The image contains a mathematical problem written in Spanish. Here is the transcription and translation of the problem:

Transcription:

"Se un bloque rectangular de masa m cae del reposo desde una altura determinada con respecto al suelo, determinando el trabajo hecho por el peso en su caída y tomando la base de energía cinética con expresiones en función de m, h y g. Considere una aceleración de la gravedad de 9.8 m/s²."

Translation:

"If a rectangular block of mass $m$ falls from rest from a certain height with respect to the ground, determine the work done by the weight during its fall, using kinetic energy as a basis, with expressions in terms of $m$, $h$, and $g$. Consider a gravitational acceleration of 9.8 m/s²."

Solution:

To solve this problem, let's start by understanding the relationship between gravitational potential energy and kinetic energy.

1. Gravitational Potential Energy (GPE): $\text{GPE} = mgh$ where:

   • $m$ is the mass of the block,
   • $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
   • $h$ is the height from which the block falls.

2. Kinetic Energy (KE): As the block falls, the gravitational potential energy is converted into kinetic energy. When the block reaches the ground, all the potential energy has been converted to kinetic energy. Therefore, $\text{KE} = \frac{1}{2}mv^2$

3. Work Done by the Weight: The work done by the weight of the block during the fall is equal to the change in gravitational potential energy, which is given by: $W = \Delta \text{GPE} = mgh$ This is the same as the work-energy principle, where the work done by all forces (in this case, only gravity) equals the change in kinetic energy: $W = \Delta \text{KE}$

Thus, the work done by the gravitational force as the block falls is $mgh$.

Summary:

• The work done by the weight of the block during its fall is $W = mgh$.
• This work equals the change in the block's kinetic energy.

Would you like more details on any part of this solution, or do you have any specific questions?

Further Questions:

1. What is the relationship between potential energy and kinetic energy in other scenarios?
2. How would the work done change if the block were on an inclined plane?
3. Can you derive the velocity of the block just before it hits the ground?
4. What if the block were falling through a medium with air resistance? How would that affect the work done?
5. How is the work-energy principle applied in different areas of physics?

Tip:

Always double-check units when working with physical equations, especially when converting between different systems (e.g., from SI to imperial units).