models and rules (gravitation + cm)

Question 1

what is a radian?

Answer 1

1 radian is the angle subtended at the centre of the circle by the arc of the circle where length of arc (s) = radius of circle (r)

(the angle where 2 radii join and s = r)

Question 2

how to convert from radians to degrees and vice versa?

Answer 2

radians -> degrees:
(360 / 2π) x radian you’re converting

degrees -> radians:
(2π / 360) x degree you’re converting

lesson from this: put the form you’re converting to on top and other on bottom then multiply by value you’re converting

Question 3

[equation] arc length?

Answer 3

arc length, s = rθ

Question 4

what happens when θ is very small?

Answer 4

when θ is very small, sinθ = tanθ = θ

Question 5

explaining centripetal force

Answer 5

• consider particle moving along circular path
• velocity is changing as particle is changing direction (bc velocity is vector quantity)
• even though speed is constant, there is still acceleration as force is still applied to particle
• force is centripetal which produces centripetal acceleration
• centripetal means acting towards centre of orbit (/ circle)

note:
• centripetal force is NOT new force, it is component of net force directed towards centre of orbit
• centripetal force could be due to reaction force, spring force, tension, friction, weight / gravitational pull, etc.

Question 6

equations for centripetal force?

Answer 6

centripetal force,
F = (mv^2) / r

centripetal acceleration,
a = v^2 / r

Question 7

def for angular displacement, Δθ?

Answer 7

angular displacement (Δθ) is the angle moved through relative to a specific axis

Question 8

def for angular velocity, ω?

Answer 8

angular velocity (ω) is the rate of change of angular displacement with respect to time

Question 9

equations for angular velocity, ω?

Answer 9

ω = Δθ / Δt

ω = 2πf

ω = 2π / T

Question 10

equation that relates linear v (v) and angular v (ω)?

Answer 10

linear velocity = angular velocity x r

v = ωr

Question 11

why does v = ωr relate linear velocity (v) and angular velocity (ω)?

Answer 11

• if particle is moving in a circular orbit w a linear (tangential) velocity (v), it moves the distance s = rθ in time (t):

Note:
• r is taken outside of the differential because it is a constant multiplier

Question 12

what are some other equations for centripetal force?

Answer 12

F = mr(ω^2)

a = (ω^2)r

Question 13

equation for gravitational force?

Answer 13

F grav = -(GMm) / r^2

where:
• G = gravitational constant (= 6.67x10^-11)
• negative sign is because force due to gravity is attractive force
• M and m are different masses in orbit
• r is distance between centre of the mass concerned + centre of orbit

Question 14

what is a test mass?

Answer 14

a test mass is a mass small enough so that it does not affect the surrounding gravitational field

Question 15

what happens if an object is put into a gravitational field?

Answer 15

if an object is put into a gravitational field, the object is subject to a force - NOT it feels a force

Question 16

explaining gravitational field strength, g

Answer 16

• gravitational field strength (g) is the magnitude and direction of the force on 1kg at a given point in a gravitational field

Note:
• g varies with an inverse square law as shown by the equation g = -(GM) / r^2

Question 17

equation for gravitational field strength, g?

Answer 17

g = -(GM) / r^2

Note:
• g varies with an inverse square law as shown by the equation

Question 18

what is a uniform field?

Answer 18

a uniform field is a field where there is the same magnitude and direction of field everywhere, eg) near the earth’s surface

Question 19

features of a geostationary satellite?

Answer 19

• stay in the same perceived place in the sky
• orbits the earth above the equator once a day
• orbit that is too low will result in satellite moving too fast, orbit that is too high will result in moving too slow
• stable orbit will be achieved if velocity of satellite = rate of earth’s rotation…
•…this occurs if centripetal force = gravitational force

Question 20

what is kepler’s third law?

Answer 20

Kepler’s third law is that r^3 ∝ T^2

Question 21

equation for orbital period of geostationary satellite?

Answer 21

equation for orbital period of geostationary satellite:

T = 2π√(r / g)

Question 22

what is gravitational potential (V)?

Answer 22

• gravitational potential is the work done in moving a unit mass from infinity to a position in a gravitational field
or
• gravitational potential (V) is the gravitational energy per kg of material

units = Jkg^-1

Question 23

equation for gravitational potential (V)?

Answer 23

V = -(GM) / r

Question 24

what are equipotential surfaces?

