Physics
Proofs
To show the relationship between linear speed and angular velocity:
To show
v
=
rω
θ=s
r
Now divide both sides by ‘t’
θ=s
t tr
θ
t
=
s
t
x
1
r
But
ω
Therefore
=
θ/t
and
ω
=
v
x
1
r
Hence v
=
rω
F = ma
is a special case of Newton’s second law:
v
=
s/t
F
is proportional to
change in Momentum
Time taken for change
F
is proportional to
Final Momentum
Time taken for change
F
is proportional to
mv
t
F
=
-
mu
Km(v-u)
t
As a = (v-u)
t
F
=
Kma
F
=
ma
o
H
(K = 1)
rizontal Component:
Vertical Component:
o
vC
vSinθ
sθ
Initial Momentum
Resistors in Series:
R
=
R1
+
R2
+
R3
By Ohm’s Law:
V
=
IR
V1
=
IR1
+
IR2
+
IR3
V
=
V1
+
V2
+
V3
V
=
IR1
+
IR2
+
IR3
V
=
I( R1
+
R2
+
R3 )
V
=
IR
=
R1
+
R2
1
R2
+
1
R3
I2
=
V
R2
Therefore
R
Resistors in Parallel:
1
=
1
+
R
R1
By Ohm’s Law:
I1
=
V
R1
+
I
=
I1
+
I2
+
I3
I
=
V
R1
+
V
R2
+
V
R3
I
=
V (
1
R1
+
1
R2
+
1 )
R3
I
=
V
R
1
R
=
1
R1
+
1
R2
Therefore
R3
I3
=
+
1
R3
V
R3
To show F =
qvB
In a time t, the amount of charge passing any point in the conductor is the amount of
charge in a length vt of the conductor.
The number of charges is given by n
The amount of charge in each particle is given by q
Therefore the amount of charge passing in time t is given by
The current
I
=
Charge passing
Time Taken
Force on length l of the conductor =
Per unit length (ie l = 1metre) =
IlB
=
qnvt
t
=
nvqlB
qnvt
=
nvqB
Thus the force on n moving charges is qvB
The Equations of Motion
A = acceleration, s = displacement, v= velocity, t = time
v = u + at
Proof : Acceleration is the rate of change of velocity
a
=
v-u
t
at = v – u
v = u + at
s= ut + ½ at2
Proof : Average velocity
u+v
2
i.e. Displacement
Time
Displacement = average velocity × time
s = (u + v) × t
2
Substituting u + at for
v
nvq
s = (u + u + at) t
2
s = (2 u + at)t
2
s = ut + ½ at2
v2 = u2 + 2as
Proof :
Squared,
v = u + at
v2 = ( u + at)2
v2 = u2 + 2uat + a2t2
v2 = u2 {ut + ½ at2}
ut + ½ at2 = s
Therefore
v2 = u2 + 2as
g = GM
d2
F = GM1m2
d2
W = mg
But weight is a force, therefore
M2g = GM1m2
d2
g = GM
d2
v2 = GM
R
F = Gm1m2
d2
Centripetal force = Force of Gravity
mv2 = GM
R
R2
v2 = GM
R
T2 = 4π2R3
GM
T = 2πR
v
T2 = 4π2R2
v2
Sub the expression for v2 :
T2 = 4π2R2
(GM/R)
T2 = 4π2R3
GM