A particle of mass m follows a path given by
r = (Xo + at^2)x + bt^3 y + ct z
where o, a, b and c are constants, t is the time. x, y , z should have hats on them showing they are position vectors (unit vectors).
1. Find the angular momentum L of the particle about the origin.
2. Find the force F required to produce this motion and verify explicitly that dL/dt = r x F
The Attempt at a Solution
L = r x p
p = mv
to get v differentiate r w.r.t time to get v = 2at.x + (3bt^2)y + cz
Subsitute into the equation:
L = ((Xo + at^2)x + bt^3y + ctz) x m(2at.x + (3bt^2)y + cz)
Now to perform the cross product, im a bit unsure here. For the cross product i have this formula:
r x p = x(ry.pz - rz.py) - y(rx.pz - rz.px) + z(rx.py - ry.px)
where the x,y,z are unit vectors and the other x,y and z's are in subscript showing the particular component of each vector. So im a bit confused because do i just subsitute the constants from my equations of r and p or do i include their respective unit vecotrs associated with each component? But if i did that i would get for example x x y = z so would make quite a difference.
Any thought guys, thanks