Answer 24

• an equipotential surface is a continuous surface of the same gravitational potential
• in uniform field, spacing between equipotentials is equal
• when moving along equipotential, no gravitational force acts because potential energy is not changing hence no work done
• direction of gravitational field is always perpendicular to equipotentials
• change in potential between two points (on different equipotentials) is the same irrespective of route taken
• this is true for uniform and non-uniform fields

Question 25

what is escape velocity?

Answer 25

the escape velocity is the speed the object must be travelling to overcome gravitational attraction of planet (or star)

Question 26

equation for escape velocity?

Answer 26

v esc = √(2GM / r)

v esc = √(2 x orbital velocity)

Question 27

what happens if object has higher v than orbital velocity?

Answer 27

if a body has a higher velocity than the orbital velocity, it will move to a higher orbit

Question 28

what happens if orbit v is greater than escape v?

Answer 28

if the orbital velocity is greater than the escape velocity for the radius of orbit, the body will leave the planet’s orbit

Question 29

equation for orbital velocity?

Answer 29

v orb = √(GM / r)

Question 30

equation for GPE ?

Answer 30

Ep = -(GMm) / r

Question 31

what does greater spacing mean in field lines diagrams?

Answer 31

greater spacing in field lines diagrams = weaker field

Question 32

what does the area between x-axis and a g-r graph give?

Answer 32

the area between x-axis and a g-r graph gives ΔV (change in gravitational potential)

Note:
• g is always negative (acts down to earth)
• r is always positive

Question 33

what does area between x-axis and a F-r graph give?

Answer 33

the area between x-axis and an F-r graph gives ΔEp (change in gravitational potential energy)

Question 34

what does the gradient of a V-r graph give?

Answer 34

the gradient of a V-r graph gives -g

Note:
• the gradient is positive, so to find g it is the negative of the gradient (bc g is always negative)

Question 35

what does the gradient of a Ep-r graph give?

Answer 35

the gradient of an Ep-r graph gives -F

Note:
•the gradient is positive so to find F it is the negative of the gradient

Question 36

what does g, G, r, V / Vg and Eg / GPE / Ep mean?

Answer 36

• g = gravitational field strength
• G = universal gravitational constant (6.67x10-11)
• r = distance (when talking about orbits, gravity, etc) (distance ≈ radius when talking about orbits)
• V = gravitational potential
• GPE = gravitational potential energy

Question 37

features of gravitational potential wells?

Answer 37

• gravitational potential wells are used in calculations concerning celestial bodies (0 is set as potential energy per kg at infinity)
• gravitational potential increases the further you get from Earth
• more massive object has more negative potential at its surface

EG)
for spacecraft travelling from point of net gravitational neutrality to Earth, it’s KE ⬆️ and it’s Vg ⬇️ as it ‘falls down’ potential well

Note:
• point of net neutrality is where there is zero net force acting

Question 38

how to help in escaping potential well?

Answer 38

to help in escaping potential well, you gain energy

Question 39

explaining gravitational field inside the Earth

Answer 39

explaining the gravitational field inside the earth:
• the grav field is proportional to the distance from the centre
• the mass of Earth above a point inside the planet does not have a gravitational effect

Question 40

what is the Potential for a falling particle in a Uniform Field?

Answer 40

the potential for a falling particle in a uniform field is:

Δgh = Δ(1/2)v^2

derivation:
=> ΔEp = ΔEk
=> Δmfg = Δ(1/2)mv^2
=> Δgh = Δ(1/2)v^2

Question 41

explaining potential gradient

Answer 41

potential gradient:
• if a mass moves a distance Δr (not along equipotential), it’s potential changes by ΔV
• the force required to move the mass vertically upwards at a constant velocity is:

g = dV / dr

•^this is the potential gradient (for a radial field)
• the radial component of the field at any point in space equals the NEGATIVE of the potential gradient at that point

derivation:

Question 42

what is gravitational field strength, g?

Answer 42

gravitational field strength is the force acting per unit mass

Question 43

what do the lines pointing towards the body of mass represent in a field line diagram?

Answer 43

lines pointing towards the body of mass in a field line diagram represent the direction and magnitude of the gravitational field strength

Question 44

what is Kepler’s First Law?

Answer 44

⋅ Kepler’s first law states that ‘the orbit of a planet is an ellipse, with the sun at one of the two foci.’

Question 45

what is Kepler’s Second Law?

Answer 45

⋅ Kepler’s second law states that ‘A line segment joining the Sun to a planet sweeps out equal areas in equal time intervals.’