According to the textbook, the ball will nearly climb the same height (a little less but never greater), why? I came across a question in SS Krotov where it was asked why does the ball not climb the same height in the double inclined plane when it goes from one end to the other.
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2$\begingroup$ Hi new reader! Please post the name of the text book & the page(s) where this is discussed. That allows us to see whether the book is wrong or you have misinterpreted some of the statements. $\endgroup$– Carl WitthoftCommented Jul 26, 2021 at 13:38
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$\begingroup$ What is the question number in Krotov's book? $\endgroup$– nasuCommented Jul 26, 2021 at 14:55
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4 Answers
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Of course, air resistance is another form of friction and will lower the total energy of the system. If however we exclude air resistance, I think some energy will be lost anyway when transitioning from one inclined plane to the other because the transition will not be smooth and the ball has to hit "hard" the second plane. This wouldn't happen on a half-pipe, say, and the final height would be the same in that case.
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Even if the surface is frictionless, there is still air resistance. This will dissipate some of the energy and thus the ball will not return to exactly the same height.
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At the initial position, the object has some potential energy. At the other end it should have the same potential energy or less (no kinetic energy because its velocity is zero at the maximum height). If it is said that object goes higher than the initial position, it means that the mechanical energy is increased, without any work. It is not possible, because we have to do some work to increase the energy of a system. So you can observe the ball will climb a little less as some part of the energy is spent against friction and air resistance. If there are no opposing forces, the ball will climb the same height.
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If the ball rolling down one plane is to roll up the other without bouncing, the collision with the other plane has to be inelastic. This means energy is lost to sound, heat, and vibration in the collision. If the planes are at right angles and the ball rolls straight down the slope perpendicular to the line of intersection, the ball hits the bottom and stops dead. All the energy is lost.
If you are talking about something like this page, then the "almost" is referring to the fact that real experiments are not the same as the mathematical idealisations. Even if you join the planes with a smooth curve (which of course means it is not really a 'double inclined plane'), real surfaces have friction.
