Answer:
Explanation:
At resonance ω₀L = 1 / ω₀C , L is inductance and C is capacitance .
C = 1 / ω₀²L , ω₀ = 5.1 x 10⁵ . ( given )
voltage over resistance = R I , R is resistance and I is current
voltage over inductance = Iω₀L
R I / Iω₀L = 60 / 40
R / ω₀L = 3 / 2
L = 2 R / 3 ω₀
= 2 x 121 / 3 x 5.1 x 10⁵
= 15.81 x 10⁻⁵
C = 1 / ω₀²L
= 1 / (5.1 x 10⁵)² x 15.81 x 10⁻⁵
= .002432 x 10⁻⁵
= 24.32 x 10⁻⁹ F
Let the angular frequency required be ω
Tan 45 = (ωL - 1 / ωC) / R
ωL - 1 / ωC = R
ω²LC - 1 = R ωC
ω²LC = 1 + R ωC
ω² x 15.81 x 10⁻⁵ x 24.32 x 10⁻⁹ = 1 + 121 x ω x 24.32 x 10⁻⁹
ω² x 384.5 x 10⁻¹⁴ = 1 + 2942.72 x10⁻⁹ω
ω² - 7.65 x 10⁶ ω - 1 = 0
ω = 7.65 x 10⁶
frequency = 7.65 x 10⁶ / 2π
= 1.22 x 10⁶ Hz
Newton's 2nd law of motion:
F = ma
F is force, m is mass, and a is acceleration.
Explanation:
u = 13m/s
s = 36.2m
a = 9.81m/s² (positive because it is acting on the same direction as the velocity)
Using Kinematics, we have v² = u² + 2as = (13)² + 2(9.81)(36.2) = 879.244m²/s². => v = 29.65m/s.
Statement :- We assume the orthagonal sequence
in Hilbert space, now
, the Fourier coefficients are given by:

Then Bessel's inequality give us:

Proof :- We assume the following equation is true

So that,
is projection of
onto the surface by the first
of the
. For any event, 
Now, by Pythagoras theorem:


Now, we can deduce that from the above equation that;

For
, we have

Hence, Proved