Rotational Motion Flashcards
Derivation of v=u+at
a=dv/dt a dt = dv define limits and integrate both sides at = v (where limit is t and 0 on left, and v and u on right) at-0 = v - u rearrange to be v=u+at
Derivation of s=ut+1/2 (at^2)
ds/dt = u+at ds = (u+at) dt define limits and integrate both sides s = ut + 1/2 (at^2) where limits are s and 0 on the left, and t and 0 on the right s = ut + 1/2 (at^2)
Derivation of v^2 = u^2 + 2as
v = u + at square both sides v^2 = u^2 + 2uat + (a^2)(t^2) take common factor v^2 = u^2 + 2a(ut+1/2 (at^2)) substitute s=ut+1/2 (at^2) v^2 = u^2 + 2as
How do you find acceleration, velocity or displacement through differentiating?
a=dv/dt
v=ds/dt
remember the number of variables decreases as you go from displacement to velocity to acceleration, so you know whether to integrate or differentiate to get your answer
What are equations that link linear and angular motion?
s=rθ
v=rω
a=rα
What is the formula for angular velocity?
ω=2pi/T
or 2pi x no of revolutions/t
What are the formulas for central force?
F=mv^2/r
F=mrω^2
Note that tension/central force travels into a circle
What is torque and its formula?
the turning effect a force can produce
T=Fr
or T=Frsinθ if the force is applied at an angle
What is inertia and its formula?
an objects resistance to angular acceleration
I=mr^2
What is the formula linking torque and inertia?
T=Iα
What are the formulas for angular momentum?
L=Iω
L=mr^2ω
L=mvr
the right hand rule can be used for angular momentum. the thumb is the direction of angular momentum and the fingers are curled in the direction of rotation.
What is the formula for the total kinetic energy of a system that is moving linearly and rotationally?
Ek=1/2(Iω^2) + 1/2 (mv^2)
What is the formula for the force of gravity between two objects?
F=GMm/r^2
What is the formula for the gravitational force on a small object on earths surface?
w=mg F=GMm/r^2
mg=GMm/r^2
g=GM/r^2
What is a gravitational field?
the force per unit mass at that point
What is gravitational potential and its formula?
the work done in moving a unit mass from infinity to that point. Measured in V or Jkg^-1
v= -GM/r
What is the formula for potential energy in a gravitational field?
E=Vm
What is the formula for a satellites velocity?
v=(GMe/r)^0.5
What is escape velocity and its formula?
the minimum velocity neeed for a mass to escape a gravitational field
v=(2GM/r)^0.5
this can be derived by:
Ek+Ep=0
substitute kinetic energy formula and Vm and rearrange
What is the formula for a satellite changing orbits?
E=GMeM/2r
What is general relativity?
the ideas behind a non inertial (accelerating) frame of reference eg equivalence principle or time dilation
What is a black hole?
a dense star with an escape velocity greater than the speed of light
What is the formula for the schwarzchild radius and its derivation?
escape velocity: v=(2GM/r)^0.5 since the escape velocity > c c=(2GM/r)^0.5 c^2=2GM/r r=2GM/c^2
What is the formula for power of a star?
P=kAT^4
What is the formula for the luminosity of a star?
L=4pi.r^2.k.T^4
What is the formula for the brightness of a star?
b=L/4pi.r^2
How is a star formed?
- gravitational forces pull gas and dust into the centre
- accumulation of mass increases gravitational attraction, drawing in more hydrogen gas
- gravitational potential energy converts to heat. the temperature and pressure of the core increases
- hydrogen nuclei gains sufficient kinetic energy to overcome electrostatic repulsion and a fusion reaction begins.
What are the steps to the proton-proton chain?
step 1: two hydrogen nuclei fuses to form deuterium, a positron and a neutrino.
step 2: deuterium fuses with a proton to form a helium-3 nucleus and a gamma ray photon
step 3: two helium-3 nuclei fuses to form a helium-4 nucleus and two free protons.
These protons can start a new proton-proton chain
What is a hertzsprung russell diagram?
where luminosity of stars is plotted against surface temperature. most of a stars life will be spent in the main sequence until it evolves
- in the top right corner are red supergiants
- just below the supergiants are red giants
- from the left top corner to bottom right corner is the main sequence
- in bottom left corner are white dwarfs
How are red supergiants created?
outward pressure from fusion would be a greater force than the inward gravitational pull. therefore the outer layers get pushed outwards and the star becomes a sueprgiant and leaves the mains sequence.
How are white dwarfs created?
the cores temperature is too low to fuse elements so it contracts because of gravitational attraction. The star ejects the outer layers as nebulae and leaves behind a white dwarf, eventually forming a black dwarf.
This is the death of a small star
How are black holes created?
when very big stars die they explode in a supernova. It eventually collapses int a neutron star or black hole
