# Conservation of angular momentum with no net torque

Suppose there is a point mass with mass m1 orbiting around the z axis at a radial distance r1 with tangential velocity v1. So I would say the angular momentum of this point mass is m1•v1•r1. Then I exert forces on m1 in the radial directions such that m1 is now orbiting the z axis and 2r1. I would say now the angular momentum is m1•v1•2r1. Clearly the angular momentum has increased by a factor of 2. However there was no torque applied to this system about the z axis as the forces that moved m1 to 2r1 were completely in the radial direction. Isn’t this a violation of the conservation of angular momentum?

$$v$$ will not be constant in this example. The new angular momentum would be $$m_1\cdot v_2\cdot 2r_1$$, where $$v_2\neq v_1$$.
If you insist on keeping $$v$$ constant, then you would actually have to apply a torque to stop the particle from changing its speed to conserve angular momentum, so there is no contradiction here.
If you don't believe that $$v$$ will change, the classic example to visualise is a figure skater spinning around. When they have their arms tucked in, they spin at speed $$v_1$$. As they stretch their arms out, they slow down to conserve angular momentum (try it yourself on a spinning chair, but be careful of your surroundings!)