This section ends with a discussion of the
theorem of Pappus for volume , which allows us to find the volume of particular kinds of solids by using the centroid. (There is also a theorem of Pappus for surface area, but it is much less useful than the theorem for volume.)
Theorem of pappus for volume
Let
R be a region in the plane and let
l be a line in the plane that does not intersect
R . Then the volume of the solid of revolution formed by revolving
R around
l is equal to the area of
R multiplied by the distance
d traveled by the centroid of
R.
Proof
We can prove the case when the region is bounded above by the graph of a function
and below by the graph of a function
over an interval
and for which the axis of revolution is the
y -axis. In this case, the area of the region is
Since the axis of rotation is the
y -axis, the distance traveled by the centroid of the region depends only on the
x -coordinate of the centroid,
which is
where
Then,
and thus
However, using the method of cylindrical shells, we have
So,
and the proof is complete.
□
Using the theorem of pappus for volume
Let
R be a circle of radius 2 centered at
Use the theorem of Pappus for volume to find the volume of the torus generated by revolving
R around the
y -axis.
The region and torus are depicted in the following figure.
The region
R is a circle of radius 2, so the area of
R is
units
2 . By the symmetry principle, the centroid of
R is the center of the circle. The centroid travels around the
y -axis in a circular path of radius 4, so the centroid travels
units. Then, the volume of the torus is
units
3 .
Let
R be a circle of radius 1 centered at
Use the theorem of Pappus for volume to find the volume of the torus generated by revolving
R around the
y -axis.
Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system.
For point masses distributed along a number line, the moment of the system with respect to the origin is
For point masses distributed in a plane, the moments of the system with respect to the
x - and
y -axes, respectively, are
and
respectively.
For a lamina bounded above by a function
the moments of the system with respect to the
x - and
y -axes, respectively, are
and
The
x - and
y -coordinates of the center of mass can be found by dividing the moments around the
y -axis and around the
x -axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line.
The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region.
Questions & Answers
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you.
Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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