Work, Energy, and Conservation of Energy: Study Notes

Energy and Work

Definition of Energy

Energy is a fundamental concept in physics, defined as the capacity to do work. It is measured in Joules (J). Energy can exist in various forms and is essential for all physical processes.

• Work is done when a force causes displacement of an object.

• When work done by a force is positive, energy is added to the object.

• When work done by a force is negative, energy is taken away from the object.

Examples of Energy

• Kinetic energy: Energy of motion (e.g., a moving car).

• Potential energy: Energy due to position (e.g., a ball held at a height).

• Chemical energy: Stored in chemical bonds (e.g., food, batteries).

• Nuclear energy: Stored in atomic nuclei (e.g., the sun).

Conservation of Energy

Principle of Conservation

The law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another. The total energy of an isolated system remains constant over time.

• Energy transformations: Nuclear energy in the sun becomes sunlight, which is converted by plants to chemical energy, consumed as calories, and used to do work.

• Historically, the reason for energy conservation was not understood until the 20th century.

Noether's Theorem

Emmy Noether (1882-1935), a Jewish-German mathematical physicist, proved that conservation of energy is a result of time-invariance symmetry—the laws of physics do not change over time.

Mechanical Energy

Forms of Mechanical Energy

Mechanical energy is the sum of energy due to an object's position and motion. It is classified into two main types:

• Kinetic Energy (KE): Energy of an object in motion. $KE = \frac{1}{2}mv^2$

• Potential Energy (PE): Energy stored due to an object's position. Gravitational Potential Energy: $PE = mgy$ where $m$ is mass, $g$ is acceleration due to gravity, and $y$ is height above a reference point.

Conservation of Total Mechanical Energy

Energy Transformation Example

When a ball is thrown straight upward:

• On the way up, kinetic energy is converted to potential energy ($mgy$ increases).

• At the top, kinetic energy is zero, and potential energy is maximum.

• On the way down, potential energy is converted back to kinetic energy, and the ball speeds up.

Tabular Example: Dropped Ball

A 1 kg ball is dropped from a 100 m tall building. The table below shows the changes in potential and kinetic energy at different heights:

Height	PE	KE	v
100 m	980 J	0 J	0 m/s
75 m	735 J	245 J	7 m/s
50 m	490 J	490 J	10 m/s
25 m	245 J	735 J	12 m/s
0 m	0 J	980 J	14 m/s

Additional info: The values for KE and v at intermediate heights are inferred using $PE + KE = 980 \text{ J}$ and $KE = \frac{1}{2}mv^2$ for $m = 1 \text{ kg}$.

Work-Energy Theorem

Statement and Equation

The work-energy theorem states that any change in the kinetic energy of an object is the result of work done on that object by external forces.

• Equation: $W_{\text{total}} = \Delta KE$

• If a moving car comes to a stop, the force doing the work is friction (from brakes), and the total work done is negative because the force opposes the car's motion.

Example: Applying the Brakes

• When brakes are applied, friction does negative work, reducing the car's kinetic energy to zero.

• The stopping distance depends on the initial kinetic energy and the magnitude of the friction force.

Types of Forces: Conservative and Non-Conservative

Conservative Forces

Conservative forces are those for which the work done is independent of the path taken and depends only on the initial and final positions.

• Examples: Gravity, spring force

• When only conservative forces act, total mechanical energy remains constant.

• If the starting and ending points are the same, the total work done by a conservative force is zero.

Non-Conservative Forces

• Examples: Friction, drag force

• When non-conservative forces act, total mechanical energy changes (energy is transformed into other forms, such as heat).

Potential Energy

Gravitational Potential Energy

Potential energy is energy stored in a system due to its position. For gravity near Earth's surface:

• $U_{\text{grav}} = mgy$

• The work done in elevating an object is the same regardless of the path taken.

Energy Transformation Example

• If a drawn bow has 50 J of potential energy and the arrow receives 40 J of kinetic energy, the remaining 10 J is transformed into heat (warming the bow and target).

Summary Table: Conservative vs. Non-Conservative Forces

Type of Force	Effect on Mechanical Energy	Examples
Conservative	Mechanical energy remains constant	Gravity, spring force
Non-Conservative	Mechanical energy changes (usually decreases)	Friction, drag force

Key Equations

• Kinetic Energy: $KE = \frac{1}{2}mv^2$

• Gravitational Potential Energy: $PE = mgy$

• Work-Energy Theorem: $W_{\text{total}} = \Delta KE$

Applications and Problem Solving

• The work-energy theorem is a powerful tool for solving motion-related problems, such as calculating stopping distances and energy transformations.

• Doubling the speed of a car quadruples its kinetic energy, thus requiring four times the stopping distance if the same friction force is applied.