Elongation of a rod hinged at the end

Suppose a rod is rotating in a horizontal frictionless plane, hinged at one of its ends. If the body is non rigid, it would change its length, but I am not sure whether it would elongate or get compressed. According to me, it should get compressed, as centripetal force acts towards the centre leading to compression of the rod. If I am wrong, feel free to correct me.

• Do you mean the rod is rotating? Aug 22, 2020 at 3:23
• Yes the rod is indeed rotation about hinged end. Aug 22, 2020 at 3:27
• You must consider centrifugal force. Aug 22, 2020 at 3:34
• Why do i need to consider centrifugal force. I am seeing the whole motion from an inertial frame and according to that Particles tends to have acceleration towards centre . Moreover as far as i know centrifugal force is not a real force . How can It provide real effecfs like elongation If it's just a mathematical trick. Aug 22, 2020 at 3:38
• @AdrianHoward you do not need to invoke centrifugal force. My answer provides an explanation that does not cite centrifugal force. Aug 22, 2020 at 3:40

Centripetal force is not some separate force that acts on objects in circular motion, rather it is a term used to refer to whichever force happens to act in the radial direction to keep the object on a circular trajectory. In this case, the tension in the rod would provide the centripetal force, and since the rod must stretch (elongate) in order to have tension, its length increases.

See the rod will elongate.This is shown below:

Consider dx element.There are two tensions acting on it as shown above,both due the masses adjacent to the dx element. Note how the tensions are unbalanced and net force is dT.This force dT provides centripetal acceleration for this dx element.

However the forces responsible for elongation, T and T-dT individually are acting outwards for each dx element.They tend to pull apart each dx element causing it to elongate.

Further when we try to calculate the actual elongation using Young's Modulus, we use force as T for both ends of the dx element. This 'error' is taken care of by a mathematical trick called Calculus.

I am inserting this part for @user266897

By your argument, the net force here should be zero and hence there should be no elongation. However, when talking about rigid body, we do not use Force at centre of mass to calculate elongation.This is because the concept of centre of mass is to convert a rigid body to a point mass.

For calculating elongation,we use Force at their given locations on the rigid body.

Again this rod elongates because forces at both ends tend to pull it apart.

P.S. I have tried my best to explain the matter to you.If you still have any doubt comment below.

• This dT tension i agree that it provides centrepetal acceleration but we can directly see in your diagram that dT actually direction is towards centre ( if You remove the negative sign by take dT on to other side of rod so that it direction is towards the centre) and dT being towards at centre the particles should get compressed i think. Aug 22, 2020 at 3:45
• @user266897 dT is not towards centre.It is acting at the centre but it is towards the wall.Note when you are trying to find elongation,you do not use net force acting at the centre of mass. You use all the forces acting on the body at their respective locations.Think over it. I have inserted an analogy in my answer to help you understand. Aug 22, 2020 at 4:06