Define mechanical energy.

Mechanical energy is the sum of potential and kinetic energy in a system, which remains constant if no external forces act on it.

Mechanical energy is a fundamental concept in physics, particularly in the study of physical systems and their movements. It is a form of energy associated with the motion (kinetic energy) and position (potential energy) of an object. In other words, it is the energy that is possessed by an object due to its motion or due to its position relative to other objects.

Kinetic energy is the energy of an object in motion. It depends on both the mass and speed of the object, with the formula KE = 1/2 mv^2, where m is the mass and v is the velocity. Potential energy, on the other hand, is the energy stored in an object due to its position in a force field, such as gravity or electromagnetism, or due to its configuration. The most common types of potential energy are gravitational potential energy (GPE = mgh, where m is mass, g is the acceleration due to gravity, and h is height) and elastic potential energy (stored in a stretched or compressed spring, for example).

The principle of conservation of mechanical energy states that, in an isolated system (that is, a system not affected by external forces), the total mechanical energy remains constant. This means that energy can be transferred between potential and kinetic forms, but the total amount of energy in the system does not change. For example, when a pendulum swings, at the highest point of its swing, it has maximum potential energy and minimum kinetic energy. At the lowest point, it has maximum kinetic energy and minimum potential energy. However, the sum of the potential and kinetic energy at any point in the swing is always the same.

Understanding mechanical energy and its conservation is crucial in physics as it helps us predict how objects will move under certain conditions, and it underpins many of the most important principles and equations in the field.