Forces and Fields (6)

In the most fundamental equations about the universe, we find fields.

Black holes, the Aurora Borealis, and microwave ovens all are understood in terms of fields.

Fields are abstract, but quite real.

Mr. KlapholzShaker Heights

High School

True or False?

• There is a force of attraction at the earth’s surface.• The force of attraction between an object and the

earth depends on the mass of the object.• The acceleration of a falling object depends on the

mass of the object.• The force of attraction between an object and the

earth is called the weight of the object.

Gravitational Force and Field

True or False?

• On a mountaintop, the acceleration of falling objects is less than at the earth’s surface.

• If you go high enough, the force of attraction becomes zero.

• There is no gravity in space.• Every object attracts every other object.• The farther two objects are apart, the less the

gravitational force of attraction.

Gravitation Basics

M1 M2Forceon 1

Forceon 2

R

Gravitation Basics

• The force on object One equals the force on object _ _ _. Recall the _ _ _ _ _ law.

Gravitation Basics

• The force on object One equals the force on object Two. Recall the third law.

Gravitation Basics

• The force on object One equals the force on object Two. Recall the third law.

• The force is proportional to the mass of object _ _ _.

Gravitation Basics

• The force on object One equals the force on object Two. Recall the third law.

• The force is proportional to the mass of object One. F M1

Gravitation Basics

• The force on object One equals the force on object Two. Recall the third law.

• The force is proportional to the mass of object One. F M1

• The force is also proportional to the mass of object _ _ _.

Gravitation Basics

• The force on object One equals the force on object Two. Recall the third law.

• The force is proportional to the mass of object One. F M1

• The force is also proportional to the mass of object Two. F M2

Force varies dramatically with Distance

Distance Dist2 Force

1 1 1

2 4 1/4 = 0.25

3 9 1/9 = 0.11

4 16 1/16 = 0.04

Gravitation and Distance (R)

The farther apart the objects,the _ _ _ _ the attraction.

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F∝1

R2

Gravitation and Distance (R)

The farther apart the objects,the less the attraction.

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F∝1

R2

Put all our ideas together

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F ∝M1M2

R2

Change the proportionality to an equality by including a constant.

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F =GM1M2

R2

G

• The universal gravitation constant was measured by Cavendish and Jolly.

• G is very _ _ _ _ _.

G = 6.67 x 10-11 Nm2/kg2

G

• The universal gravitation constant was measured by Cavendish and Jolly.

• G is very small.

G = 6.67 x 10-11 Nm2/kg2

Can you put this law into words?

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F =GM1M2

R2

Strictly speaking, this law applies only to ‘point masses’. Still it works perfectly for spherical masses too.

€

F =GM1M2

R2

And now, a quiet moment to contemplate what keeps the moon in earth orbit.

Period of the earth’s motion around the sun

Mass of the sun: 1.99 x 1030 kg Mass of the earth: 5.97 x 1024 kg Distance between earth and sun: 1.49 x1011mCalculate the amount of force that the sun puts on the earth. From this, calculate the speed of the earth.From this calculate the period of the earth.

Period of the earth’s motion around the sun

F = GMm / R2

F = (6.67x10-11)(1.99x1030)(5.97x1024) / (1.49 x1011)2

F = 3.56 x1022 N

Circular motion: a = v2 / R

F = Ma = Mv2 / R

3.56 x1022 N = (1.49 x1011) v2 / (1.49 x1011)

v = 29850 m s-1

v = Dist / Time = circumference / T = 2pR / T

T = 2pR / v = 2(3.14)(1.49 x1011) / (29850)

T = 3.14 x 107 s = 1 year

Gravity Man, don’tfeel so sad.

“But with such a small constant, I’m the weakest force (…sob…) in the

universe!”Yes, it’s true that for two small charged particles, Gravity is less than the Electrical force. But for uncharged big things like planets and galaxies, the electrical forces

cancel out and you, …you rule the Universe!

