The law of conservation of energy states that energy cannot be created or destroyed; it can only change forms. This concept is foundational in physics and explains the energy transformations observed in various systems.

When the ball is dropped, its gravitational potential energy is converted into kinetic energy as it accelerates towards the ground. This basic transformation demonstrates the principle of energy conversion due to gravity.

Gravitational potential energy is the energy stored in an object because of its position relative to the ground. The higher the object, the more gravitational potential energy it possesses.

Kinetic energy quantifies the energy an object possesses due to its motion, which is directly related to its speed and mass. This concept is central to problems involving moving objects.

A frictionless pendulum converts energy between gravitational potential and kinetic forms without any loss. Thus, the total mechanical energy (the sum of potential and kinetic energy) remains constant throughout its motion.

At the top of the hill, the roller coaster car has maximum gravitational potential energy, which is then converted into kinetic energy as it descends. Understanding this transformation is key to analyzing motion in roller coaster dynamics.

In real-world scenarios, forces such as air resistance and friction act on a swinging pendulum, converting some of its mechanical energy into sound and heat. This gradual energy loss causes the pendulum to eventually come to rest.

A car engine does not utilize all the energy from fuel combustion efficiently because friction and imperfect combustion processes dissipate energy as heat. This energy loss is why engines are not 100% efficient in converting fuel energy into mechanical work.

Gravitational potential energy is directly proportional to the height of an object above the ground. As the height increases, the potential energy increases in a linear fashion, described by the equation U = mgh.

In an inelastic collision, some of the mechanical energy is not conserved in its original form but is transformed into other forms like sound, heat, or deformation energy. Despite this, the overall energy is still conserved when all forms are considered.

In a hydroelectric dam, water stored at an elevated height has gravitational potential energy which is converted into kinetic energy as it flows downward. This kinetic energy is then transformed into electrical energy using turbines, demonstrating multiple stages of energy conversion.

The compressed spring stores elastic potential energy. When the spring is released, this energy is converted into kinetic energy that propels the toy car forward.

Friction is a force that converts some of the mechanical energy into thermal energy, reducing the total mechanical energy available in the system. This conversion is why moving objects eventually slow down if no additional energy is supplied.

As the ball bounces, it continuously changes energy forms: kinetic energy is converted into gravitational potential energy as it rises, and the potential energy is then converted back into kinetic energy as it falls. This cycle illustrates the conservation of mechanical energy, even though some energy may be lost to non-conservative forces.

Non-conservative forces such as friction and air resistance convert a portion of mechanical energy into other energy forms like thermal energy. This conversion must be taken into account when solving energy conservation problems, as it affects the total mechanical energy of the system.

In an isolated system, no energy enters or leaves, and the total energy remains constant. This constancy enables predictions about the system's behavior by equating the energy before and after processes, even when multiple energy transformations occur.

To determine the block's speed on the rough surface, you first use conservation of energy to find the energy available from the drop. Then, subtract the work done against friction along the horizontal surface to find the remaining kinetic energy, from which the speed can be calculated.

Air resistance is a non-conservative force that slowly converts some of the pendulum's mechanical energy into thermal energy. This energy loss is observable as a gradual decrease in the swing amplitude until the motion eventually ceases.

Cyclic processes involve exchanges of energy through both work and heat. To accurately account for the energy transformation in these processes, both components must be considered as dictated by the first law of thermodynamics.

The first law of thermodynamics is a formal statement of energy conservation for thermodynamic systems. It relates the change in internal energy to the net heat and work interactions, ensuring that all energy exchanges within the closed system are properly accounted for.