There is a clever kitchen gadget for drying lettuce leaves after you wash them. It consists of a cylindrical container mounted s
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Answer:
I(x) = 1444×k ×
I(y) = 1444×k ×
I(o) = 3888×k ×
Explanation:
Given data
function = x^2 + y^2 ≤ 36
function = x^2 + y^2 ≤ 6^2
to find out
the moments of inertia Ix, Iy, Io
solution
first we consider the polar coordinate (a,θ)
and polar is directly proportional to a²
so p = k × a²
so that
x = a cosθ
y = a sinθ
dA = adθda
so
I(x) = ∫y²pdA
take limit 0 to 6 for a and o to for θ
I(x) = y²p dA
I(x) = (a sinθ)²(k × a²) adθda
I(x) = k da ×
(sin²θ)dθ
I(x) = k da ×
(1-cos2θ)/2 dθ
I(x) = k ×
I(x) = k × × (
I(x) = k × ×
I(x) = 1444×k × .....................1
and we can say I(x) = I(y) by the symmetry rule
and here I(o) will be I(x) + I(y) i.e
I(o) = 2 × 1444×k ×
I(o) = 3888×k × ......................2
The energy stored by a system of capacitors is given by
where Ceq is the equivalent capacitance of the system, and V is the voltage applied.
In the formula, we can see there is a direct proportionality between U and C. This means that if we want to increase the energy stored by 4 times, we have to increase C by 4 times, if we keep the same voltage.
Calling
the capacitance of the original capacitor, we can solve the problem by asking that, adding a new capacitor with 
, the new equivalent capacitance of the system 
must be equal to 
. If we add the new capacitance X in parallel, the equivalent capacitance of the new system is the sum of the two capacitance
and since Ceq must be equal to 4 C1, we can write
from which we find
where Ceq is the equivalent capacitance of the system, and V is the voltage applied.
In the formula, we can see there is a direct proportionality between U and C. This means that if we want to increase the energy stored by 4 times, we have to increase C by 4 times, if we keep the same voltage.
Calling
and since Ceq must be equal to 4 C1, we can write
from which we find