CIRCULAR MOTION AND NEWTON’S LAW OF GRAVITATION

I. Speed and Velocity

Speed is distance divided by time…is it any different for an object moving around a circle?

The distance around a circle is C = 2πr, where r is the radius of the circle

So average speed must be the circumference divided by the time to get around the circle once

v =CT

=2πrT

one trip around the circle is known as the period. We’ll use big T to represent this time

SINCE THE SPEED INCREASES WITH RADIUS, CAN YOU VISUALIZE THAT IF YOU WERE SITTING ON A SPINNING DISK, YOU WOULD SPEED UP IF YOU MOVED CLOSER TO THE OUTER EDGE OF THE DISK?

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These four dots each make one revolution around the disk in the same time, but the one on the edge goes the longest distance. It must be moving with a greater speed.

Remember velocity is a vector. The direction of velocity in circular motion is on a tangent to the circle. The direction of the vector is ALWAYS changing in circular motion.

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II. ACCELERATION

If the velocity vector is always changing in circular motion,

THEN AN OBJECT IN CIRCULAR MOTION IS ACCELERATING.

THE ACCELERATION VECTOR POINTS INWARD TO THE CENTER OF THE MOTION.

Pick two v points on the path. Subtract head to tail…the resultant is the change in v or the acceleration vector

-vi vf

a

Note that the acceleration vector points in

vi

vf

a

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EQUATIONS:

acceleration in circular motion can be written as,

a=v2

r

a=4π 2rT 2

Assignment: check the units and do the algebra to make sure you believe these equations

vi

vf

a

If the acceleration vector points inward, there must be an inward force as well.

III. Centripetal and Centrifugal Force

The INWARD FORCE on an object in circular motion is called CENTRIPETAL FORCE

Without centripetal force, an object could not maintain a circular motion path. The velocity vector is telling you the object wants to go straight.

Centripetal force acts in a direction perpendicular to the velocity vector (the force, as it must be, is in the same direction as the acceleration).

Centripetal force causes the acceleration NOT by increasing (or changing) speed but by changing direction

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The outward force is fictitious. It is called CENTRIFUGAL force.

FAST merry-go-round

you

You feel like you are being pushed outward when you are on a circular motion path.

This is only because your moving body WANTS to travel tangentially to the circular path.

The only thing keeping you in a circle is YOU holding on.

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car

you in the turn

direction before curve

you before curve

Banking into a left curve, you feel like you are being thrown by an outward force to the right.

That’s not a real force. That’s you, wanting to continue in your straight line path (your inertia resists the change) meeting the side of the car that is moving in a circle.

You stay in a circular path thanks to your seat belt and the closed door of the car.

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III. Some Equations for Circular Motion

We know from Newton’s 2nd Law that F = ma

Here’s how the equation applies to circular motion

F = mv2

r

F = m4π 2rT 2

Using these equations, you can solve problems involving force on objects moving in a circular path.

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Example problem

A945‐kgcarmakesa180‐degreeturnwithaspeedof10.0m/s.Theradiusofthecirclethroughwhichthecaristurningis25.0m.DeterminetheforceoffricEonacEnguponthecar.

This problem is way easier than it sounds. You just need to realize what the source and direction of the friction force is.

A car “in a 180 degree turn” just means it is in a circular motion path.

The frictional force points inward. It is the centripetal force required to keep the car on the road in a circle.

Ffrict

Fnorm

Fgrav

Fnorm = F

grav

F

frict = Fnet

Ffrict = mv2

r

Ffrcit = 945kg(10 m

s)2

25mFfrict = 3780N

25 m

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IV. NEWTON”S LAW OF UNIVERSAL GRAVITATION

•  Newton applied his ideas about why objects fall to earth to planetary orbits

•  Basically he hypothesized that the Moon falls around the Earth for the same reason an object (like an apple?) falls toward the Earth…an attractive force he called GRAVITY

Assuming no air resistance…

Newton reasoned that if one could launch a projectile with sufficient speed that its horizontal path followed the Earth’s curvature, the force of gravity would cause it to “fall around” the Earth. The projectile would always fall toward the Earth without ever striking it.

Launch speeds too low, as we know from experience, crash to the Earth due to gravity

path of projectile without gravity

According to Newton, the projectile falling around the Earth was an analogy for the Moon orbiting the Earth

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•  Newton knew that objects near the Earth accelerated by 9.8 m/s2 •  He also knew the Moon accelerated toward the Earth by 0.00272 m/s2

• WHY THE DIFFERENCE IF GRAVITY CAUSES AN OBJECT TO FALL AND THE MOON TO ORBIT THE EARTH?

Look at the ratio of the accelerations:

9.8 ms2

0.00272 ms2

=1

3600

Now look at the ratio of the distances of from the surface of the Earth to its center and the Moon to the center of the Earth:

6378km382680km

=160

9.8 ms2

0.00272 ms2

= ( 6378km382680km

)2 = 13600

THIS SAYS THAT GRAVITY DECREASES AS THE INVERSE SQUARE OF DISTANCE

COMPARE THE TWO RATIOS ABOVE

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Earth’s gravity force represented with vectors. The arrows are smaller further away from Earth but still directed to the center of Earth

NEWTON SHOWED:

Force of attraction between two objects is inversely proportional to the square of the distance separating them

The Moon, being very far from the Earth, is less influenced by its gravity and so accelerates more slowly toward the Earth.

This not explain the source of the Moon’s velocity

F∝1r2

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Newton’s Law of Universal Gravitation is:

Fgrav = G

m1m2

r2The force of attraction between two objects is

DIRECTLY PROPORTIONAL TO THE PRODUCT OF THEIR MASSES INVERSELY PROPORTIONAL TO THE SQUARE OF THE DISTANCE BETWEEN THEM

The universal gravitational constant, G, was ingeniously determined by Cavendish in the 1700s. The value is use today is

G = 6.67428 ×10−11N( mkg)2

m1 m2 F1 F2

r

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V. Computing gravitational acceleration

Using the Universal Gravitational equation, we can derive an equation that allows you to compute the acceleration of gravity for other objects (besides Earth)

Fgrav = G

m1mEarth

r2

m1g = Gm1mEarth

r2

g = GmEarth

r2

generally, g = Gmplanet

r2r is the radius of the planet

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VI. SATTELITE MOTION

Remember the projectile falling around the Earth earlier in the notes? Remember the discussion of the Moon falling around the Earth?

This type of motion is the same as satellite motion and we can derive an equation for the velocity of a satellite orbiting the Earth using our equations for circular motion, centripetal force and universal gravitation.

A satellite in circular orbit, like a car rounding a curve MUST experience a centripetal force.

Fnet = msat

v2

r

Fgrav = G

msatmEarth

r2

Fnet = Fgrav

msatv2

r= G

msatmEarth

r2

v2 = GmEarth

r

v = GmEarth

r

velocity of satellite orbiting Earth a distance r from the center

…and the acceleration of the the satellite is like the acceleration due to gravity equation on the last page…

g = GmEarth

r2

asat = GmEarth

r2