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
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
- ŗ . {ŗ dot} = h = constant
- put into unit vector form
- 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
