Chapter 11 Flashcards
Converting degrees to radians
=2π/ 360 x angle in degrees
Angular speed
-The angle an object rotate through per second
Angular speed is defined as…
angle ÷ time
ω = θ ÷ t
Units for angular speed =
radians per second => rads^-1
Equation for linear speed using radius and angular speed?
v = rω
Or v= 2πr/ T
Circular motion: frequency and period
-Frequency is the number of revolutions per second (revs^-1)
Equations for omega ω
ω= 2πf or ω= 2π/ T
Centripetal acceleration
- An object moving in a circular path at a constant speed is accelerating because it is changing direction
- The acceleration is always directed towards the centre of the circle
Equations for acceleration involving velocity, radius and omega
a = v^2/ r
a = ω^2r
Centripetal force
-If there is a centripetal acceleration, there must be a centripetal force acting towards the centre of the circle
Centripetal force: equations
F = mv^2/ r
F= mω^2r
If you remove the centripetal force…
The object would fly off at a tangent
Keplers rules
1-A planet moves in an ellipse with the sun at one focus
2-The line fromt he sun to the planet sweeps out equal areas in equal times
3-Square of orbital time is proportional to cube of orbital radius. T^2 ∝ r^3
How does an object move in a circular path?
-To make an object move in a circular path, a force must act perpendicular to velocity
Where does acceleration act?
Acceleration acts towards the centre of velocity
Where does centripetal force act?
- Force always acts at right angles to displacement along a circular path
- Perpendicular to velocity
No work is done when…
A force on a body acts perpendicular to its motion
When is the gravitational force the strongest on a planet?
-When it is closest to the sun
Working out the arc length between two points on a circular object using speed
= 2vsinθ/ t
and if t= 2θr/v
then => (v^2/r)x(sinθ/θ)
Expressed by words, what is the gravitational law?
all the particles in the universe attract all other particles
Expressed by an equation, what is the gravitational law?
F ∝ m1m2/ r^2
Gravitational force equation
F = -Gm1m2/ r^2
Equation for acceleration in a circle
a = v^2/ r inverse square law
Where does the minus sign come from in the equation for the gravitational force?
The minus sign says that the force is always attractive
The minus sign indicates that the force acts towards the mass M
Newton’s universal law of gravitation
-All particles in the universe attract all other particles
(force of attraction; proportional to mass of each, proportional to inverse square of distance between them)
-Attractive force obeys inverse square law
(F = -GmM/ r^2)
Why do planets have a slightly elliptical orbit around the sun?
- The inverse square law
- As the planet moves around the sun there is a gravitational pull towards the sun, and there is a greater pull as it nears the sun.
Gravitational field lines
- They act evenly around a planet and point inwards towards the planet
- Field direction is shown by the direction of the lines
- Field strength is shown by the closeness of the lines
- The field has a vector value (direction + magnitude)
What are a few phenomenons that Newton’s gravitational law accounts for?
- The spherical shapes of planets + stars
- The planets attract each other
Equation for gravitational field strength
g= -GM/ r^2
Geostationary orbits
Geostationary or geosynchronous orbits are high orbits that allow satellites to match the Earth’s rotation and the satellite appears virtually still over one spot; it stays at the same longitude, but its orbit may be tilted, or inclined, a few degrees north or south.
How to calculate the radius of an orbit of a satellite using two force equations
-Centripetal force is equal to gravitational force on sat
so mv^2/ r = GMm/ r^2
through cancellation leaves you with v^2 = GM/R
Then is equal to speed in orbit v^2 = 4π^2 R^2/ T^2
Then can rearrange for R
=> R^3 = GMT^2/ 4π^2
Momentum equation
Momentum is mass x velocity. Momentum is a vector quantity. The SI unit of momentum is
kg m s–1.
m= mv
Define gravitational field strength
Gravitational field strength is a vector quantity in the direction of the gravitational force.
Gravitational field strength units
The SI unit of gravitational field strength is N kg-1 or equivalently m s-2.
The force on a point is
F = mg
Gravitational field lines near the earth
Close to the surface of the Earth, the gravitational field is almost uniform. The lines of force are parallel and at right angles to the Earth’s surface.
Newton’s law of gravitation states that ..
