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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?

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$v$ 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!)

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    $\begingroup$ There is no force in the tangential direction at anytime so how is it possible that the object decelerates in the tangential direction? $\endgroup$
    – Blue5000
    Sep 2, 2023 at 11:57
  • $\begingroup$ If you're working in the frame of reference of the particle, there actually is a pseudo Coriolis force which makes its tangential speed change. This comes from coordinate transforming from a stationary frame to a non-inertial rotating frame. Physics can be a little weird in accelerating frames! $\endgroup$
    – Garf
    Sep 2, 2023 at 11:59

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