models and rules (gravitation + cm) Flashcards
what is a radian?
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)
how to convert from radians to degrees and vice versa?
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
[equation] arc length?
arc length, s = rθ
what happens when θ is very small?
when θ is very small, sinθ = tanθ = θ
explaining centripetal force
• 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.
equations for centripetal force?
centripetal force,
F = (mv^2) / r
centripetal acceleration,
a = v^2 / r
def for angular displacement, Δθ?
angular displacement (Δθ) is the angle moved through relative to a specific axis
def for angular velocity, ω?
angular velocity (ω) is the rate of change of angular displacement with respect to time
equations for angular velocity, ω?
ω = Δθ / Δt
ω = 2πf
ω = 2π / T
equation that relates linear v (v) and angular v (ω)?
linear velocity = angular velocity x r
v = ωr
why does v = ωr relate linear velocity (v) and angular velocity (ω)?
• 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
what are some other equations for centripetal force?
F = mr(ω^2)
a = (ω^2)r
equation for gravitational force?
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
what is a test mass?
a test mass is a mass small enough so that it does not affect the surrounding gravitational field
what happens if an object is put into a gravitational field?
if an object is put into a gravitational field, the object is subject to a force - NOT it feels a force
explaining gravitational field strength, g
• 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
equation for gravitational field strength, g?
g = -(GM) / r^2
Note:
• g varies with an inverse square law as shown by the equation
what is a uniform field?
a uniform field is a field where there is the same magnitude and direction of field everywhere, eg) near the earth’s surface
features of a geostationary satellite?
• 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
what is kepler’s third law?
Kepler’s third law is that r^3 ∝ T^2
equation for orbital period of geostationary satellite?
equation for orbital period of geostationary satellite:
T = 2π√(r / g)
what is gravitational potential (V)?
• 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
equation for gravitational potential (V)?
V = -(GM) / r
what are equipotential surfaces?
• 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
what is escape velocity?
the escape velocity is the speed the object must be travelling to overcome gravitational attraction of planet (or star)
equation for escape velocity?
v esc = √(2GM / r)
v esc = √(2 x orbital velocity)
what happens if object has higher v than orbital velocity?
if a body has a higher velocity than the orbital velocity, it will move to a higher orbit
what happens if orbit v is greater than escape v?
if the orbital velocity is greater than the escape velocity for the radius of orbit, the body will leave the planet’s orbit
equation for orbital velocity?
v orb = √(GM / r)
equation for GPE ?
Ep = -(GMm) / r
what does greater spacing mean in field lines diagrams?
greater spacing in field lines diagrams = weaker field
what does the area between x-axis and a g-r graph give?
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
what does area between x-axis and a F-r graph give?
the area between x-axis and an F-r graph gives ΔEp (change in gravitational potential energy)
what does the gradient of a V-r graph give?
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)
what does the gradient of a Ep-r graph give?
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
what does g, G, r, V / Vg and Eg / GPE / Ep mean?
• 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
features of gravitational potential wells?
• 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
how to help in escaping potential well?
to help in escaping potential well, you gain energy
explaining gravitational field inside the Earth
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
what is the Potential for a falling particle in a Uniform Field?
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
explaining potential gradient
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:
what is gravitational field strength, g?
gravitational field strength is the force acting per unit mass
what do the lines pointing towards the body of mass represent in a field line diagram?
lines pointing towards the body of mass in a field line diagram represent the direction and magnitude of the gravitational field strength
what is Kepler’s First Law?
⋅ Kepler’s first law states that ‘the orbit of a planet is an ellipse, with the sun at one of the two foci.’
what is Kepler’s Second Law?
⋅ Kepler’s second law states that ‘A line segment joining the Sun to a planet sweeps out equal areas in equal time intervals.’
