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1st year Maths > mechanics flashcards > Flashcards

mechanics flashcards

1
Q

What are the 7 base quantities

A 
Mass
Length
Time
Current
Temperature
Luminous Intensity
Amount of Substance
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2
Q

dimensions

A

how a quantity is related to the base quantities

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3
Q

principle of dimensional homogeneity

A

An equation derived from physical principles must be dimensionally homogeneous, that is

  • the dimensions on the LHS must be the same as the dimensions on the RHS
  • The dimensions of additive terms must be the same
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4
Q

What is a neccessary (But not sufficient) for an equation to be valid

A

Principle of dimensional homogeneity

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5
Q

kinematics

A

how things move. represented by position, velocity and acceleration.

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6
Q

frame of reference

A

A frame of reference is an abstract co-ordinate system and the set of physical reference points that fix it in the frame.

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7
Q

particle

A

A mathematical object that has mass but not size or shape

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8
Q

determine the shape of a particle path

A

extract three equations for x,y,z in terms of t for the three vectors. Eliminate t from the equations.

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9
Q

Newtons first law

A

In the absence of any resultant forces, a particle moves with constant velocity x

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10
Q

Newtons second law

A

If a net force F acts on a particle of mass m, then the acceleration a of the particle is related to F and m by: F=ma

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11
Q

alternative Newtons second law

A

Force = rate of change of momentum

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12
Q

Newtons third law

A

For every force there is an equal and opposite reaction force

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13
Q

General Strategy for solving

A
  1. set up a frame of reference and co-ordinate system
  2. use newtons law to write down governing equations
  3. solve/analyse the equations
  4. determine what happens (shape of particle path/ now quickly do you move/ where(when) does an event occur)
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14
Q

Solving a homogeneous differential equation

A
  1. substitute i x = ce^λt
  2. cancel out to characteristic eqn
  3. solve for λ
  4. put into correct form of eqn
  5. use initial conditions to get constants
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15
Q

Solving a non-homogeneous eqn

A
  1. choose the form of x꜀ depending on the form of f(t)
  2. differentiate xₚ and sub into original eqn to find the constants
  3. x = x꜀ + xₚ
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16
Q

form of solution for a non-homogeneous eqn

A

x = x꜀ + xₚ

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17
Q

for of a particular solution when f(t) is a polynomial of degree n

A

A₀ + A₁t + A₂t² + … + Aₙtⁿ

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18
Q

for of a particular solution when f(t) is a exponential

A

Ae^αt

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19
Q

for of a particular solution when f(t) is a Trig function (either sinαt OR cosαt)

A

Acos(αt)+Bsin(αt)

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20
Q

Form of the solution of the characteristic equation when there are two distinct roots (a,b)

A

x = C₁eᵃᵗ + C₂eᵇᵗ

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21
Q

Form of the solution of the characteristic equation when there is one repeated root (a)

A

x = eᵃᵗ(C₁+C₂)

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22
Q

Form of the solution of the characteristic equation when there is a complex conjugate pair of roots a+bi

A

x = eᵃᵗ(C₁cos(bt)+C₂sin(bt))

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23
Q

Notes of solving second order differential equations

A
  1. if f(t) is made of multiple forms from the f(t) table then use a trial solution that is a multiple of the relevant trial solutions
  2. if the trial solution xₚ is already contained in the characteristic equation x꜀, multiply xₚ by t
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24
Q

hookes laws

A

f=-kx

25
Q

frame of reference propostion

A

Suppose that S is an inertial fram with origin O then a frame of reference S’ whose origin is moving at speed u with respect to S is also an inertial frame

26
Q

order of finding constants when solving non-homogenous DEs

A
  • find particular integral constants first by subbing in

- find homogenous parts solutions after you have full x = x꜀ + xₚ

27
Q

How may frames of reference move

A
  • the frame may be rotated

- the frame may be translated

28
Q

Newtons gravitational law

A

The attractive force between two gravitating bodies of mass m and M is given by:
F = GMm/r²

29
Q

Keplers first law

A

The planets move about the sun in an elliptical orbit with the sun at one focus

30
Q

Keplers second law

A

The straight line joining a planet and the sun sweeps out equal areas in equal time

31
Q

Kepler third law

A

The square of the period of the orbit is equal to the cube of the semi-major axis of the orbit

32
Q

Central force

A

A central force F acting on a particle P depends only on the distance of that particle from the fixed central origin in an inertial frame, and is direction along the line joining the particle and the origin

33
Q

unit vectors

A

Unit Vectors are vectors with magnitude 1.
e.g. i j
eᵣ and eθ

34
Q

eᵣ and eθ

A

non constant unit vectors in the direction of increasing r and θ.
Their direction changes but magnitude is constant.
They are functions of time

35
Q

eᵣ =

A

cos(θ)i + sin(θ)j

36
Q

eθ =

A

-sin(θ)i + cos(θ)j

37
Q

the position vector, ŗ =

A

reᵣ

38
Q

derivative of ŗ (vector reᵣ)

A

= r {theta dot} eθ + {r dot} eᵣ

39
Q

derivative of eᵣ

A

{theta dot} eθ

40
Q

second derivative of ŗ (vector)

A

({r double dot} - r {theta dot}²)eᵣ + (1/r d/dt(r²{theta dot}))eθ

41
Q

derivative eθ

A
  • {theta dot} eᵣ
42
Q

expression for F(r)

A

F(r) = m( {r double dot} - r{theta dot}²)

43
Q

h =

A

r²{theta dot}

44
Q

second order ODE in u

A

d²u/dθ² + u = F(1/u)/mh²u²

45
Q

du/dθ in terms of r and h

A

du/dθ = - {r dot}/h

46
Q

angular momentum

A

The moment of momentum = ŗ x (m{ŗ double dot})

47
Q

moment of a Force

A

ŗxF

where ŗ is the location of the point where the force is applied to relative to 0

48
Q

momentum of a body

A

P = m{ŗ dot}

49
Q

For a central force angular momentum is…

A

constant

50
Q

showing angular momentum for a central force is constant

A

show that the derivative is zero.

51
Q

showing motion takes place in only one plane

A

dot the equation (ŗ x {ŗ dot}) = h with ŗ
ŗ.(ŗx{ŗdot}) = ŗ.h
=> ŗ.h = 0
(eqn for a plane)

52
Q

showing r²{theta dot} is conserved and is the angular momentum

A
  1. ŗ . {ŗ dot} = h = constant
  2. put into unit vector form
  3. rearrange
53
Q

when is max PE and min KE

A

when the ball it at the top about to fall back down

54
Q

when is min PE and max KE

A

when the ball is at the bottom (just been thrown)

55
Q

Kinetic energy

A

1/2mv²

56
Q

Potential energy

A

V(x) = - ∫F(x)dx

57
Q

energy is a

A

scalar quantity

58
Q

Net energy is…

A

Conserved

59
Q

Energy conservation leads to a…

A

First order ODE

1st year Maths (24 decks)

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