Newton’s law of gravitation states that the force of gravitational attraction F of a mass M on another mass m obeys an inverse square law:
F = -GMm/ r^2
where r is the distance from the centre of M to m
What is the value of the gravitational constant G
The measured value of the Universal Gravitational Constant G is
6.67 × 10–11 N m2 kg-2
The time T taken to move once round the circular path is
T = 2 π r / v
Centripetal force explained
An object moving in a horizontal circle at constant speed changes its direction of motion continuously. Its velocity is not constant because its direction of motion is not constant. The resultant force is directed towards the centre of the circle
Newtons first law
- Newton’s first law of motion states that an object remains at rest or moves with constant velocity unless acted on by a resultant force.
- Newton’s first law defines what a force is, namely any physical effect that is capable of changing the motion of an object. If an object is at rest or in uniform motion, either no force acts on it or forces do act on it and the resultant force is zero.
Newton’s second law
Newton’s second law of motion states that the rate of change of momentum of an object is equal to the resultant force on the object.
Equation for momentum and a resultant force.
That is, F = dp / dt, where p = mv is the momentum of an object acted on by a resultant force, F.
For an object of constant mass m, acted on by a force
F= m dv/dt = ma
What are the units of force
The SI unit of force is the newton (N). 1 N is the force that gives a 1 kg mass an acceleration
of 1 m s–2
Newtons third law
Newton’s third law of motion states that when two objects interact, there is an equal and opposite force on each.
General rule for momentum
Momentum before = momentum after
Equation for the conservation of momentum
(m1v1 + m2v2)before = (m1v1 + m2v2)after
Equation for momentum
p = mv
Between two masses, their ratio determines their changes in velocity
v1/v2 = m1/m2
The larger the moment transferred…
The bigger force on each
Softening the blow
- The time also impacts the force
- The shorter the time, and the faster the rate of changes of momentum, the larger the force.
How can you ‘soften the blow’?
-By spreading out the momentum change over a longer time reduces the force.
How could a parachuter ‘soften the blow’?
-By bending their legs and relaxing and letting yourself fall over, it makes the contact time with the ground as long as possible so the momentum can be absorbed by the ground.
The force is…
The rate of change of the moment
F = Δmv/ Δt
since v/t = a this is also f=ma (newtons second law)
Equation for change in momentum
FΔt
therefore to reduce F you must increase t
FΔt is also known as impulse
How can you show that forces come in pairs?
If momentum is conserved Δp1 = -Δp2 so ΔF1t = -ΔF2t then F1= -F2
so forces come in equal and opposite pairs.
If the mass is constant then…
If the mass is constant this can be expressed as ‘force = mass × acceleration’ because
acceleration is rate of change of velocity.
Thrust
The thrust on a rocket of the jet of gases that it ejects is equal to the rate at which the jet carries away momentum. This is given by the mass ejected per second x the velocity of the
jet.
Equation for thrust
Thrust F = Δp/ Δt = -v Δm/Δt
Units for momentum
kgms^-1
Momentum has…
Size (magnitude) and direction
Explaining rocket propulsion by momentum
- A rocket will be propelled forward when it expels exhaust gases
- The momentum in the forward direction is equal to the momentum of the exhaust gases backwards
- Using the conservation of energy m1v1 + m2v2 = mv (or a variation) you can work it out.
If the forces aren’t balanced…
The overall resultant force will cause the body to accelerate (change in direction or speed or both)
What does the equation F =mv / t show or tell us?
- It shows that the more force you have acting on a certain mass, the more acceleration you get.
- It tells us that for a given force the more mass you have, the less acceleration you get.
- A small force acting for a long time can causes the same change in momentum as a large force acting for a short time.
When is F = ma appropriate?
When the mass of an object is constant, then the bigger the force acting on it, the greater its acceleration.
Masses in a gravitational field will…
Experience a force of attraction
-Only objects with a large mass have a significant effect (e.g. the force of the moon on earth affects the tides)
In the equation for f = -GMm/ r^2, what is r?
r is the radius which is the distance between the centre of m and M
The law of gravitation is an inverse square law (F ∝ 1/r^2) so…
- If the distance r between the masses increases then the force will decrease
- If the distance double the force will be one quarter the strength of the original force
What do the field lines look like close to the earth surface?
-The gravitational field is almost uniform, so the lines are almost parallel.
Gravitational field strength
g = F/ m
- The force per unit mass
- g is the acceleration of a mass in a gravitational field; acceleration due to gravity (-9.81 ms^-2)
A mass in a uniform gravitational field
- The mass will experience a constant force, given by mg.
