Monday, October 19, 1998

Chapter 7: Newton’s Law of Gravity Gravitational Potential Energy Kepler’s Laws

Hint: Be able to do the homework (both theproblems to turn in AND the recommended ones)you’ll do fine on the exam!

Friday, October 23, 1998 in classChapters 5 - 7 inclusive

You may bring one 3”X5” index card (hand-writtenon both sides), a pencil or pen, and a scientificcalculator with you.

I will put any constants, math, and Ch. 1 - 4formulas which you might need on a singlepage attached to the back of the exam.

When an object experiences a centripetalacceleration, there must be a force actingwhich results in such an acceleration.

Fc = m ac

For objects moving in a circle, Newton’s 2nd Lawtakes the form

F F mamv

rnet c c tan2

What kind of forces might result ina centripetal force?

Force of Tension

What makes this carturn to the right?

Frictional Forceof Tires on Road

Component of Normal Forceof Road on Car

What if the car now goes aroundan inclined bend?

Frictional Forceof Tires on Road

Could point up or downthe road. On what willthe direction of thefrictional force depend?

Angle of the incline

Speed of the car

And for a given velocity, thereexists one angle at which thefrictional force exerted by theroad on the tires is 0.

Component of Normal Forceof Road on Car

FN

Fg

F F mgN gcos

F Fmv

rN csin 2

mgmv

r

sin

cos

2

tan 12v

rg

Component of Normal Forceof Road on Car

FN

Fg

Notice that in this case, the Normal Forceexerted by the road on the car is greater inmagnitude than the weight of the car!

WHY?Think about the direction ofthe tangential velocity andthe shape of the road in frontof the car...

Every massive particle in the Universeattracts every other massive particle inthe Universe with a force that is directlyproportional to the product of theirgravitational masses and inverselyproportional to the square of theirseparation distance.

Fmm

rg 1 2

122

To make this proportional relationship anequality, we simply multiply by a constantof proportionality...

FGmm

rg 1 2

122

The proportionality constant, G, is called theconstant of universal gravitation, and has beenexperimentally determined to be

G 6 673 102

31011

2

210

2

2.

Nm

kg

Nm

kg

The vector pointsin the direction fromone mass to the other.

To make our life simpler in applying theuniversal law of gravitation, we are aidedby the fact that the gravitational forceexerted by spherical objects acts as ifall the mass were concentrated at thecenter of the object!

So the gravitational forcethese astronauts feel highabove the Earth’s surfacedepends upon how far awayfrom the CENTER of theEarth they are!

And the force they feel hoveringabove the Earth would beexactly the same as the forceexerted by a point mass equalto the mass of the Earth andlocated at the very center ofthe Earth!

The planets inour solar systemmove around theSun in roughlycircular orbits.

This means thatsome centripetalforce must beacting on theplanets.

What force is responsible?

Given that the Earth (m = 6 X 1024 kg)orbits the Sun (m = 2 X 1030 kg) inroughly a circular orbit (r = 1.5 X 1011 m)once per year, calculated the mean orbitalspeed of the Earth.

F Fc G m v

r

Gm m

rE

ES

E S

ES

2

2

vGm

rS

ES

2 vGm

rS

ES

Given that the Earth (m = 6 X 1024 kg)orbits the Sun (m = 2 X 1030 kg) inroughly a circular orbit (r = 1.5 X 1011 m)once per year, calculated the mean orbitalspeed of the Earth.

vGm

rS

ES

( . )( )6 67 10 2 1011 2

230Nm

kg

11

kg

1.5 10 m

v 29 8. km / sSpaceship Earth travels through the Cosmosat a surprisingly high speed!