Respuesta :
Answer:
Explanation:
This problem is about a inclined plane where we have to apply the conservation of energy theorem.
So, we know that the ice cube has a mass of 50g, which is 0.05kg (dividing by 1000). Also, the spring constant is [tex]k=25\frac{N}{m}[/tex] and the spring is compressed 0.100m the ice cube reach it.
So, if we consider the ice cube in the top of the slope, we know that the total amount of energy there is potential, that is, [tex]E_{total} = U_{height} = mgh[/tex]. When the when the ice cube is in the final position, in the spring, again the total amount of energy is potential, but this time is due to the spring, that is, [tex]U_{spring} =\frac{1}{2} kx^{2}[/tex].
Then, using the conservation of energy theorem, we know that the energy at the beginning is the same at the end of the movement period, that is:
[tex]U_{height} = U_{spring}[/tex]; now we replace all values and solve for h.
[tex]mgh = \frac{1}{2} kx^{2} \\h=\frac{kx^{2} }{2mg} = \frac{25(0.10)^{2} }{2(0.05)(9.8)} = \frac{0.25}{0.98}=0.26m[/tex]
This results means that the ice cube will reach 0.26m of heigh from the floor. However, the problem is asking the distance it's gonna travel up the slope, to find that we use trigonometric reasons, because the inclined plane forms a right triangle.
In the right triangle, we know the angle of the slope, the height and the unknown variable is the hypotenuse, which is the distance on the slope. In this case, we apply the sin trigonometric reason.
[tex]sin25\°=\frac{h}{d}\\ d=\frac{h}{sin25\°} = \frac{0.26}{0.42}=0.62m[/tex]
But, we have to subtract the contraction of the spring (0.100m). Therefore, the ice cube travels up the slope 0.62 - 0.10 = 0.52m
On the other hand, the initial and final potential energies are the same, we can calculate them using one of those equations:
[tex]U_{height} = mgh = 0.05kg(9.8\frac{m}{s^{2} } )(0.26m)=0.13 J[/tex]
Therefore the energy in the initial and final point of the trajectory is 0.13J.
