Question

Bigldea What is the necessary condition for the mechanical energy of a system to be conserved?

Short answer

Mechanical energy is conserved when no non-conservative forces do work on the system.

Step-by-step solution

Understand Mechanical Energy

Mechanical energy in a system is the sum of its kinetic energy and potential energy. It can be represented as \( E = K + U \), where \( K \) is the kinetic energy and \( U \) is the potential energy. For mechanical energy to be conserved, there must not be any net work done by external forces.

Define Conservation of Mechanical Energy

When mechanical energy is conserved, the total mechanical energy (the sum of kinetic and potential energies) remains constant in an isolated system. This implies \( E_{initial} = E_{final} \).

Identify Necessary Condition

The necessary condition for the conservation of mechanical energy is that there are no non-conservative forces (such as friction or air resistance) doing work on the system. This ensures that all forces are conservative, such as gravity or spring force, which do not change the total mechanical energy of the system.

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. Whenever something moves, it gains kinetic energy. The faster an object moves, the more kinetic energy it has. Mathematically, kinetic energy can be expressed using the formula: \[ K = \frac{1}{2}mv^2 \] where \( m \) is the mass of the object and \( v \) is its velocity. Here, we see that both mass and velocity play crucial roles in determining how much kinetic energy an object will have.

• A larger mass increases kinetic energy, assuming the velocity stays the same.
• A higher velocity increases kinetic energy even more significantly due to the velocity being squared in the equation.

Kinetic energy is always a positive value because mass and the square of velocity are always positive. It's important to remember that when discussing the conservation of mechanical energy, the change in kinetic energy is linked to changes in potential energy within a system.

Potential Energy

Potential energy is the stored energy in an object due to its position or arrangement. It's like energy waiting to be used. One of the most common forms of potential energy is gravitational potential energy, which depends on an object's height above the ground. The formula for gravitational potential energy is:\[ U = mgh \] where \( m \) is the mass, \( g \) is the gravity constant (approximately \( 9.81\, m/s^2 \)), and \( h \) is the height. This formula shows that an object will have more potential energy if it is higher off the ground. Another type of potential energy happens when an object is stretched or compressed, known as elastic potential energy. A good example is a spring, which stores energy when stretched or compressed. Potential energy is crucial in conservation of mechanical energy because any decrease in potential energy within a system could lead to an increase in kinetic energy, keeping the total energy constant.

Non-Conservative Forces

Non-conservative forces are forces that cause energy to be lost from a system, usually as heat or other forms of energy that are not useful for mechanical purposes. These forces are considered non-conservative because they depend on the path taken. Common examples include:

• Friction: Causes energy loss as heat due to the rubbing between surfaces.
• Air resistance: Opposes motion through air, also causing energy loss.

Non-conservative forces can cause the total mechanical energy in a system to change. This means if these forces are present, the mechanical energy isn't conserved because the system does work against these forces, losing energy in the process. For instance, when you slide a book across the table and it comes to a stop, the kinetic energy initially in the book is converted into thermal energy due to friction.

Conservative Forces

Conservative forces are forces where the work done is independent of the path taken and depend only on the initial and final positions of an object. Excellent examples include gravitational force and spring force. These forces do not change the total mechanical energy of a system. In simple terms:

• They can convert kinetic energy into potential energy and vice versa without loss.
• The total mechanical energy in a system influenced only by conservative forces remains constant.

In the absence of non-conservative forces, systems where only conservative forces are acting can conserve mechanical energy. If these forces do work, the energy they utilize is stored within the system itself. For example, when you lift a book onto a shelf, you're using kinetic energy, which gets stored as potential energy due to gravitational forces. Hence, the total energy within the system doesn't change, demonstrating this concept quite well.