r

netF

(Fnet)t

2

2net r rr

mvF F ma m rr

net t ttF F ma

net 0z zzF F ma


at

PHY1012F

34

VERTICAL CIRCLES

Motion in a vertical circle is NOT uniform.

Only at the top and bottom is at =

0.

As a result of gravity, on its way down, the body speeds up; on the way up it slows down...

going down, at & vt are

parallel;

going up, at & vt are

antiparallel.

at = 0

at = 0

Elsewhere, the net acceleration is given by .

at

at

ar

ar

at

at

at

ar

ar

ar ar

The magnitude and direction of this net acceleration change continuously in a complex way…

net r ta a a

neta

neta

neta

neta

neta

neta


…but at the top and bottom, where at = 0, is centripetal.

netF

The net force, , which produces this

acceleration, is made up of the body’s weight and the tension force provided by the string.

Like the acceleration,

varies around the circle…

PHY1012F

35

VERTICAL CIRCLES

Note:

At the top of the circle is the sum of and .

At the bottom, is given by the difference of the two.

w

T

w T

w

T

netF

netFw

w

w

T

netF

netF

T

T

netF

netF

netF

netFnetF


36

vbot

At the bottom of a vertical circle…r

t

VERTICAL CIRCLES

Apparent weight is actually a sensation arising from the contact forces which support you, rather than an awareness of the gravitational force of the Earth which acts simultaneously on every part of you.

2bot

app

m vw n w

r

w

n

2bot

r

m vma

r r r rF n w n w

The extra force required to achieve this is what “adds to your g’s” in a bottom turn.


37

vtop

At the top of a vertical circle… t

r

VERTICAL CIRCLES

2top

app

m vw n w

r

If (because of lack of speed) this term becomes too small (i.e. < w), n disappears, the body “comes unstuck” and goes into free fall.

wn

2

topr

m vma

r r r rF n w n w

The speed at which n = 0 is called the critical speed, vc:

2c0

m vmg

r c c

gv rg

r and

w

w

vc


38

ACCELERATION DUE TO GRAVITY

We are now in a position to understand why the measured value of g = 9.80 m/s2 is less than the value we calculated from Newton’s law of universal gravitation (g = 9.83 m/s2).Objects on the rotating Earth are in circular motion, so there must be a net force towards the centre. Thus wapp = mgapp = n < Fgrav.

At mid-latitudes the reduction is about 0.03 m/s2, hence the measured value gapp = 9.80 m/s2.


39

MOTION IN A CIRCLE

Learning outcomes:At the end of this chapter you should be able to…

Apply kinematics and dynamics knowledge, skills and techniques to circular motion.

Manipulate angular quantities and formulae against the background of an angular (rtz-) coordinate system.


NEWTON’S LAWS

The goals of Part I, Newton’s Laws, were to…

Learn how to describe motion both qualitatively and quantitatively so that, ultimately, we could analyse it mathematically.

Develop a “Newtonian intuition” for the explanation of motion: the connection between force and acceleration.

Documents

NEWTON’S LAWSMOTION IN A CIRCLE PHY1012F CIRCULAR MOTION Gregor Leigh [email protected]