Rotational Motion

Question 1

Derivation of v=u+at

Answer 1

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

Question 2

Derivation of s=ut+1/2 (at^2)

Answer 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)

Question 3

Derivation of v^2 = u^2 + 2as

Answer 3

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

Question 4

How do you find acceleration, velocity or displacement through differentiating?

Answer 4

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

Question 5

What are equations that link linear and angular motion?

Answer 5

s=rθ
v=rω
a=rα

Question 6

What is the formula for angular velocity?

Answer 6

ω=2pi/T

or 2pi x no of revolutions/t

Question 7

What are the formulas for central force?

Answer 7

F=mv^2/r

F=mrω^2

Note that tension/central force travels into a circle

Question 8

What is torque and its formula?

Answer 8

the turning effect a force can produce

T=Fr
or T=Frsinθ if the force is applied at an angle

Question 9

What is inertia and its formula?

Answer 9

an objects resistance to angular acceleration

I=mr^2

Question 10

What is the formula linking torque and inertia?

Answer 10

T=Iα

Question 11

What are the formulas for angular momentum?

Answer 11

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.

Question 12

What is the formula for the total kinetic energy of a system that is moving linearly and rotationally?

Answer 12

Ek=1/2(Iω^2) + 1/2 (mv^2)

Question 13

What is the formula for the force of gravity between two objects?

Answer 13

F=GMm/r^2

Question 14

What is the formula for the gravitational force on a small object on earths surface?

Answer 14

w=mg F=GMm/r^2

mg=GMm/r^2
g=GM/r^2

Question 15

What is a gravitational field?

Answer 15

the force per unit mass at that point

Question 16

What is gravitational potential and its formula?

Answer 16

the work done in moving a unit mass from infinity to that point. Measured in V or Jkg^-1

v= -GM/r

Question 17

What is the formula for potential energy in a gravitational field?

Answer 17

E=Vm

Question 18

What is the formula for a satellites velocity?

Answer 18

v=(GMe/r)^0.5

Question 19

What is escape velocity and its formula?

Answer 19

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

Question 20

What is the formula for a satellite changing orbits?

Answer 20

E=GMeM/2r

Question 21

What is general relativity?

Answer 21

the ideas behind a non inertial (accelerating) frame of reference eg equivalence principle or time dilation

Question 22

What is a black hole?

Answer 22

a dense star with an escape velocity greater than the speed of light

Question 23

What is the formula for the schwarzchild radius and its derivation?

Answer 23

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

Question 24

What is the formula for power of a star?

Answer 24

P=kAT^4

Question 25

What is the formula for the luminosity of a star?

Answer 25

L=4pi.r^2.k.T^4

Question 26

What is the formula for the brightness of a star?

Answer 26

b=L/4pi.r^2

Question 27

How is a star formed?

Answer 27

1. gravitational forces pull gas and dust into the centre
2. accumulation of mass increases gravitational attraction, drawing in more hydrogen gas
3. gravitational potential energy converts to heat. the temperature and pressure of the core increases
4. hydrogen nuclei gains sufficient kinetic energy to overcome electrostatic repulsion and a fusion reaction begins.

Question 28

What are the steps to the proton-proton chain?

Answer 28

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

Question 29

What is a hertzsprung russell diagram?

Answer 29

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

Question 30

How are red supergiants created?

Answer 30

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.

Question 31

How are white dwarfs created?

Answer 31

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

Question 32

How are black holes created?

Answer 32

when very big stars die they explode in a supernova. It eventually collapses int a neutron star or black hole