Devil physics The baddest class on campus IB Physics...

DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS

IB PHYSICS

LSN 2-11: MOTION IN A GRAVITATIONAL FIELD

Questions From Reading Activity?

Assessment Statements

Gravitational Field, Potential and Energy

9.2.7. Explain the concept of escape speed from a planet.

9.2.8. Derive an expression for the escape speed of an object from the surface of a planet.

9.2.9. Solve problems involving gravitational potential energy and gravitational potential.

Objectives

State the definitions of gravitational potential energy, and gravitational potential.

Understand that the work done as a mass m is moved across two points with gravitational potential difference ∆V is, W = mΔV

r

MGV

r

MMGEp

21

Objectives

Understand the meaning of escape velocity and solve related problems using the equations for escape speed from a body of mass M and radius R,

R

GMvesc

2

Objectives

Solve problems of orbital motion using the equation for orbital speed at a distance r from a body of mass M,

Understand the term weightlessness.

R

GMv

Newton’s Law of Universal Gravitation

Last year we learned,

This year we look at it from an energy standpoint

2

21

r

MMGF

r

MMGFdWE 21

Gravitational Potential Energy

The gravitational force is an attractive force

Work must be done to separate two bodies in space a certain distance R

This work is converted to potential energy called the gravitational potential energy

r

MMGEp

21


For a satellite orbiting a body, its total energy is the sum of its kinetic and potential energy

r

MMGmvE 212

2

1


If the satellite is in a stable, continuous orbit, the kinetic energy is equal to its potential energy

r

MMGmv 212

2

1


Newton’s Second Law tells us that the gravitational force will be balanced by the centripetal acceleration

2

2

2

vr

GM

r

vm

r

MmG

maF c


Substituting into the traditional value for kinetic energy gives us

r

GMmE

mvE

vr

GM

k

k

2

1

2

1 2

2


And total energy becomes

r

GMmE

r

GMm

r

GMmE

EEE

r

GMmE

pk

k

2

2

1

2

1


Graph of kinetic, potential, and total energy as a function of distance for a circular orbit

Gravitational Potential

The gravitational potential at any point P in the gravitational field is the work done per unit mass in bringing a small point mass m from infinity to point P

If the work done is W, then the gravitational potential is the ratio of the work done to the mass m

m

WV


The gravitational potential due to a single mass M a distance r from the center of M is

Gravitational potential is a scalar quantity

Its units are J/kg r

GMV


If we know the gravitational potential at some point P, then the potential energy of a mass m will be

And work will be defined as

r

GMV

mVEp

VmW

Escape Velocity

Total energy of a mass m moving near a large mass M is given by

We assume the only force acting on m is the gravitational force created by M

r

MMGmvE 212

2

1

Escape Velocity

We want to know if a mass m is launched from the surface of M, will it escape M’s gravitational field?

22

02

1

2

1 mv

r

MmGmv

Escape Velocity

If total energy is greater than zero, m escapes

If total energy is less than zero, m will eventually return to the surface of M

r

MMGmvE 212

2

1

Escape Velocity

The separation point is when V∞ is equal to zero

r

GMv

r

MmGmv

mvr

MmGmv

2

2

1

02

1

2

1

0

2

0

22

0

Escape Velocity

This is the minimum velocity needed to exceed the gravitational attraction of M and is called the escape velocity

What happens if we double the value of m?

r

GMvesc

2

Orbital Motion

The law of gravitational attraction combined with Newton’s second law show that the orbit of any body due to gravitational attraction will follow the path of an ellipse or a circle (circles are ellipses with both foci at the same point).

Orbital Speed

2

2

2

vr

GM

r

vm

r

MmG

maF c

Period of Motion

The period of a planet is proportional to the 3/2 power of the orbit radius

Kepler’s Third Law of Planetary Motion

32

2

2

2

4

2

2

rGM

T

T

r

r

GM

T

rv

vr

GM

Period of Motion

For two planets orbiting the same body, the ratio of their periods squared to their mean distance from the attracting body cubed, will be equal

3

2

2

2

3

1

2

1

2

3

2

32

2

4

4

r

T

r

T

GMr

T

rGM

T

Equipotential Surfaces

Gravitational potential is given by

An equipotential surface consists of those points that have the same potential

r

GMV


The magnitude of the gravitational field is the rate of change of with distance of the gravitational potential.

r

Vg


Equipotential surfaces and field line are normal (perpendicular) to each other

r

Vg


If we have a graph showing the variation with distance of the gravitational potential, the slope (gradient) of the graph is the magnitude of the gravitational field strength

r

Vg

Σary Review

Can you state the definitions of gravitational potential energy, and gravitational potential?

Do you understand that the work done as a mass m is moved across two points with gravitational potential difference ∆V is, W = mΔV ?

r

MGV

r

MMGEp

21

Σary Review

Do you understand the meaning of escape velocity and can you solve related problems using the equations for escape speed from a body of mass M and radius R ?

R

GMvesc

2

Σary Review

Can you solve problems of orbital motion using the equation for orbital speed at a distance r from a body of mass M ?

Do you understand the term weightlessness?

R

GMv

Assessment Statements

Gravitational Field, Potential and Energy

9.2.7. Explain the concept of escape speed from a planet.

9.2.8. Derive an expression for the escape speed of an object from the surface of a planet.

9.2.9. Solve problems involving gravitational potential energy and gravitational potential.

QUESTIONS?

Part A, #1-15

Part B, #16-29

Homework

Devil physics The baddest class on campus IB Physics...

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