ECE 105 - Physics of Electrical Engineering 1
Forces and Motion
Force is a vector, and therefore includes direction. For any vector a, ▯a has the same
magnitude but opposite direction.
Given AB we can ▯nd A or B’s position based on the position of the other one
OB= O +AAB ~
for any Oxis the location of x relative to the origin.
We can break any vector into components by ▯nding the angle between it and the plane
we want to model it o▯ of.
Example: for A = [email protected]
, we can ▯nd it’s components with relation to the standard x-y
A y Acos20 ▯
A x Asin20 ▯
a = ▯v
~ ▯ ~
a = f i
a▯t = vf ▯ ~i
~f= ~i+~ a▯t
1 For the position vector d;d =fd + vi ~i▯t + 1a(▯t) 2
2 2 ▯ ▯
vf = v~i+ 2~ a ▯d ~
For any three objects a;b; and c
vca= v~cb v~ ba
read "the velocity of c with respect to a is equal to the velocity of c with respect to b plus
the velocity of b with respect to a.
▯ ▯ r for very close points.
For circle with center O and radius r ;r ::: connected to object on circumferance with velocity
0 v~ ▯v~
~0;v~1::: tangent to circumferance, a = 1t 0. For arc length between object (at di▯erent times
t ;t :::) s, ▯ =s. For small ▯ ▯ 1;jv ~ ▯ v~ j = ▯v:
0 1 r 1 0
jaj = v▯
a is perpendicular to ~v
v = r1▯r0 = r▯
) ~a = v▯ = v2
s = r▯
= v = r
! = dt
Types of Forces
A force is a push or pull interaction between two objects, reponsible for changing motion.
When unstretched, no spring forces exist. When a string is pushed from equilibrium, its
spring force pushes back toward equilibrium.
F s ▯k▯x
The tension force pulls an object toward a rope and a rope toward an object. Ropes can
The normal force "pushes back" against other objects via molecular electromagnetism. It
is always perpendicular to the surface for any surface-to-surface contact. Technically, it is a
type of spring force.
Friction is the interaction between an object and a surface. It is a real force which acts
opposite the direction of sliding, and is always tangent to surface.
f / N is an experimental fact. f = ▯N, where ▯ is the coe▯cient of friction. ▯ is dependant
on the type of objects and must be determined experimentally.
Kinetic friction is when objects are sliding relative to each other and static friction is
when objects are not yet sliding
f s ▯ Ns
Example: A 50kg person is in a 1000kg elevator at rest. When the elevator begins to rise,
the person notices her weight is 600N. How far does the elevator move in 3s?
▯F = m~ a
~a = Fn▯ mg
600 ▯ 50g
= 2:2m=s 2
d = ~ t + at2
= 0 + (2:2)9
An object can be said to have a total energy equal to the sum of the various forms of energy
it may posess.
The kinetic energy of an object is determined by its mass and velocity
For any object with a changing velocity
vf= v i 2ad
mv f = mv i + mad
K f K + iFd
▯K = ▯Fd ~
Potential Gravitational Energy
Potential gravitational energy is a measure of stored energy of an object based on its
height. It is essentially non-sensical to determine an object’s "absolute" potential gravita-
tional energy, thus we often simply solve for the di▯erence in energy.
4 For a distance hfabove a reference height h i
U = mg(h ▯ h )
g f i
thus if an object moves from h io h f
▯U g U gf▯ U gi
= mgh ▯fmgh i
A spring’s energy is based on its spring constant k and how far it is compressed from its
U s 2
If a collision is isolated, then energy is conserved. Elastic collisions also conserve energy.
For all real or inelastic collisions, energy is lost.
Just as energy is a way of keeping track of motion, work is a mechanical means for transfering
energy equal to the applied force multiplied by the distance it operates along
dW = Fds ~
It can be used to compute the change in energy of a system between two states, as the
total work done by non-conservative forces (ie friction) will be equal to the work done by
conservative forces (ie gravity, springs, motion)
For a system involving friction, motion, gravity, and a spring, we have
▯E th = ▯K + ▯U + ▯g s
or, if we compute the value of the thermal work done by friction as energy (using U = ▯Nd,
where d is the distance during which the object undergoes friction), we get
0 = ▯K + ▯U + ▯g + ▯U s f
5 Rotation (of a non-deformable, rigid bodied object)
For any point on an object in circular rotation
s = r▯