M5, C3 Gravitational Fields

Question 1

define gravitational field

Answer 1

a force field generated by any object with mass which causes any other object with mass to experience an attractive force

Question 2

what are gravitational field lines

how would you draw them

Answer 2

known as lines of force
show the direction of the force that masses would feel in a gravitational field

drawn towards the edge of the object with arrows pointing inwards

Question 3

what is Newton’s law of gravitation

Answer 3

the force acting between two point masses is proportional to the product of their masses and inversely proportional to the square of the distance between their centres of mass

Question 4

F = - GMm / r^2

what does this equation mean

Answer 4

F = force acting on mass m due to mass M
G = gravitational constant (= 6.67 X 10^-11 Nm^2kg^-2)
M = mass causing the force
m = mass the force is acting on
r = distance between the centre of the two masses

only use the minus sign in front of the equation when you are interested in the direction
for example if you calculate r then the minus sign is irrelevant because you don’t need a direction

the positive direction is defined as M to m
negative is m to M

Question 5

what is a field

Answer 5

the region in which a force operates

Question 6

the stronger the gravitational field, the _______ the force on the mass

Answer 6

larger

Question 7

relating to Newton’s law of gravitation what happens to the force if the distance between the masses doubles

Answer 7

inverse square law

F α 1/r^2

so if r doubles then F will be 1/4 of the original force

Question 8

The gravitational force between 2 objects 10m apart is 0.291N.
What will the gravitational force between them be if they move to 25m apart?

Answer 8

25/10 = 2.5
inverse square law F α 1/r^2

0.291/2.5^2
= 4.7 X10^-2 N

Question 9

define gravitational field strength

Answer 9

force acting per unit mass

Question 10

what does this equation mean
g = F/m

what are the units of g

Answer 10

gravitational field strength = force experience by a mass in the gravitational field / mass

units of g are Nkg^-1 or ms^-2 (acceleration due to gravity)

Question 11

derive the formula g = -GM / r^2

Answer 11

use F = - GMm / r^2 and F = mg

-GMm/r^2 = mg

cancel m

-GM/r^2 = g

Question 12

sketch a graph of distance r (x) against g (y)

Answer 12

g is negative
rapidly increases
then decreases rapidly as distance increases
levels off

(page 66 of year 2 textbook)

(reflect in y=0 line for positive g)

Question 13

2 particles of mass 0.2kg and 0.3kg are placed 0.15m apart. A third particle of mass 0.05kg is placed between them.
Calculate F acting on the third particle if it’s placed 0.05m from the 0.3kg mass.

Answer 13

Need to work out the two forces either side of the mass then get a resultant force.

Use equation F = GMm / r^2

F1 = 6.67X10^-11 X 0.2 X 0.05 / 0.1^2
= 6.67X10^-11 N

F2 = 6.67X10^-11 X 0.3 X 0.05 / 0.05^2
= 40.2 X10^-11 N

F = F2 - F1 = 33.5 X10^-11 N

Question 14

define gravitational potential at a point

Answer 14

the work done in moving a unit mass from infinity to that point

Question 15

what does the equation
V_g = -GM / r
mean

Answer 15

V_g = gravitational potential
G = gravitational constant
M = mass of the object causing the gravitational field
r = distance from the centre of the object

Question 16

At an infinite distance from the mass, the gravitational potential will be _____.

Answer 16

zero

Question 17

What equation could you use to determine the amount of energy needed to move an object against gravity

Answer 17

∆W = m∆V_g

because V_g has the units of Jkg^-1 and mass is kg so when multiplying, it leaves J

Question 18

what are the units of gravitational potential (V_g)

Answer 18

Jkg^-1

Question 19

On a graph of force (y) against distance, r (x)

what does the area under the graph between 2 points equal

Answer 19

the work done to move the object between those two points

Question 20

how are satellites kept in orbit

Answer 20

the gravitational ‘pull’ of the mass they’re orbitting

in our solar system the gravitational force is the centripetal force as they’re undergoing circular motion

Question 21

derive an equation for orbital speed

Answer 21

F = GMm / r^2 and F = mv^2 / r

GMm/r^2 = mv^2/r
GM/r = v^2

v^2 = GM / r

Question 22

derive an equation for orbital period:

T^2 = (4π^2r^3 / GM)

Answer 22

v = d/t distance for a circular orbit = 2πr so v = 2πr/T
T = 2πr / v

derive the equation for orbital speed to get v^2 = GM/r
sub in

T = 2πr / (√(GM/r)
square it

T^2 = (4π^2/GM)(r^3)

Question 23

What are Kepler’s 3 laws of planetary motion

Answer 23

1) Each planet moves in an ellipse around the sun, with he sun at one focus.
2) A line joining the sun to a planet will sweep out equal areas in equal times.

3) The period of the orbit and the mean distance between the sun and the planet are related by
T^2 α r^3

Question 24

A value for gravitational potential is always ___________.

Answer 24

negative

Question 25

sketch a graph of gravitational potential (y) against radius (x)

Answer 25

pg 68 of year 2 textbook

Question 26

what value would you get from the tangent of a gravitational potential - radius graph
what can be determined

Answer 26

-g

g = -∆V_g / ∆r

Question 27

derive an equation for gravitational potential energy

Answer 27

it will be the gravitational potential multiplied by its mass

so E = mV_g = -GMm / r

Question 28

what are geostationary satellites

Answer 28

orbit directly over the equator and are always above the same point on Earth
it travels from west to east
travels at the same angular speed as earth
has a period of 24 hours

Question 29

what are the uses of geostationary satellites

Answer 29

communication - TV and telephone signals

monitor and track weather

Question 30

If you have 2 planets with the same volume but different masses, the planet with the larger mass will have a thicker atmosphere. Why?

Answer 30

At any distance above its surface, the gravitational force will be greater than at the same distance above the planet with the lower mass.
So the larger mass planet can stop more atmosphere particles escaping into space, leading to a thicker atmosphere.

Question 31

Two objects orbit a planet.
Object A is 4.22 X10^5 km away from the centre of the planet. Object B is 6.71 X10^5 km away from the centre of the planet.
Object A completes one orbit in 42.5 hours.
Find the orbital period of planet B, assuming both orbits are circular.

Answer 31

Using Kepler’s Third Law

A: 42.5^2 / (4.22X10^5)^3 = 2.403 X10^-14

that equals B: T^2 / (6.71 X10^5)^3

T^2 = 2.403X10^-14 X (6.71X10^5)^3 = 7261.194
T = 85.2 hours
= 85 hours

Question 32

define escape velocity

Answer 32

the velocity needed so an object has just enough kinetic energy to escape a gravitational field

Question 33

derive an equation for escape velocity

Answer 33

the kinetic energy lost = gravitational potential energy gained so

0.5mv^2 = GMm/r

cancel out m and rearrange for v

v^2 = 2GM / r

Question 34

for calculating the gravitational potential energy of a satellite, why can’t you use E = mgh

Answer 34

because the gravitational field strength isn’t constant