- Near the surface of earth, the value of g is the same in all locations, so the force experienced by a mass due to gravity is constant.
Gravitational potential
The gravitational potential at a point is the potential energy per unit mass of a small object placed at that point. This is the work done per unit mass to move a small object from infinity to that point.
Difference between gravitational potential and gravitational potential energy
Ep = the amount of energy required to bring a particle from one point to another Vp = gravitational potential is the GPE per unit mass
What is the equation for Ep?
The gravitational potential energy Ep of a point mass m is given by Ep = m Vp, where Vp is
the gravitational potential at that point.
Units for gravitational potential?
The SI unit of gravitational potential is J kg–1. Gravitational potential is a scalar quantity.
What is an equipotential?
An equipotential is a surface of constant potential. No change of potential energy occurs when an object is moved along an equipotential. The lines of force are therefore always perpendicular to the equipotential surfaces.
Equation for the gravitational field
In an inverse square gravitational field, the field strength is:
g = - GM/ r^2
Equation for the gravitational potential (Vp)
Vp = - GM/ r
What does a graph that shows the variation of gravitational potential with distance from the centre of a spherical body look like?
curve that is in negative Y and positive X quarter of a graph. Plateaus towards the the x-axis
Equation for the gravitational potential energy difference in a uniform field
mg ∆h
What is gravitational potential?
Gravitational potential energy per unit mass
V = Egrav/ m
How would you find gravitation field strength on a graph?
- gravitational potential gradient
What are some purposes for satellites?
- Mapping the earth, oceans and atmosphere
- Communications
- Military reconnaissance + navigation
- Astronomical observation
Gravitational field lines
- If they are close to the earth then they will be parallel (almost)
- Field is nearly uniform
- The gravitational force (mg) hardly varies with height so the change mg ∆h in Ep is the same for equal increases ∆h in height
Contour lines on a map
If they are close together then it indicates a steep hill
How does the centripetal and gravitational force vary depending on how far away you are from earth?
The closer you are from the earth, the stronger the Fc and Fg so the harder it is to get away from Earth’s gravitational potential ‘hill’.
Equipotentials=
- A map of the gravitational potential is a set of constant potential.
- Equipotentials show all the points in a field which have the same period.
- If you travel along a line of equipotential you don’t lose or gain energy
- Equipotentials and field lines are perpendicular.
Mass in a uniform gravitational field
- Field strength = g
- Force on mass is a gravitational field = mg
- Field = force/ mass = g
How do you find g from a graph of Ep per kg against displacement upwards?
- It will be the gradient of the graph
- -slope ∆Vgrav/ ∆r
The higher you are the in atmosphere…
The larger change in Ep because of ∆h
Potential energy change = force x distance = mg ∆h
Gravitational potential energy equation
Epotential = Etotal - Ekinetic
Vgrav energy equation
Vgrav = constant - 1/2v^2
How do you find gravitational potential from a graph of gravitational field against ∆r?
- ∆V = Area g ∆r
- The difference in potential is the area under graph
What would need to be true to escape the Earth’s potential well?
Ekinetic + mVgrav ≥ 0
What does gravity assist mean?
-To take energy from another planet’s motion
Equation for time period using r and G
T = 2π √ r^3 / GM
Equation for r using ω and g and R
r^3 = GR^2/ ω^2
Work…
Work is done whenever energy is transferred
Work = force x distance
Work means the amount of energy transferred from one form to another
Units for work done
Work = Joules, J
How can you find the work done in a graph of force against distance?
Work done is the area under the graph
Equation for work done when the direction of movement is different from the direction of force
W = Fs cosθ
Where θ is the angle between the direction of force and the direction of motion
What is the relationship between g and r^2?
In a radial field g is inversely proportional to r^2
Egrav equation
Egrav = -GMm/ r
If you move away from earth you gain…
Gravitational potential energy
How to calculate the speed of an orbit
Fg = -GMm/r and this is equal to the Fc = mv^2/ r
So mv^2/ r =GMm/ r
which can rearrange to v = √GM/ r
Or v = ωr ( if given a time period)
Graph of g against r
-The strength of the gravitational field decreases the larger r is.
The area under the graph give a value for Vp
g = -GM/ r^2
What does the minus sign indicate in a Vpotential equation?
- Potential energy is 0 at an infinite distance away from the earth.
- That gravity is always attractive
