Energy and Its Conservation

Energy

The concept of energy is one of the most fundamental and unifying ideas in physics. It is often described as the capacity to do work. When work is done on an object, its energy changes. Similarly, when an object does work, its energy decreases. Energy exists in various forms, and it can be transformed from one form to another, but the total amount of energy in a closed system remains constant.

Energy is a scalar quantity. The SI unit of energy is the same as that of work, the joule (J).

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Forms Of Energy

Energy manifests itself in numerous forms in the physical world. These different forms are essentially different ways in which systems can store or transfer energy. Some of the common forms of energy include:

• Mechanical Energy: The energy associated with the motion or position of an object. It is the sum of kinetic energy and potential energy.
• Kinetic Energy: Energy due to motion.
• Potential Energy: Stored energy due to position or configuration (e.g., gravitational potential energy, elastic potential energy).
• Heat Energy (Thermal Energy): Energy associated with the random motion of atoms and molecules within a substance. It is related to temperature.
• Light Energy (Radiant Energy): Energy carried by electromagnetic waves, such as visible light, infrared, ultraviolet, etc.
• Sound Energy: Energy transmitted through vibrations in a medium.
• Chemical Energy: Energy stored in the bonds of chemical compounds. It is released or absorbed during chemical reactions (e.g., energy in food, batteries, fuels).
• Electrical Energy: Energy associated with the flow of electric charge (electric current) or the presence of electric fields (electric potential).
• Nuclear Energy: Energy stored within the nucleus of an atom, released during nuclear reactions like fission or fusion.

These forms are not isolated; they can often be converted into one another.

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Kinetic Energy

Kinetic energy ($KE$) is the energy possessed by an object due to its motion. Any object that is moving has kinetic energy. The faster an object moves, the more kinetic energy it has. Also, a more massive object moving at the same speed has more kinetic energy than a less massive one.

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Formula for Kinetic Energy

The kinetic energy of an object of mass $m$ moving with a velocity $v$ is given by the formula:

$ KE = \frac{1}{2} m v^2 $

where:

• $m$ is the mass of the object (in kilograms, kg).
• $v$ is the magnitude of the velocity (speed) of the object (in metres per second, m/s).
• $KE$ is the kinetic energy (in joules, J).

Since velocity is squared, the kinetic energy is always non-negative. It depends on the *speed* ($v$), not the *velocity vector* ($\vec{v}$), making it a scalar quantity.

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Derivation of the Kinetic Energy Formula (using constant force)

Consider an object of mass $m$ starting from rest ($v_{initial} = 0$). A constant force $\vec{F}$ acts on it, causing it to accelerate uniformly with acceleration $\vec{a}$ over a displacement $\vec{d}$. Let the final velocity be $v_{final} = v$.

From Newton's Second Law, $\vec{F} = m\vec{a}$. Assuming force and displacement are in the same direction, the work done by the force is $W = Fd = (ma)d$.

From kinematics, for uniform acceleration starting from rest, the final velocity $v$ after displacement $d$ is related by the equation:

$ v^2 = v_{initial}^2 + 2ad $

Since $v_{initial} = 0$, we have:

$ v^2 = 0^2 + 2ad $

$ v^2 = 2ad $

We can express $ad$ as $\frac{v^2}{2}$.

Substitute this into the work equation:

$ W = ma d = m \left(\frac{v^2}{2}\right) = \frac{1}{2} m v^2 $

This work done by the force is stored in the object as kinetic energy, starting from zero kinetic energy at rest. Thus, the kinetic energy gained by the object is $KE = \frac{1}{2} m v^2$.

More generally, if an object's velocity changes from $v_1$ to $v_2$ due to a net force doing work $W_{net}$, the Work-Energy Theorem states $W_{net} = \Delta KE = KE_{final} - KE_{initial} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$. This holds true even for variable forces and complex motion.

Answer:

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Potential Energy

Potential energy ($PE$) is the energy stored in an object or a system due to its position or configuration. It is "potential" because it has the potential to be converted into other forms of energy, such as kinetic energy, or to do work.

There are different types of potential energy depending on the force field or interaction:

• Gravitational Potential Energy: Energy stored due to an object's position in a gravitational field.
• Elastic Potential Energy: Energy stored in an elastic material (like a spring or a stretched rubber band) when it is deformed from its equilibrium position.
• Electric Potential Energy: Energy stored due to the position of a charge in an electric field.
• Chemical Potential Energy: Energy stored in chemical bonds.

Potential energy is associated with conservative forces. A conservative force is one for which the work done in moving an object between two points is independent of the path taken. Gravity and the force exerted by an ideal spring are examples of conservative forces. Friction and air resistance are non-conservative forces, and we cannot define a potential energy associated with them.

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Potential Energy Of An Object At A Height

This section specifically discusses gravitational potential energy ($PE_g$), which is the potential energy stored in an object because of its position relative to a reference level in a gravitational field.

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Definition and Concept

When an object is lifted against the force of gravity, work is done on it. This work done is stored in the object as gravitational potential energy. When the object is allowed to fall, this stored energy is converted into kinetic energy.

The reference level is an important concept for potential energy. Gravitational potential energy is always measured relative to some chosen zero reference level (e.g., the ground, a tabletop). The choice of the zero reference level is arbitrary, as only the change in potential energy matters physically.

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Formula for Gravitational Potential Energy

The gravitational potential energy ($PE_g$) of an object of mass $m$ at a height $h$ above a chosen reference level is given by:

$ PE_g = m g h $

where:

• $m$ is the mass of the object (in kilograms, kg).
• $g$ is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$ on Earth, can vary slightly with location).
• $h$ is the vertical height of the object above the reference level (in metres, m). If the object is below the reference level, $h$ is negative, and $PE_g$ is negative.
• $PE_g$ is the gravitational potential energy (in joules, J).

