Angular Momentum, Energy and Work

Question 1

What is angular momentum?

Answer 1

The angular momentum of a particle about a point is the moment of the momentum of the particle about the same point

K = r x mv

Question 2

What is the result if we differentiate K with respect to time?

Answer 2

It is the sum of the cross products of the distances and the forces.
The time rate of change of the angular momentum about a
point equals the sum of the moments of forces acting upon it,
about the same point

Question 3

Using polar coordinates, what is the equation for K?

Answer 3

K = r x mv = r x m(r dot * u(r) + r * theta dot * u(theta)
K = r x m * r dot * u(r) + r x mr * theta dot * u(theta)

The first component of the final equation is equal to 0 because the vectors are parallel

Question 4

What is the equation of the trajectory in polar coordinates?

Answer 4

We will denote that u = 1/r where r is the radius
We can also use Newton’s Second Laws in polar coordinates to help
(d^2 u) / (d theta^2) + u = F / (m * h^2 * u^2)

Question 5

What is Newton’s Second Law in polar coordinates?

Answer 5

m * (r double dot - r (theta double dot)^2) = -F
m * (r * theta double dot + 2 * r dot * theta dot) = 0

Question 6

What is the mechanical work done by force F?

Answer 6

W = s F cos (x) where x is the angle between the force and the surface of travel
W = F dot product s

Question 7

What is the kinetic energy of a body?

Answer 7

Kinetic energy = 1/2 * m * v^2

Question 8

What is the power?

Answer 8

Power is the rate at which work is performed
Average Power = Total Work / Total Time

Question 9

What is the equation for power if the forces acting upon an object reduce to a single moment?

Answer 9

Power = Moment * Angular Velocity

Question 10

What is gravitational potential energy?

Answer 10

Potential Energy = Mass * Height * Gravitational Field Strength

Question 11

What is the elastic potential energy?

Answer 11

Elastic Potential Energy = 1/2 * Spring Constant * Displacement^2