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 , what must the distance from the mass m1 to the hole be if the mass m2 is to remain at rest?

0

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:

rir

MmF

2

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.

is a gravitational constant

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 ?