# Is it necessary for force to be zero for constant angular velocity? [closed]

According to a few articles I read, they say Newton's first law for rotational inertia is that if net torque on body is zero it keeps on rotating with same angular velocity .

But will the body rotating with constant angular velocity can have zero force (by force I mean net force on the body hence only external force) also? I know the necessary and sufficient condition for angular velocity to be zero is that torque acting on it must be zero but My question is when the torque is zero is there a force acting on the the body because if torque is zero it is not necessary that force is also zero ? ie in other words if torque is zero that means body rotates with constant angular velocity but is there a force MAINTAINING this constant angular velocity or is it just like inertia in linear motion ?

(Also the body is rotating about a fixed axis and is rigid)

• Your wording is somewhat confusing. 0 net torque is the requirement to ensure it has a constant angular velocity. – JMac May 4 '17 at 10:55
• If a force is applied to the center of gravity, it will not exert torque on the body, hence the angular momentum will stay constant. – noah May 4 '17 at 10:57
• I have edited my question I hope now its clear – Matt May 4 '17 at 11:02
• @Michael okay but if torque is zero that means body rotates with constant angular velocity but is there a force MAINTAINING this constant angular velocity or is it just like inertia in linear motion ? – Matt May 4 '17 at 11:10
• It is just like inertia. – noah May 4 '17 at 11:11

Think of angular velocity $\omega$, angular acceleration $\alpha$, moment-of-inertia $I$ and torque $\tau$ in the exact same way as you think of normal (translational) velocity $v$, acceleration $a$, inertia (mass) $m$ and force $F$. They have their two equivalent laws combining them:

$$\sum F=ma\qquad , \qquad \sum \tau=I\alpha$$

And you can trust these laws.

• If there is a net torque $\sum \tau$, there is an angular acceleration $\alpha$. Regardless of what else is happening at the same time.

• Just like, if there is a net force $\sum F$ then there is an (translational) acceleration $a$. Regardless of other simultaneous effects and things.

So for example: If a force pushes at a point on an object that does not cause a torque, or if it causes a balanced torque, then there is zero net torque and no angular acceleration. Whatever angular velocity it has, is not changed. Simple. Other forces might or might not be present - the only important thing is if they cause a net torque.

So, when you ask...

But will the body rotating with constant angular velocity can have zero force

... the answer is yes, a body rotating (constant angular velocity or not) can easily have a net force acting on it, as long as that net force doesn't cause a torque (which it doesn't, if it pushed in the centre-of-mass e.g.).

And when you ask...

if torque is zero that means body rotates with constant angular velocity but is there a force MAINTAINING this constant angular velocity or is it just like inertia in linear motion ?

... the answer is no, there is no need to maintain a constant velocity (rotational or translational) with any force or torque. A motion only requires torque or force to change (and how much torque or force depends on it's inertia), but not to be constant.

• But the linear velocity of particles is constantly changing (in direction) so how can there be no net force ? – Matt May 4 '17 at 11:35
• @RaghavSingal Remember that we talked about net force. Yes, all particles are pulled towards the centre by a force in order to rotate around and do the circle motion around the centre. But that force on a particle in one side of the object, is balanced by the same opposite force on another particle in the other side of the object. Each particle (except the centre-of-mass) experiences a net force, but in the object as a whole it all cancels out and balances. – Steeven May 4 '17 at 11:38
• But wont that be valid only for symmetrical systems ? For example we require an external torque to sustain uniform motion of a half wheel about an axis passing through its centre of mass and perpendicular to it . – Matt May 7 '17 at 9:03
• @RaghavSingal. A bike wheel must be a symmetrical object, yes. Otherwise the torques are not balanced out on either side, as explained in the previous comment. Then the centre-of-mass will be somewhere else - at the new point where all the torques cancel out on either side. But there will always be such a point, it just doesn't have to be the geometric centre. And when an object turns about this point, it needs no help to sustain that motion. – Steeven May 7 '17 at 15:27
• So we need external torque to keep it in uniform motion. – Matt May 7 '17 at 15:28

Yes, there is a possibility that force is applied but angular velocity does not change. The requirement for this is - "torque" about the axis of rotation should be 0. And you can imagine how it can be done - one of the ways is - hit the body at it's axis of rotation. The body might do some translational acceleration, but angular velocity of rotation will not change.