Question

Apply The potential energy of an object decreases by \(10 \mathrm{~J}\). What is the change in the object's kinetic energy, assuming there is no friction in the system?

Short answer

The object's kinetic energy increases by 10 J.

Step-by-step solution

Understanding Energy Conservation

In physics, energy is conserved in an isolated system. This means that the total energy within such a system remains constant, though it can switch between different forms like potential energy and kinetic energy.

Analyzing the Problem

The problem states that the potential energy of the object decreases by 10 J. This is the only change in energy mentioned, and it indicates that energy must have converted to another form due to the conservation principle.

Applying Conservation of Energy

As potential energy decreases, kinetic energy increases to conserve total energy. If 10 J of potential energy is lost, it is converted to kinetic energy, assuming no energy is lost to other forms like heat (no friction implies lossless conversion).

Calculating the Change in Kinetic Energy

Since there has been a decrease of 10 J in potential energy, by the conservation of energy, this same amount of energy will be gained by the kinetic energy. Therefore, the change in the object's kinetic energy is equal to 10 J.

Key concepts

Potential Energy

Potential energy is one of the main forms of energy, often linked to an object's position or configuration. Consider a book on a shelf. The book has potential energy because of its height relative to the ground. This form of energy is stored and can be converted into other types. The potential energy of an object is influenced by:

\( \cdot \) Height: Potential energy is directly proportional to height. An increase in height results in an increase in potential energy.
\( \cdot \) Mass: More massive objects have higher potential energy if at the same height as lighter objects.
\( \cdot \) Gravity: The strength of the gravitational field affects potential energy. On Earth, gravity is typically constant, which simplifies many calculations.

Its formula is given by: \[ PE = mgh \]where:

• \( PE \) is the potential energy
• \( m \) is the mass in kilograms
• \( g \) is the acceleration due to gravity (approx. \(9.8 \text{ m/s}^2\) on Earth)
• \( h \) is the height above a reference point in meters

Understanding potential energy is crucial for predicting how energy will transform under different circumstances.

Kinetic Energy

Kinetic energy is associated with an object's motion. Any moving object has kinetic energy. It's the energy due to motion and depends on both the mass and speed of the object. Consider a moving car. The faster and heavier the car, the more kinetic energy it possesses. Key factors that influence kinetic energy include:

\( \cdot \) Speed: Kinetic energy is directly proportional to the square of the speed. Doubling an object's speed will quadruple its kinetic energy.
\( \cdot \) Mass: More massive objects have more kinetic energy if they move at the same speed as lighter objects.

The formula for kinetic energy is:\[ KE = \frac{1}{2}mv^2 \]where:

• \( KE \) is the kinetic energy
• \( m \) is the mass in kilograms
• \( v \) is the velocity in meters per second

Recognizing how kinetic energy is calculated helps in understanding various processes, like the motion of vehicles or athletes in movement.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In many physical situations, potential energy converts to kinetic energy and vice versa. This is a cornerstone of the conservation of energy principle. A classic demonstration of energy conversion is a pendulum. As it swings, its energy continually shifts between potential and kinetic forms. Here are some important points about energy conversion:

\( \cdot \) Conservation Principle: In an isolated system, total energy remains constant. Energy only changes forms, which means no energy is created or destroyed.
\( \cdot \) Efficiency: In a perfect world, all potential energy would convert to kinetic energy. However, in reality, factors like friction can transform some energy into heat, making conversions less than 100% efficient.
\( \cdot \) Practical Examples: This concept is seen in roller coasters, where potential energy at the highest points converts into kinetic energy during descents, providing thrilling speeds.

Understanding energy conversion allows for predicting how energy will flow in systems, aiding in the design of machines and assessing potential impacts of energy usage.