Going in circles

Why is circular motion cool?

you get accelerated! (due to change in direction)

a

v

Circular Motion in Our Daily Lives

Driving around curves & banks

Amusement park rides (loops & circles)

Weather patterns (jet streams, coriolis effect)

Horizontal Circles (Rotor)

Bart’sweight

Friction betweenBart and wall

wall pushing in on Bart

The inward wall force keeps Bart in the circle.Friction keeps him from falling down.

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vertical circles

Track provides centripetal force

You feel heavier at bottom since larger centripetal force needed to battle gravity

You feel lighter on top since gravity helps the track push you down

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Angled turns –

wings provide centripetal force

feel heavier if go faster in a tighter turn

Earth rotates in a tilted circle -high speed (800 mph), but small acceleration (adds .1% extra gravity)

- -west to east motion - curves south

(Coriolis effect- )

Uniform circular motion

R

v

The speed stays constant, but the direction changes

The acceleration in this case is called

centripetal acceleration, pointed toward the center!

a

8

Uniform Circular Motion: Period

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

• Distance = circumference = 2r

• Velocity = distance / time

• Period = time for one circle

Centripetal acceleration

• centripetal acceleration

• V is the tangential velocity(constant number with changing direction)

• F= ma is now…… F = mv2/r

2

C

va =

R

big R

little R

for the same speed, the tighter turn

requires moreacceleration

Wide turns and tight turns

Example• What is the tension in a string used to twirl a

0.3 kg ball at a speed of 2 m/s in a circle of 1 meter radius?

• Force = mass x acceleration [ m aC ]• acceleration aC = v2 / R = (2 m/s)2/ 1 m

= 4 m/s2

• force = m aC = 0.3 4 = 1.2 N• If the string is not strong enough to handle

this tension it will break and the ball goes off in a straight line.

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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.

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Examples of centripetal force

• Tension- ball on a string• Gravity- planet motion• Friction- cars • Normal Force- coasters & banked cars

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.

What’s this Centrifugal force ? ?• The red object will make the

turn only if there is enough friction on it

• otherwise it goes straight• the apparent outward force is

called the centrifugal force• it is NOT A REAL force!• an object will not move in a

circle until something makes it!

object onthe dashboard

straight lineobject naturally

follows

Work Done by the Centripetal Force

• Since the centripetal force on an object is always perpendicular to the object’s velocity, the centripetal force never does work on the object - no energy is transformed.

• W= Fd cos(90)=0v

Fcent

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Direction of Centripetal Force, Acceleration and Velocity

With 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.

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Tension Can Yield a Centripetal Acceleration:

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

F= Tension = mv2/r

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

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Friction Can Yield a Centripetal Acceleration:

F= friction = u*mg = mv2/r

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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?

F= mMG/r2 = mv2/r

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Banked CurvesWhy exit ramps in highways are banked?

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Artifical Gravity

F= Normal force = mv2/r

If v2/r = 9.8, seems like earth!

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horizontal Circular Motion(normal force always same)

F= Normal force = mv2/r (doesn’t matter where)

Like center of a vertical circle

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Vertical Circular Motion(normal force varies)

Top: mg + normal = mv2/r (normal smallest, v same)

side: normal = mv2/r (weight not centripetal, v same)

bottom: normal - mg = mv2/r (normal largest, v same)

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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.

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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.

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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.

The Apple & the Moon Isaac Newton realized that the motion of

a falling apple and the motion of the Moon were both actually the same motion, caused by the same force - the gravitational force.

Universal GravitationNewton’s idea was that gravity was a

universal force acting between any two objects.

At the Earth’s SurfaceNewton knew that the gravitational

force on the apple equals the apple’s weight, mg, where g = 9.8 m/s2.

W = mg

Weight of the MoonNewton reasoned that the centripetal

force on the moon was also supplied by the Earth’s gravitational force.

Fc = mg?

Law of Universal Gravitation In symbols, Newton’s Law of

Universal Gravitation is:

Fgrav = ma = G

Where G is a constant of proportionality. G = 6.67 x 10-11 N m2/kg2

Mmr 2

An Inverse-Square Force

Gravitational Field Strength(acceleration)

Near the surface of the Earth, g = F/m = 9.8 N/kg = 9.8 m/s2.

In general, g = GM/r2, where M is the mass of the object creating the field, r is the distance from the object’s center, and G = 6.67 x10-11 Nm2/kg2.

Gravitational Force If g is the strength of the gravitational

field at some point, then the gravitational force on an object of mass m at that point is Fgrav = mg.

If g is the gravitational field strength at some point (in N/kg), then the free fall acceleration at that point is also g (in m/s2).

Gravitational Field Inside a Planet

The blue-shaded partof the planet pulls youtoward point C.

The grey-shaded partof the planet does not pull you at all.

Black HolesWhen a very massive star gets old and

runs out of fusionable material, gravitational forces may cause it to collapse to a mathematical point - a singularity. All normal matter is crushed out of existence. This is a black hole.

Earth’s Tides2 high tides and 2 low tides per day.The tides follow the Moon.Differences due to sun not signficant

Why Two Tides? Tides due to stretching of a planet. Stretching due to difference in forces

on the two sides of an object. Since gravitational force depends on distance,

there is more gravitational force on the side of Earth closest to the Moon and less gravitational force on the side of Earth farther from the Moon. Not much difference from the Sun since it’s much further awayI

Why Two Tides?Remember that