# Conservation of Momentum not universal?

So I calculate the momentum of a body moving with a constant speed along a circular path with mass m, tangential velocity v and radius r. Its angular momentum is mvr.kk. Good. Now what if I calculate the angular momentum from a point on the path of the circle. A simple calculation shows that the angular momentum about a point vertically below the center of the circle and which is also on the path of the circle is different for the particle at two different points. e.g. at the the point where the radius of the circle makes an angle with the positive x axis and the point where the radius of the circle makes an angle zero with the x axis. at the 90 degree point it is 2mvr and at the 0 degree point it is mvr. So does this mean that angular momentum is not conserved in every reference frame?

For circular motion there must be a constant force of $\frac{mv^2}{r}$ towards the center of the circle. When you take your reference frame as the center of the circle then there is no net torque since the force vector is parallel to the position vector, however when you move your reference frame to any other point there will be net torques over parts of the motion.