Fields• This is an abstract idea, to be sure.• The field describes how an object affects the

space around it. • Fields have ‘strength’ (‘intensity’ or ‘magnitude’).• Fields have direction; fields are vectors. • The direction of the field is the direction in which

an object (often called ‘”a mass”) would be forced, if there was an object there.

• There might not be an object in the field, but the field is still there!

Direction of the Gravitational Field due to the earth (or even a point mass).

Direction of the Gravitational Field due to the earth, near the earth.

Strength of the Gravitational Field at a distance of R, made by mass M:

g = { GMm/R2 } / m

Field = Force / Mass

Strength of the Gravitational Field at a distance of R, made by mass M:

g = GM / R2

g = { GMm/R2 } / m

Field = Force / Mass

The uptight definition of the Gravitational Field:

g = limm0 F / m

In the Problem solving section, include a problem that uses v2/R

• Use GMM/R2 to get the force on the earth.• Use Centripetal force to get velocity.• Use R to turn that into period and see that it’s

one year.

A 2 kg object is 6.38 x 106 m from a 5.98 x 1024 kg object.

What is the force of attraction?

What is the acceleration ofthe 2 kg object?

Practice calculating force, and introduction to field.

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F =GM1M2

R2= N

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F =GM1M2

R2=19.598 N

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a =ΣFM

=19.598

2= ?

Field = Force / Mass

a = _._ m/s2 !!!

• Why the familiar result?• The large mass given must be the mass of the

_ _ _ _ _ .• The distance given must be the _ _ _ _ _ _ of

the _ _ _ _ _.

a = 9.8 m/s2 !!!

• Why the familiar result?• The large mass given must be the mass of the

_ _ _ _ _ .• The distance given must be the _ _ _ _ _ _ of

the _ _ _ _ _.

a = 9.8 m/s2 !!!

• Why the familiar result?• The large mass given must be the mass of the EARTH .

• The distance given must be the _ _ _ _ _ _ of the _ _ _ _ _.

a = 9.8 m/s2 !!!

• Why the familiar result?• The large mass given must be the mass of the EARTH .

• The distance given must be the RADIUS of the EARTH.

Repeat the problem, except replace the 2 kg with 3 kg.

What is the new force?

What is the new acceleration?

Please do this practice problem…

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F =GM1M2

R2= 29.397 N

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a =29.397

3= _ ._ m/s2

Field = Force / Mass

Why is the acceleration of all falling objects the same?

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a =

GMem

R2

m=GMe

R2= 9.8

Field = Force / Mass

The acceleration of m does not depend on m!

(And the acceleration of M does not depend on M)

In the previous example we saw that the force depends on the mass, but the field

does not. This is a hint that fields are

important.

Gravitational Field ProblemA 1000.0 kg sphere is at the origin, and a 100.0 kg sphere is on the x axis at x = 5.00 meters. Draw the system. Find the magnitude and direction of the field: (a) halfway between the masses, and (b) at the point: x = -1.00 m, y = 0.

Gravitational Field Problem Solution a)g = GM / R2

Due to big mass: g = (6.67x10-11)(1000) / 2.502

= 1.07 x 10-8 N (towards the Left)

Due to small mass: g = (6.67x10-11)(100) / 2.502

= 1.07 x 10-9 N (towards the Right)

Total field = g = 1.07 x 10-8 - 1.07 x 10-9

= 9.63 x 10-9 N kg-1 Towards the Left.

Gravitational Field Problem Solution b)Due to big mass: g = (6.67x10-11)(1000) / 1.002

= 6.67 x 10-8 N (towards the Right)

Due to small mass: g = (6.67x10-11)(100) / 6.02

= 1.85 x 10-10 N (towards the Right)

Total field = g = 6.67 x 10-8 + 1.85 x 10-10

= 6.69 x 10-8 N kg-1 Towards the Right.