Imagine a rod in Free space having sticky nature....Now hit at the centre with some object....since momentum should be conserved,so both the object and rod together move as one with the same momentum...Now if we again do it but this time hitting it at one end, both the object and rod would again move conserving the linear momentum..But this time it should rotate as well in order to conserve angular momentum...Since both the momenta (angular as well as linear) are independent, it seems ok...but now we have some rotational kinetic energy as well....Why does this energy exceeds the one in former case as in former case we have only kinetic....My question is if the initial momentum and energy is same in both cases?...why the 2nd one seems to violate Conservation of Energy?..I m missing Something important....Please help
2 Answers
Kinetic energy isn't a conserved quantity. It's total energy that's conserved. That includes "thermal energy" - energy associated with heating the rod and object up.
In an inelastic (sticky) collision, some kinetic energy will be turned into thermal energy. The rod and object will both get slightly hotter during the collision.
Your observation is correct - there is more kinetic energy in the system after the collision when the object hits the rod on the side. This means less kinetic energy is turned into thermal energy. When the object hits the rod on the side, they heat up less than when the object hits the rod in the center.
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$\begingroup$ Sir,which law tells us that there shd b more heat when objects hits the rod at centre than that when hit at end....How could we predict the outcome of the whole process? $\endgroup$ Commented Feb 16, 2023 at 6:40
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$\begingroup$ You can find the motion of the rod after the impact using conservation of momentum and angular momentum. After finding the motion of the rod, you can calculate the change in kinetic energy of the system during the impact. However much the kinetic energy falls, the thermal energy rises, due to conservation of energy. $\endgroup$ Commented Feb 16, 2023 at 12:01
Total energy is conserved in both cases, but the energy gets divided up in different ways. Because the object and rod stick together after the collision, kinetic energy is not conserved. It gets converted into some other form. In the first case, where there is no rotation after the collision, the kinetic energy lost in the collision will end up as work done to deform the shape of the rod or object and as heat produced in the collision. In the second case, less deformation and less heat will be produced with the difference going into the rotational kinetic energy. The linear kinetic energy in both cases will be the same since the masses and linear speeds are the same.
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$\begingroup$ But sir there was no mention of heat or any other form of energy while formulating the laws of mechanics....i.e kinematics.....how do we know there is more heat transfer in former case....? $\endgroup$ Commented Feb 14, 2023 at 14:21
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$\begingroup$ @SuhailSarwar I mention heat because the missing energy has to be accounted for. Here's a simpler situation: two objects of equal mass and equal speed collide head on and stick together. The total momentum is zero before and after the collision, so the stuck-together objects are not moving after the collision. Where did the kinetic energy go? Thermal energy, vibrational energy, sound energy, light energy, and on and on. This is why kinetic energy is not conserved except in special circumstances. Since total energy is conserved, we have to look for it based on the details of the collision. $\endgroup$– Mark HCommented Feb 15, 2023 at 2:49
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$\begingroup$ Yes sir...i understand that...but what was the explanation before atomic or molecular theory....a rod was considered as a continuous 1....so without considering the internal structure,how can we apply basic newtons laws....and what if there is no heat in some case....like the elastic collision....what if the rod is non sticky $\endgroup$ Commented Feb 15, 2023 at 5:31
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$\begingroup$ @SuhailSarwar The idea of conservation of energy wasn't formalized until the 1840s, 150 years after Newton's Principia. Here's a three-part story of the history of the conservation of energy by a physicist: Part 1, Part 2, and Part 3. $\endgroup$– Mark HCommented Feb 15, 2023 at 9:03