A small block of mass M = 0.10 kg is released from rest at point 1 at a height H = 1.8 m above the bottom of a track, as shown in the diagram. It slides
down the track and around the inside of a loop of radius R = 0.6 m. The speed of the block is 2.5 m/s at point 3. Which of the following claims about the
situation is correct?
A) The gravitational potential energy of the block-Earth system at point 3 is less than the gravitational potential energy at point 2.
B) The kinetic energy of the block at point 3 is greater than the kinetic energy of the block at point 2.
The mechanical energy of the block-Earth system at point 3 is less than the mechanical energy of the block-Earth system at point 1.
D
The mechanical energy of the block-Earth system at point 2 is equal to the gravitational potential energy of the block-Earth system at point 1.

A) is not correct, because the gravitation potential energy will depend on the height the block is located at. It will be calculated with the formula:

U=mgh.

If we take the ground as a zero height reference, then on point 2 the potential energy will be:

[tex]U_{2} = 0.10kg(9.81 m/s^{2})(0.6m)[/tex]

[tex]U_{2}=0.59 J[/tex]

While on point 3, the potential energy will be greater.

[tex]U_{3}=0.10kg(9.81 m/s^{2})(1.2m)[/tex]

[tex]U_{3}=1.18 J[/tex]

B) is not the right answer because the kinetic energy will vary with the height the block is located at in the fact that the energy is conserved (this is if we don't take friction into account or air resistance) so in this case:

[tex]U_{2}+K_{2}= U_{3}+K_{3}[/tex]

We already know the potential energy at point 2. We can calculate the kinetic energy at point 3 like this:

[tex]K_{3} =\frac{1}{2}mv_{3}^{2}[/tex]

[tex]K_{3} =\frac{1}{2}(0.10kg)(2.5 m/s)^{2}[/tex]

[tex]K_{3} =0.31 J[/tex]

So the kinetic energy at point 2 is given by the equation:

[tex]K_{2} =U_{3}-U_{2}+K_{3}[/tex]

so:

[tex]K_{2} = (1.18J)-(0.59J)+0.31J[/tex]

[tex]K_{2} =0.9J[/tex]

As you may see the kinetic energy at point 2 is greater than the kinetic energy at point 3.

C) Is not correct because according to the first law of thermodinamics, energy is not lost, only transformed. So, since we are not taking into account friction or any other kind of loss, then we can say that the amount of mechanical energy at point 1 is exactly the same as the mechanical energy at point 3.

D) Because of what we talked about on part C, this will be the true situation, because the mechanical energy of the block will be the same no matter theh point you measure it at.