Consider two massless springs connected in series. Spring 1 has a spring constant k1, and spring 2 has a spring constant k2. A c
2 answers:
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<h2>Answers:</h2><h2>(a) </h2>
According to Newton's Law of Gravitation, the Gravity Force is:
(1)
This expression can also be written as:
(2)
If we derive this force respect to the distance
between the two masses:
(3)
Taking into account are constants:
(4)
Or
(5)
In other words, this means how much does the Gravity Force changes with the distance between the two bodies.
More precisely this change is inversely proportional to the distance elevated to the cubic exponent.
As the distance increases, the Force decreases.
<h2>(c) The minus sign indicates that the bodies are being forced in the negative direction. </h2>This is because Gravity is an attractive force, as well as, a central conservative force.
This means it does not depend on time, and both bodies are mutually attracted to each other.
<h2>(d) </h2>In the first answer we already found the decrease rate of the Gravity force respect to the distance, being its unit :
(5)
We have a force that decreases with a rate 1 when
:
(6)
Isolating :
(7)
In addition, we have another force that decreases with a rate 2 when
:
(8)
Isolating :
(9)
Making (7)=(9):
(10)
Then isolating :
Solving and taking into account the units, we finally have:
>>>>This is how fast this force changes when
1) Linear charge density of the shell:
2) x-component of the electric field at r = 8.7 cm: outward
3) y-component of the electric field at r =8.7 cm: 0
4) x-component of the electric field at r = 1.15 cm: outward
5) y-component of the electric field at r = 1.15 cm: 0
Explanation:
1)
The linear charge density of the cylindrical insulating shell can be found by using
where
is charge volumetric density
A is the area of the cylindrical shell, which can be written as
where
is the outer radius
is the inner radius
Therefore, we have :
2)
Here we want to find the x-component of the electric field at a point at a distance of 8.7 cm from the central axis.
The electric field outside the shell is the superposition of the fields produced by the line of charge and the field produced by the shell:
where:
where
is the linear charge density of the wire
r = 8.7 cm = 0.087 m is the distance from the axis
And this field points radially outward, since the charge is positive .
And
where
And this field points radially inward, because the charge is negative.
Therefore, the net field is
in the outward direction.
3)
To find the net electric field along the y-direction, we have to sum the y-component of the electric field of the wire and of the shell.
However, we notice that since the wire is infinite, for the element of electric field produced by a certain amount of charge
along the wire there exist always another piece of charge
on the opposite side of the wire that produce an element of electric field
, equal and opposite to
.
Therefore, this means that the net field produced by the wire along the y-direction is zero at any point.
We can apply the same argument to the cylindrical shell (which is also infinite), and therefore we find that also the field generated by the cylindrical shell has no component along the y-direction. Therefore,
4)
Here we want to find the x-component of the electric field at a point at
r = 1.15 cm
from the central axis.
We notice that in this case, the cylindrical shell does not contribute to the electric field at r = 1.15 cm, because the inner radius of the shell is at 2.7 cm from the axis.
Therefore, the electric field at r = 1.15 cm is only given by the electric field produced by the infinite wire:
where:
is the linear charge density of the wire
r = 1.15 cm = 0.0115 m is the distance from the axis
This field points radially outward, since the charge is positive . Therefore,
5)
For this last part we can use the same argument used in part 4): since the wire is infinite, for the element of electric field produced by a certain amount of charge
along the wire there exist always another piece of charge
on the opposite side of the wire that produce an element of electric field
, equal and opposite to
.
Therefore, the y-component of the electric field is zero.
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