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Derivation of the Gravitational Potential Energy Formula

Consider lifting an object of mass $m$ vertically upwards from a reference height (say, $h_1 = 0$) to a height $h_2 = h$. Assume we lift it slowly or at a constant velocity, so the upward force applied is approximately equal to the gravitational force, which is $F = mg$. The displacement is $\vec{d}$ upwards, with magnitude $h$. The force is upwards, and the displacement is upwards, so $\theta = 0^\circ$.

The work done by the applied force is $W = F d \cos\theta = (mg)(h)( \cos 0^\circ) = mgh$.

This work done against gravity is stored as the increase in potential energy. If we choose the potential energy at the initial height ($h_1=0$) to be $PE_1 = 0$, then the potential energy at height $h$ is $PE_2$.

Change in potential energy = Work done against gravity

$ PE_2 - PE_1 = W $

$ PE_h - 0 = mgh $

$ PE_h = mgh $

So, the gravitational potential energy of an object at height $h$ above the zero reference level is $mgh$.

Answer:

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Are Various Energy Forms Interconvertible?

Yes, absolutely! One of the most important principles in physics is that energy can be converted from one form to another. Energy transformations are happening all around us constantly.

Here are some examples of energy interconversion:

• In a hydroelectric power plant: Potential energy of water stored at a height is converted into kinetic energy as it flows down, which is then converted into mechanical energy by the turbine, and finally into electrical energy by the generator. ($PE \rightarrow KE \rightarrow \text{Mechanical Energy} \rightarrow \text{Electrical Energy}$)
• In a light bulb: Electrical energy is converted into light energy and heat energy. ($Electrical Energy \rightarrow Light Energy + Heat Energy$)
• In a car engine: Chemical energy stored in fuel (petrol or diesel) is converted into heat energy through combustion, which is then converted into mechanical energy to move the car. ($Chemical Energy \rightarrow Heat Energy \rightarrow \text{Mechanical Energy}$)
• When an object falls: Gravitational potential energy is converted into kinetic energy. As it hits the ground, this kinetic energy is converted into heat energy, sound energy, and possibly deformational energy. ($PE_g \rightarrow KE \rightarrow \text{Heat Energy} + \text{Sound Energy} + ...$)
• In a battery-powered device: Chemical energy in the battery is converted into electrical energy, which is then used to power other components (e.g., light in a torch, sound in a speaker, mechanical motion in a toy car). ($Chemical Energy \rightarrow Electrical Energy \rightarrow \text{Light/Sound/Mechanical Energy}$)
• In photosynthesis in plants: Light energy from the sun is converted into chemical energy stored in glucose. ($Light Energy \rightarrow Chemical Energy$)

These examples demonstrate the dynamic nature of energy and its ability to change forms. The total amount of energy involved in these transformations is governed by the principle of conservation of energy.

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Law Of Conservation Of Energy

The Law of Conservation of Energy is one of the most fundamental laws of physics. It states:

Energy cannot be created or destroyed; it can only be transformed from one form to another.

In simpler terms, the total amount of energy in an isolated system remains constant over time. An isolated system is one that does not exchange energy (or matter) with its surroundings.

While energy can change forms (e.g., kinetic to potential, chemical to thermal), the total energy within that system is always conserved. If energy appears to be 'lost' in a system (for instance, a bouncing ball eventually stops), it is usually because it has been converted into forms that are not easily usable or observed, such as heat and sound, which have dissipated into the environment. The total energy of the ball-Earth-surroundings system is conserved.

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Conservation of Mechanical Energy

A common application of the law of conservation of energy is in mechanics, specifically for systems where only conservative forces (like gravity and spring force) are doing work. In such cases, the total mechanical energy ($E$), which is the sum of kinetic energy ($KE$) and potential energy ($PE$), is conserved.

$ \text{Total Mechanical Energy} = KE + PE = \text{Constant} $

This means if $KE$ increases, $PE$ must decrease by the same amount, and vice versa, such that their sum remains constant.

For example, consider a ball falling freely under gravity (neglecting air resistance):

• At the highest point (maximum height), velocity is zero ($KE=0$), potential energy is maximum ($PE_{max}$). Total Energy $E = 0 + PE_{max}$.
• As it falls, height decreases, so $PE$ decreases. Velocity increases, so $KE$ increases. The decrease in $PE$ is equal to the increase in $KE$.
• At the lowest point (just before hitting the ground, assuming ground is zero PE level), height is zero ($PE=0$), velocity is maximum ($v_{max}$). Total Energy $E = KE_{max} + 0 = \frac{1}{2}mv_{max}^2$.

Throughout the fall, $KE + PE$ remains constant, equal to the initial total energy $E$.

$ KE_{initial} + PE_{initial} = KE_{final} + PE_{final} $

$ \frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2 $

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The Law in a Wider Context

The law of conservation of energy is a fundamental principle that applies to all forms of energy and all physical processes. Even when non-conservative forces like friction are present, the total energy (including thermal energy generated by friction) is conserved, although mechanical energy might not be conserved.

In the realm of nuclear physics, Einstein's famous mass-energy equivalence formula, $E = mc^2$, reveals that mass itself is a form of energy. This expanded view means that in some processes (like nuclear reactions), mass can be converted into energy (or vice-versa), but the total mass-energy of the system is conserved. This is the most general statement of the law of conservation of energy and mass combined.

The law of conservation of energy has profound implications and is a cornerstone for understanding and analysing various physical phenomena, from simple mechanical motion to complex thermodynamic systems and nuclear reactions.