A particle of mass m is constrained to move on the inside surfaceof a cone of half-angle a, under the influence of constantgravitational force. In the cylindrical coordinates (r, phi, z) theconstraint that the particle is moving on the surface of the coneis given by z=r cot a . [do not use lagrangian mothod)
a) Find the net force acting on the particle and derive theequation of motion for r and phi using Newton's 2nd law. Show thatthe angular momentum L is a constant of motion. (use a incylindrical coordinateshttp://mathworld.wolfram.com/CylindricalCoordinates.html
see eq 72)
b) Find the total energy in terms of r, r dot and L
My attempt:
a) I found all the forces acting on the particle and set them equalto the forces I get from using the acceleration in cylindricalcoordinates
so, F(in phi direction) = 0 = m (2(r dot)(phi dot) + r (phidoubledot))
F(r) = -N cos a = m( (r double dot - r* (phi dot)^2) [N = normalforce]
F(z) = -mg + N sin a = m (z double dot)
How should I proceed next ? I have to find the equations of motionfor r and phi.
And how should I show that L is a constant of motion?
b) I have no idea how to do this part!
A particle of mass m is constrained to move on the inside surfaceof a cone of half-angle a, under the influence of constantgravitational force. In the cylindrical coordinates (r, phi, z) theconstraint that the particle is moving on the surface of the coneis given by z=r cot a . [do not use lagrangian mothod)
a) Find the net force acting on the particle and derive theequation of motion for r and phi using Newton's 2nd law. Show thatthe angular momentum L is a constant of motion. (use a incylindrical coordinateshttp://mathworld.wolfram.com/CylindricalCoordinates.html
see eq 72)
b) Find the total energy in terms of r, r dot and L
My attempt:
a) I found all the forces acting on the particle and set them equalto the forces I get from using the acceleration in cylindricalcoordinates
so, F(in phi direction) = 0 = m (2(r dot)(phi dot) + r (phidoubledot))
F(r) = -N cos a = m( (r double dot - r* (phi dot)^2) [N = normalforce]
F(z) = -mg + N sin a = m (z double dot)
How should I proceed next ? I have to find the equations of motionfor r and phi.
And how should I show that L is a constant of motion?
b) I have no idea how to do this part!
