# Is energy conserved on a ball sliding without friction?

If a ball is sliding without friction (down an inclined surface) is energy conserved?

I can’t think of why it wouldn’t be, but my intuition on this stuff is really bad.

In Newtonian mechanics, energy is always conserved (as are angular and linear momentum, separately). When the ball slides down the slope, it loses gravitational potential energy and it gains the equivalent in kinetic energy. Both the horizontal and vertical velocity components increase. Due to the lack of friction between the ball and the slope, there is no torque applied to the ball and the angular kinetic energy and angular momentum remains constant. If there was friction between the ball and the slope, total energy is still conserved and when the angular kinetic energy of the ball increases, the gain in linear kinetic energy would be correspondingly lower and this is why objects with a high moment of inertia accelerate slower down a slope (when there is friction). If the slope was completely frictionless, all these objects in the animation would arrive at the bottom of the slope at the same time.

In the context of General Relativity, energy is not always conserved in the classical sense, such as in cases like the expansion of the universe where the metric is not necessarily time symmetric, as per Noether's theorem.

The thing that's always true* about energy is that it's neither created nor destroyed. When we talk about whether or not it is conserved, then we have to describe a "system," and in order to fully describe the system, we have to define a boundary.

Energy is "conserved" in any system that does not allow energy to enter or leave across the boundary.

Imagine a perfectly inelastic collision: A ball of putty is dropped onto the floor, and it sticks there. Initially, the system had potential energy. Then, prior to the collision, all of that potential energy was converted to kinetic kinetic energy. But in the final state, the kinetic energy and the potential energy have both gone to zero. Was energy conserved?

If we define our system such that the thermal energy of the ball and the floor are within its "boundary," then yes, energy was conserved. But if we ignore the thermal energy, then we say, no. Energy was not conserved in that collision. In either case, energy was neither created nor destroyed, but in the second case, energy escaped from the system-as-we-defined-it.

Your system is frictionless: It has no means for converting between kinetic or potential energy and heat. So, in your system, it doesn't really matter whether you include thermal energy or not. It doesn't play any role.

* At least, it's always close enough true when you're dealing with classical theory and you're not worried about relativistic motion.