Mechanical energy

Key takeaways

Mechanical energy belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mechanical energy to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mechanical energy from memory before moving on to harder problems.

Reference excerpt

In physical science, mechanical energy is the sum of macroscopic potential and kinetic energies. The principle of conservation of mechanical energy states that if an isolated system or a closed system is subject only to conservative forces, then the mechanical energy is constant. If an object moves in the opposite direction of a conservative net force, the potential energy will increase; and if the speed (not the velocity) of the object changes, the kinetic energy of the object also changes. In all real systems, however, nonconservative forces, such as frictional forces, will be present, but if they are of negligible magnitude, the mechanical energy changes little and its conservation is a useful approximation. In elastic collisions, the kinetic energy is conserved, but in inelastic collisions some mechanical energy may be converted into thermal energy. The equivalence between lost mechanical energy and an increase in temperature was discovered by James Prescott Joule. Many devices are used to convert mechanical energy to or from other forms of energy, e.g. an electric motor converts electrical energy to mechanical energy, an electric generator converts mechanical energy into electrical energy and a heat engine converts heat to mechanical energy.

General Energy is a scalar quantity, and the mechanical energy of a system is the sum of the potential energy (which is measured by the position of the parts of the system) and the kinetic energy (which is also called the energy of motion):

E mechanical = U + K {\displaystyle E_{\text{mechanical}}=U+K}

The potential energy, U, depends on the position of an object subjected to gravity or some other conservative force. The gravitational potential energy of an object is equal to the weight W of the object multiplied by the height h of the object's center of gravity relative to an arbitrary datum:

U = W h {\displaystyle U=Wh}

Potential energy is the energy stored in an object due to its position relative to a conservative force field, such as gravity or a spring. It increases when work is done against the force—meaning when the object is moved in the direction opposite to that of the force. If F represents the conservative force and x the position, the potential energy of the force between the two positions x1 and x2 is defined as the negative integral of F from x1 to x2:

U = − ∫ x 1 x 2 F → ⋅ d x → {\displaystyle U=-\int _{x_{1}}^{x_{2}}{\vec {F}}\cdot d{\vec {x}}}

The kinetic energy, K, depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them. It is defined as one half the product of the object's mass with the square of its speed, and the total kinetic energy of a system of objects is the sum of the kinetic energies of the respective objects:

K = 1 2 m v 2 {\displaystyle K={1 \over 2}mv^{2}}

The principle of conservation of mechanical energy states that if a body or system is subjected only to conservative forces, the mechanical energy of that body or system remains constant. The difference between a conservative and a non-conservative force is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path. On the contrary, when a non-conservative force acts upon an object, the work done by the non-conservative force is dependent of the path.

Conservation of mechanical energy

According to the principle of conservation of mechanical energy, the mechanical energy of an isolated system remains constant in time, as long as the system is free of friction and other non-conservative forces. In any real situation, frictional forces and other non-conservative forces are present, but in many cases their effects on the system are so small that the principle of conservation of mechanical energy can be used as a fair approximation. Though energy cannot be created or destroyed, it can be converted to another form of energy.

Swinging pendulum

In a mechanical system like a swinging pendulum subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible, energy passes back and forth between kinetic and potential energy but never leaves the system. The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points. However, when taking the frictional forces into account, the system loses mechanical energy with each swing because of the negative work done on the pendulum by these non-conservative forces.

Irreversibilities

… excerpt ends here. Continue reading the full article.

Illustrations

Mechanical energy: An example of a mechanical system: The only force acting on a satellite orbiting the Earth is its own weight; its mechanical energy is therefore conserved. The satellite's acceleration is represented by the green vector and its velocity is represented by the red vector. The potential energy of the satellite, and its kinetic energy, may vary with time but their sum remains constant. — An example of a mechanical system: The only force acting on a satellite orbiting the Earth is its own weight; its mechanical energy is therefore conserved. The satellite's acceleration is represented by the green vector and its velocity is represented by the red vector. The potential energy of the satellite, and its kinetic energy, may vary with time but their sum remains constant.

Mechanical energy: A swinging pendulum with the velocity vector (green) and acceleration vector (blue). The magnitude of the velocity vector, the speed, of the pendulum is greatest in the vertical position and the pendulum is farthest from Earth in its extreme positions. — A swinging pendulum with the velocity vector (green) and acceleration vector (blue). The magnitude of the velocity vector, the speed, of the pendulum is greatest in the vertical position and the pendulum is farthest from Earth in its extreme positions.

Worked examples

Example 1 — a first encounter with Mechanical energy

Start with the simplest possible case. Write down what Mechanical energy claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mechanical energy before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mechanical energy ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mechanical energy

In research: Mechanical energy appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mechanical energy in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mechanical energy is common in secondary-school and first-year university syllabi. It links to neighbouring topics Energy (physics), Mechanical quantities, so understanding it makes those chapters shorter.
In everyday life: Look for Mechanical energy outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Mechanical energy in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mechanical energy means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mechanical energy out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mechanical energy in simple terms?

In physical science, mechanical energy is the sum of macroscopic potential and kinetic energies. The principle of conservation of mechanical energy states that if an isolated system or a closed system is subject only to conservative forces, then the mechanical energy is constant.

Why does Mechanical energy matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mechanical energy?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mechanical energy.