Various systems of coordinates

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Various systems of coordinates

• Cartesian
• Spherical
• Cylindrical
• Elliptical
• Parabolic

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Spherical coordinates
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Coordinates on a sphere latitude and longitude
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Celestial sphere
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Cylindrical coordinates
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Polar coordinates
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A ball of mass m is swung around a circle at the
end of a string of length L. The string will
break if the tension in it exceeds a critical
value, Tc. What is the largest constant angular
velocity the ball can have without breaking the
string? What is the largest period the ball can
have without the string becoming slack?
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Conical pendulum
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Platform rotates with a constant angular velocity
?0. At t 0 it starts rotating with angular
acceleration ?(t)?t. At the same time a man
starts a distance L from the center and walks in
along a straight line painted on the platform
towards the center. He decreases his distance
from the center at a constant rate, V0. What
force does the platform exert on the man, as a
function of his distance from the center?
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A mass m1 is going around in a circle on a string
on a frictionless table and the string goes
through a hole where it is attached to a hanging
mass m2. If the mass m1 is going around with
constant , what must the distance from
the mass m1 to the hole be if the mass m2 is to
remain at rest?
m1
m2
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Playing with weight

• A car on an arched bridge
• Your weight in a rotating space station
• or on the rotating Earth

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A satellite of mass m is attracted to the Earth
of mass M with a force of gravity proportional to
the inverse square of the distance to the Earth
center, r
? is a gravitational constant
Calculate the work done by this force a) If the
satellite moves out along a radius vector from B
(rB from the origin) to C (rC from the
origin). b) if the satellite moves from A to B
where A and B are points on a circle, centered at
the origin Find the velocity, the kinetic
energy, and the potential energy of a satellite
on a circular orbit of radius r.
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A race track designer wants to have the cars able
to maintain a speed vmax without skidding on a
circular track. If the track is flat with a
coefficient of friction ? what does the radius
have to be?
A race track designer wants to have the cars able
to maintain a speed vmax without skidding. At
what angle must the track of radius R be banked
assuming no friction? Assuming a coefficient of
friction ??