# Why does the magnitude of linear momentum of a particle in circular motion change with radius? [duplicate]

My problem is with linear momentum of a particle in circular motion. If we imagine a particle moving around a circle, if there are no torques acting, then we can say its angular momentum is conserved, i.e. $mvr=k$ (where $m$ is the mass, $v$ the velocity, $r$ the radius and $k$ a constant value). It means that if the radius decreases, then the velocity will increase and viceversa.

Since during all the process the force (which is centripetal) is only responsible for changing the direction of the velocity the particle (it makes it move around a circle) and is always perpendicular to it, the magnitude of linear momentum should not change. Then why does the velocity (or momentum) change with a change in the radius (by conservation of angular momentum)? Why is linear momentum not conserved in circular motion? Correct me if something of what I said is wrong. If you are able to explain this, please try to give an intuitive explanation.

## marked as duplicate by Emilio Pisanty, ja72, user36790, Community♦Apr 17 '16 at 16:36

• Well, take into account the linear momentum of the thing that keeps the circularly moving thing moving in a circle! – ACuriousMind Apr 17 '16 at 15:19
• Because torque and linear momentum are not related, torque changes the angular momentum. Furthermore, if you have a fixed centre (instead of considering a two body problem) then it is obvious that there is no conservation of linear momentum – linear momentum is only conserved if the Langrangian shows translation invariance, but a fixed centre point (occurring in the potential) breaks translation invariance. – Sebastian Riese Apr 17 '16 at 15:30
• This has been answered here many times. Links anyone? – ja72 Apr 17 '16 at 16:14

For linear momentum to be a constant, its value should not change. Being linear momentum a vector quantity, it's value is completely described by specifying both magnitude and direction. In a circular motion, even though the speed remains a constant, the direction of velocity (which is tangential to the point on a circle) is changing throughout the motion as shown:

Change in direction of velocity means the velocity is changing (no matter whether its magnitude changes or not). So linear momentum is not a constant in circular motion. But it is possible to have a uniform acceleration in circular motion if we keep the rate of change in velocity a constant.

Now, centripetal force guarantees the circular motion of the particle. If the force is a constant, then the acceleration of the particle will be a constant. By Newton's second law, the rate of change in linear momentum of the particle is equal to the centripetal force acting on it

(m$v^2$)/r = dp/dt = m dv/dt

From this equation, it is clear that, for the linear momentum or linear velocity to be conserved in a circular motion, it's time derivative

dp/dt=0

or that implies the centripetal force

(m$v^2$)/r=0.

But if the centripetal force is zero, then the particle can no longer execute circular motion, which means a circular motion with uniform linear velocity or constant linear momentum is not possible. However, if no external torque acts on the system, then the angular momentum remains conserved