Please answer part B if you can 4. A particle of mass m is constrained to move o
ID: 1884564 • Letter: P
Question
Please answer part B if you can 4. A particle of mass m is constrained to move on a circle of radius R. The circle in tur rotates with constaut angular velocity about an axis orthogonal to its plane and passing through some point P on its circumference. Show that (a) The motion of the particle with respect to an axis passing through P and the center of the circle is that of a planar pendulum in uniform gravitational field. (b) Calculate the total energy E and angular momentum L. Show that E -L-s is a constant of the motion. Are E&L themselves constants?
Explanation / Answer
4. givne particle of mass m
mopves in a circle of radius R
circle mocves at angular speed omega about a point P on its circumference
a. let the angle of the particle with the center of the circle be phi
angle of the circle with respect to the axis P and refrence axis be theta
theta' = omega
phi' = w
then
x = Rcos(theta) + Rcos(theta + phi)
y = Rsin(theta) + Rsin(theta + phi)
x' = -Rsin(theta)*omega - R*sin(theta + phi)(omega + w)
y' = Rcos(theta)*omega + Rcos(theta + phi)(omeg a+ w)
as we can see
x = R(cos(theta) + cos(theta + phi)) = 2R*cos(theta + phi/2)cos(phi/2)
y = 2R*sin(theta + phi/2)cos(phi/2)
r = sqrt(x^2 + y^2) = 2R*cos(phi/2)
hence r the position vecotr about the asxis P follows SHM equation
b. total energy = E
E = < 0.5kr^2 >
now
phi = wt
hence
E = <0.5k*4R^2*cos^2(wt/2)>
here
sqrt(k/m) = w/2
k = mw^2/4
<E> = 0.5mw^2*R^2
as average energy <E> is independent of time, the E is constant
also
L = m r x v = 2Rcos(phi/2) [cos(theta + phi)i + sin(theta + phi)j] x [x'i + y'j]
L = m r x v = 2Rmcos(phi/2) [cos(theta + phi)(Rcos(theta)*omega + Rcos(theta + phi)(omeg a+ w)) - sin(theta + phi) -Rsin(theta)*omega - R*sin(theta + phi)(omega + w)]k
L.omega = 2Rmcos(phi/2) [cos(theta + phi)(Rcos(theta)*omega + Rcos(theta + phi)(omeg a+ w)) - sin(theta + phi) -Rsin(theta)*omega - R*sin(theta + phi)(omega + w)]*omega ( both are parallel)
L.omega = 2Rmcos(phi/2) [cos(theta + phi)(Rcos(theta)*omega + Rcos(theta + phi)(omeg a+ w)) - sin(theta + phi) -Rsin(theta)*omega - R*sin(theta + phi)(omega + w)]*omega can be shown to be constant of motion by showing
d(L.omega)dt = 0
henceE - L.omega is is constnat of motion and E and L are individually conserved as well