Transcript Spherical coordinates Coordinates on a sphere

Various systems of coordinates
• Cartesian
• Spherical
• Cylindrical
• Elliptical
• Parabolic
•…
Spherical coordinates
Coordinates on a sphere: latitude and longitude
Celestial sphere
Cylindrical coordinates
Polar coordinates
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?
Conical pendulum
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?
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   0 , what must the distance from
the mass m1 to the hole be if the mass m2 is to
remain at rest?
m1
m2
Playing with weight:
• A car on an arched bridge
• Your weight in a rotating space station
or on the rotating Earth
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:

Mm 
F   2 ir
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.
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 ?