Newtonian Dynamics Flashcards
dimensional analysis
write out variables with exponents of alpha, beta, gamma
plug in dimesions
make LHS and RHS compatible
kinematics
studies how the motion takes place and describes its geometrical features
dynamics
considers the origin of the motion and investigates the interactions between physical objects
x in polar coords
rsin theta cos phi
y in polar coords
rsin theta sin phi
z in polar coords
rcos theta
trajectory
set of points traced by the particle during its motion
xt diagram
time on x axis
position on y axis
uniform motion
zero acceleration
because of viscous effects, a particle moving in a fluid or a gas feels
an acceleration that is proportional to its velocity and opposite in direction
use dimensional analysis to work out k
2 dimensions
x=
rcos theta
2 dimensions
y=
rsin theta
curvilinear coordinate
s
identifies the location of the particle along the trajectory with respect to a certain origin
velocity of particle in vectorial form
v=ds/dt ut = vut
where ur is the unit vector tangent to the trajectory
relation between cartesian and polar unit vectors
ur=costheta ux + sintheta uy
utheta = -sintheta ux + costheta uy
equation for trajectory of particle is obtained by
eliminating the time variable from x(t) and y(t)
range
maximum distance xr travelled by the particle
The relation between the positions r, r’ of the particle, measured respectively by S, S′ is given by
vectorial equation
r=r0’+r’
poisson formula
db/dt=wxb
w is angular velocity vector whose magnitude is euqal to the angular velocity of the rotation
dux/dt=
wxux
(same for y and z)
relative motion: v=
v’+v0’+wxr’
centrifugal acceleration
ac=-wx(wxR) directed outwards
coriolis acceleration
as=-2wxv’ is orthogonal to both the angular velocity vector and the velocity v’ of the particle wrt S’
responsible for cyclones etc
first law - principle of inertia
consider a body on which no net forces act.
If the body is at rest it will remain at rest
If the body is moving with constant velocity it will continue to do so
inertial reference frame
non accelerating
second law
F=ma
inertial mass
mass accounts for the inertia of the body (resistance to variation in motion)
F=ma cartesian differential equation
ma=m d2r/dt2
newton’s equation is only valid in
intertial reference frames and in the non-relativistic limit
third law
whenever two bodies A and B interact, the force Fab exerted from A on B is equal in magnitude and opposite in direction to the force Fba that B exerts on A
p=mv so F=
dp/dt
also valid when mass is not constant
impulse
J = integral of F dt
variation of the momentum in a time interval
in the case of constant mass J=m delta v
F=ma in non-inertial reference frame
additional terms appear such as centrifugal and coriolis forces
microscopically, contact forces originate from
electrostatic interactions between electric charges
ideal string
non-extendable, massless string (applies to strings and rods)
difference between rods and strings
string is only able to transmit tension forces, not compression forces
drag forces are experienced by
bodies moving in fluids
type of frictional force, ie always in opposite direction
Fdrag=-bv
work done
W= integralF.ds
power
dW/dt
P=F.ds/dt = F.v
conservative forces
line integral does not depend on the actual trajectory but only on initial and final positions
examples of conservative forces
weight
elastic forces
examples of a non-conservative force
friction
the sum of the kinetic and potential energy is a constant only when
in the presence of conservative forces only
mechanical energy
Ek+Ep
in the presence on non-conservative forces, the mechanical energy is
not conserved and the variation is equal to the work done by the non-conservative forces
The angular momentum LO of a particle, calculated with respect to the origin O, is
defined as
LO = r × p = r × mv.
If we change the origin, the angular momentum changes accordingly as
LO′ = LO + O′O × mv
where O’O is the vector from O to O’
we define the moment MO of a force, with respect to the origin
O, as
MO = r × F
the statement of the theorem of the angular momentum
dLO/dt = r × ma = r × F = MO.
central force
force that acts radially from a centre O whose magnitude is a function of the distance r=|r| from O
gravitational potential energy
Ep=-Gm1m2/r
gravitational potential generated by a mass M
V(r)=-GM/r
The gravitational field generated by a system of many particles is obtained as
the vector sum of the contributions due to each individual particle:
theorem of the centre of mass
CM of a
system of particles moves as a massive particle, with mass equal to the total mass of
the system, under a force that is given by the net external force acting on the system
itself
reference frame of the CM is identified by the properties:
its origin in the CM of the system
its axes do not rotate and are parallel to the axes of an inertial reference frame
König’s theorems
relate the values of L and Ek calculated in an inertial reference frame
with the corresponding quantities L′
,Ek’ calculated in the reference frame of the CM.
first koing theorem
The theorem states that the angular momentum L, calculated in the inertial reference
frame, is the sum of the angular momentum L’
calculated with respect to the reference
frame of the CM and the angular momentum LCM ≡ rCM × mvCM of the center of
mass thought as a material object of mass m in the inertial reference frame.
second konig theorem
The theorem states that the energy Ek of the system, calculated in the inertial reference
frame, can be expressed as the sum of the kinetic energy
of the system
with respect to the reference frame of the CM and the kinetic energy
of the CM thought as a material object of mass m in the inertial reference frame.
König’s theorems show that both Ek and L can be decomposed as the sum of two
terms:
- A component accounting for the motion of the CM, that describes the average
translation of the system. - A component accounting for the motion of the system with respect to the CM.
The general motion of an extended (rigid) body is
a roto-translation
(composition of a translation and a rotation)
