Understanding the Conversion of Potential Energy to Kinetic Energy on an Inclined Plane

Have you ever wondered what happens to the energy of an object as it slides down an inclined plane without friction? This article explores a fascinating physics problem, where the potential energy at the top of an incline is converted into kinetic energy at the bottom, aiding in a deeper understanding of the conservation of energy principle.

Problem Summary

A block of mass 2 kg slides from rest on a frictionless inclined plane 20 meters long and 10 meters high. The question at hand is to determine the kinetic energy (KE) of the block at the bottom of the slope. Let's delve into the step-by-step analysis to solve this problem.

Step 1: Calculate the Potential Energy at the Top

At the top of the inclined plane, the block possesses gravitational potential energy (PE). The formula for potential energy (PE) is given by:

[ text{PE} mgh ]

Where:

m 2 kg - the mass of the block g 9.81 m/s2 - the acceleration due to gravity h 10 m - the height of the incline

Substituting these values into the formula, we get:

[ text{PE} 2 , text{kg} times 9.81 , text{m/s}^2 times 10 , text{m} 196.2 , text{J} ]

Step 2: Determine the Kinetic Energy at the Bottom

Since the incline is frictionless, the potential energy at the top is completely converted into kinetic energy at the bottom. According to the conservation of energy principle, the total energy of the system remains constant. The kinetic energy (KE) at the bottom is therefore equal to the potential energy at the top.

[ text{KE} text{PE} 196.2 , text{J} ]

Conclusion

The kinetic energy of the block at the bottom of the slope is 196.2 joules.

Application of the Law of Conservation of Energy

This problem is a practical demonstration of the Law of Conservation of Energy. At the top of the incline, the block has gravitational potential energy. Without friction to dissipate energy, this potential energy is fully converted into kinetic energy as the block slides down the plane.

Other efficient ways to calculate the final kinetic energy include:

1. Direct Calculation using ( KE mgh )

If we assume an approximate value for ( g ), say 10 m/s2, the calculation simplifies to:

[ KE 2 , text{kg} times 10 , text{m} times 10 , text{m/s}^2 200 , text{J} ]

2. Using the General Formula for Conservation of Energy

The general formula for the conservation of energy, which states that energy cannot be created or destroyed, confirms the conversion:

[ text{GPE at top} text{KE at bottom} ]

Therefore, (text{KE at bottom} 2 , text{kg} times 9.81 , text{m/s}^2 times 10 , text{m} 196.2 , text{J}).

Importance of the Conservation of Energy Principle

This principle is fundamental not only in physics but also has numerous applications in engineering, chemistry, and even in everyday life. By understanding and applying the conservation of energy, we can solve a wide range of physical problems and optimize energy use in various systems.

This article has provided a detailed explanation of the conversion of potential energy to kinetic energy on a frictionless inclined plane. For more information on this and related topics, please visit our resources section or search for more detailed answers. Happy learning!