We have explored a number of seemingly complex polar curves in this section.
[link] and
[link] summarize the graphs and equations for each of these curves.
Access these online resources for additional instruction and practice with graphs of polar coordinates.
It is easier to graph polar equations if we can test the equations for symmetry with respect to the line
the polar axis, or the pole.
There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an equation fails a symmetry test, the graph may or may not exhibit symmetry. See
[link] .
Polar equations may be graphed by making a table of values for
and
The maximum value of a polar equation is found by substituting the value
that leads to the maximum value of the trigonometric expression.
The zeros of a polar equation are found by setting
and solving for
See
[link] .
Some formulas that produce the graph of a circle in polar coordinates are given by
and
See
[link] .
The formulas that produce the graphs of a cardioid are given by
and
for
and
See
[link] .
The formulas that produce the graphs of a one-loop limaçon are given by
and
for
See
[link] .
The formulas that produce the graphs of an inner-loop limaçon are given by
and
for
and
See
[link] .
The formulas that produce the graphs of a lemniscates are given by
and
where
See
[link] .
The formulas that produce the graphs of rose curves are given by
and
where
if
is even, there are
petals, and if
is odd, there are
petals. See
[link] and
[link] .
The formula that produces the graph of an Archimedes’ spiral is given by
See
[link] .
Section exercises
Verbal
Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.
Symmetry with respect to the polar axis is similar to symmetry about the
-axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line
is similar to symmetry about the
-axis.
What are the steps to follow when graphing polar equations?
Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at
and sketch the graph.
For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of
and then explain the differences shown in the graphs.
Both graphs are curves with 2 loops. The equation with a coefficient of
has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to
to get a better picture.
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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