mechanics flashcards
What are the 7 base quantities
Mass Length Time Current Temperature Luminous Intensity Amount of Substance
dimensions
how a quantity is related to the base quantities
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
What is a neccessary (But not sufficient) for an equation to be valid
Principle of dimensional homogeneity
kinematics
how things move. represented by position, velocity and acceleration.
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.
particle
A mathematical object that has mass but not size or shape
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.
Newtons first law
In the absence of any resultant forces, a particle moves with constant velocity x
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
alternative Newtons second law
Force = rate of change of momentum
Newtons third law
For every force there is an equal and opposite reaction force
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)
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
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
form of solution for a non-homogeneous eqn
x = x꜀ + xₚ
for of a particular solution when f(t) is a polynomial of degree n
A₀ + A₁t + A₂t² + … + Aₙtⁿ
for of a particular solution when f(t) is a exponential
Ae^αt
for of a particular solution when f(t) is a Trig function (either sinαt OR cosαt)
Acos(αt)+Bsin(αt)
Form of the solution of the characteristic equation when there are two distinct roots (a,b)
x = C₁eᵃᵗ + C₂eᵇᵗ
Form of the solution of the characteristic equation when there is one repeated root (a)
x = eᵃᵗ(C₁+C₂)
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))
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