phy1012f circular motion

NEWTON’S LAWS MOTION IN A CIRCLE

PHY1012FCIRCULAR MOTION

Gregor [email protected]

NEWTON’S LAWS MOTION IN A CIRCLEPHY1012F

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MOTION IN A CIRCLELearning 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.


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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


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Hence

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

PERIOD

v

rO

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

2 rv T

2 rv T 50 10000 rpm 6 msT

2 0.060.006v 63 m/s


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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 POSITIONIt 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, …


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ANGULAR POSITIONNotes: 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


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ANGULAR VELOCITYChange 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 velocity t

0limt

dt dt

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

r


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ANGULAR VELOCITYNotes: 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


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t

(rad/s)

–2

0

2 4 6 8 t (s)

For the first 3 s the

is

POSITION GRAPHS VELOCITY GRAPHSAngular 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


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(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


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(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


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FINDING POSITION FROM VELOCITYA 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


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THE rtz-COORDINATE SYSTEMTo 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


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

r

zt


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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

THE rtz-COORDINATE SYSTEMNotes:

A

A cos A sin


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/s] m/smrad

VELOCITY and ANGULAR VELOCITYThe 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

[


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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 VELOCITYAlthough 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

aa

a

Notes:

v

aiv

v

fv

a


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O

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

ACCELERATION and ANGULAR VELOCITY

v

v

v

a

a

aP

P'During time interval t …

rthe 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 va t r


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In vector notation:

arO

t

r

at = 0 az = 0

2va r , towards centre of circle

And since v = r… ar = 2r

vt

ACCELERATION and ANGULAR VELOCITY

Centripetal acceleration has only a radial component, ar …

2 2r

va rr


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DYNAMICS OF UNIFORM CIRCULAR MOTIONFrom Newton II…

2net

mvF 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


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DYNAMICS OF UNIFORM CIRCULAR MOTION

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

2 2

net r rrmvF F ma m rr

z t

Or v

netF

net 0t ttF F ma

net 0z zzF F ma

Necessarily so!


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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

2sr

mvF 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


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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

2r r

mvF 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

cosmgn


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vorbit

CIRCULAR ORBITSThe 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 2

orbitr

va gr

orbitv rg

w w

wr


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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 rv T

62 6.67 10 5184 s 86 min8085T

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

83.84 102 9.8T

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 rT v 2 r

rg 2 r

g


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NEWTON’S LAW OF UNIVERSAL GRAVITATIONAny 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 Gr

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

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NEWTON’S LAW OF UNIVERSAL GRAVITATION

eEarth on 2

em

M mF GR

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

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6.67 10 Nm /kg 5.98 10 kg

6.37 10 m

29.83 m/s


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e2

e( )GMg

R h

VARIATION OF g WITH HEIGHT ABOVE GROUND

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

2 2e e(1 / )

GMR h R

earth2

e(1 / )gh R

Height, h Example g (m/s2)9.83Sea level0 m9.81Kilimanjaro5 900 m9.80Jet airliner10 000 m8.85International Space Station350 000 m0.22Geosynchronous satellite35 900 000 m


on 2M mMmF Gr

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vorbit

CIRCULAR ORBITSWe can now derive more universal formulae for any satellite:

And, since ,

orbitGMv r

r

FM on mm

orbit2 r GMv T r

22 34T rGM

rma2

orbitmvr

M


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CIRCULAR ORBITSSo the correct period of the Moon is…

22 34T rGM

2 32 811 244 3.84 10

6.67 10 5.98 10T

T = 2.37 106 s = 27.4 days


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

INERTIAL and GRAVITATIONAL MASSThe 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.


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rt

t

at

NON -UNIFORM CIRCULAR MOTIONIf 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

tdva dt

The net acceleration vector, , is given by ,

where and .

2 2r ta a a 1tan t

r

aa

neta

ar

netaar

net r ta a a

neta


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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 2

net r rrmvF F ma m rr

net t ttF F ma

net 0z zzF F ma


at

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VERTICAL CIRCLESMotion 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 netaneta

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…

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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.

wT

w T

w

T

netF

netFw

w

w

T

netF

netF

T

T

netF

netF

netF

netFnetF


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vbot

At the bottom of a vertical circle…r

t

VERTICAL CIRCLESApparent 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

appm v

w n w r w

n 2bot

rm v

ma 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.


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vtopAt the top of a vertical circle… t

r

VERTICAL CIRCLES

2top

appm v

w n wr

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 2top

r

m vma r r r rF n w n w

The speed at which n = 0 is called the critical speed, vc: 2

c0m v

mgr c c gv rg r and

w

wvc


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ACCELERATION DUE TO GRAVITYWe 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.


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MOTION IN A CIRCLELearning 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 LAWSThe 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.

phy1012f circular motion

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