mechanics flashcards Flashcards by Lucy Clare

What are the 7 base quantities

Mass
Length
Time
Current
Temperature
Luminous Intensity
Amount of Substance

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dimensions

how a quantity is related to the base quantities

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principle of dimensional homogeneity

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

Principle of dimensional homogeneity

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kinematics

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

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frame of reference

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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particle

A mathematical object that has mass but not size or shape

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determine the shape of a particle path

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

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Newtons first law

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

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Newtons second law

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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alternative Newtons second law

Force = rate of change of momentum

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Newtons third law

For every force there is an equal and opposite reaction force

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General Strategy for solving

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

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Solving a homogeneous differential equation

substitute i x = ce^λt
cancel out to characteristic eqn
solve for λ
put into correct form of eqn
use initial conditions to get constants

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Solving a non-homogeneous eqn

choose the form of x꜀ depending on the form of f(t)
differentiate xₚ and sub into original eqn to find the constants
x = x꜀ + xₚ

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form of solution for a non-homogeneous eqn

x = x꜀ + xₚ

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

A₀ + A₁t + A₂t² + … + Aₙtⁿ

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

Ae^αt

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

Acos(αt)+Bsin(αt)

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

x = C₁eᵃᵗ + C₂eᵇᵗ

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

x = eᵃᵗ(C₁+C₂)

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

x = eᵃᵗ(C₁cos(bt)+C₂sin(bt))

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Notes of solving second order differential equations

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
if the trial solution xₚ is already contained in the characteristic equation x꜀, multiply xₚ by t

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hookes laws

f=-kx

frame of reference propostion

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

order of finding constants when solving non-homogenous DEs

- find particular integral constants first by subbing in | - find homogenous parts solutions after you have full x = x꜀ + xₚ

How may frames of reference move

- the frame may be rotated | - the frame may be translated

Newtons gravitational law

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

Keplers first law

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

Keplers second law

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

Kepler third law

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

Central force

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

unit vectors

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

eᵣ and eθ

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

eᵣ =

cos(θ)i + sin(θ)j

eθ =

-sin(θ)i + cos(θ)j

the position vector, ŗ =

reᵣ

derivative of ŗ (vector reᵣ)

= r {theta dot} eθ + {r dot} eᵣ

derivative of eᵣ

{theta dot} eθ

second derivative of ŗ (vector)

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

derivative eθ

- {theta dot} eᵣ

expression for F(r)

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

h =

r²{theta dot}

second order ODE in u

d²u/dθ² + u = F(1/u)/mh²u²

du/dθ in terms of r and h

du/dθ = - {r dot}/h

angular momentum

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

moment of a Force

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

momentum of a body

P = m{ŗ dot}

For a central force angular momentum is...

constant

showing angular momentum for a central force is constant

show that the derivative is zero.

showing motion takes place in only one plane

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

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

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

when is max PE and min KE

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

when is min PE and max KE

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

Kinetic energy

1/2mv²

Potential energy

V(x) = - ∫F(x)dx

energy is a

scalar quantity

Net energy is...

Conserved

Energy conservation leads to a...

First order ODE