Chapter 10: Energy and Work – Physics with Algebra Study Notes

Energy and Work

Introduction to Energy

Energy is a fundamental concept in physics, representing the capacity to do work or produce change. In mechanical systems, energy can be transferred and transformed between various forms, and its conservation is a key principle in problem-solving.

• Energy is the sum of all forms present in a system.

• Common forms include kinetic energy (motion), potential energy (position), thermal energy (internal), chemical energy (bonds), and nuclear energy (atomic nuclei).

• Energy can be transferred between a system and its environment via work and heat.

Forms of Energy

Mechanical motion involves several important forms of energy:

• Kinetic Energy (K): Energy of motion. $K = \frac{1}{2}mv^2$

• Gravitational Potential Energy (U_g): Energy due to height above ground. $U_g = mgy$

• Elastic Potential Energy (U_s): Energy stored in stretched/compressed springs. $U_s = \frac{1}{2}kx^2$

• Thermal Energy (E_{th}): Sum of kinetic and potential energies of molecules.

• Chemical Energy (E_{chem}): Energy stored in molecular bonds.

• Nuclear Energy (E_{nuclear}): Energy stored in atomic nuclei.

Energy Transformations

Energy can be transformed from one form to another within a system. For example:

• Chemical energy in muscles is converted to kinetic and gravitational potential energy when lifting weights.

• Elastic potential energy in a springboard is converted to kinetic energy, then to gravitational potential energy as a diver rises.

Energy Transfers: Work and Heat

Energy can be transferred between a system and its environment:

• Work: Mechanical transfer of energy by force acting over a distance.

• Heat: Nonmechanical transfer due to temperature difference.

Work-Energy Equation

The total energy of a system changes by the amount of work done on it:

• $\Delta E = W$

• $\Delta E = \Delta K + \Delta U_g + \Delta U_s + \Delta E_{th} + \Delta E_{chem} + \cdots$

• If no energy is transferred, the system is isolated.

Work

Definition and Calculation of Work

Work is done when a force causes displacement. Only the component of force in the direction of displacement does work.

• Work by a constant force: $W = F_{\parallel}d$

• Work is a scalar quantity, measured in joules (J): $1\,\text{J} = 1\,\text{N}\cdot\text{m}$

• If force is at an angle $\theta$ to displacement: $W = Fd\cos\theta$

Examples of Work

• Positive work: Force and displacement in same direction.

• Negative work: Force and displacement in opposite directions.

• Zero work: Force perpendicular to displacement or no displacement.

Example: Work Done by a Dog Pulling a Sled

A dog pulls a sled 4.9 m with a force of 350 N at constant speed. The work done is:

• $W = Fd = 350\,\text{N} \times 4.9\,\text{m} = 1700\,\text{J}$

• Energy goes into increasing thermal energy due to friction.

Work at an Angle

When force is applied at an angle, only the parallel component does work:

• $W = Fd\cos\theta$

• Sign of work depends on angle between force and displacement.

Forces That Do No Work

• No displacement: No work done.

• Force perpendicular to displacement: No work done.

• Internal energy transformations (e.g., pushing against a wall): No work done by external force.

Kinetic Energy

Translational and Rotational Kinetic Energy

Kinetic energy is the energy of motion. It can be translational (linear) or rotational.

• Translational: $K = \frac{1}{2}mv^2$

• Rotational: $K_{rot} = \frac{1}{2}I\omega^2$

• Work done increases kinetic energy: $W = \Delta K$

Potential Energy

Gravitational Potential Energy

Potential energy is stored energy due to position. Gravitational potential energy depends on height:

• $U_g = mgy$

• Gravity is a conservative force; potential energy depends only on height, not path.

Elastic Potential Energy

Elastic potential energy is stored in springs or elastic objects:

• Hooke's law: $F = kx$

• Elastic potential energy: $U_s = \frac{1}{2}kx^2$

Conservation of Energy

Law of Conservation of Energy

The total energy of an isolated system remains constant, though energy can be transformed between forms.

• $K_i + U_i = K_f + U_f$ (if no energy is transferred to/from environment)

• Energy conservation is used to solve motion problems.

Energy Diagrams

Interpreting Energy Diagrams

Energy diagrams graph potential energy as a function of position. The distance to the energy line represents kinetic energy.

• Turning points occur where the energy line crosses the potential energy curve.

• Maximum speed occurs where potential energy is minimum.

Energy in Collisions

Elastic and Inelastic Collisions

Collisions involve energy and momentum transfer:

• Perfectly elastic collision: Mechanical energy and momentum are conserved.

• Perfectly inelastic collision: Objects stick together; mechanical energy is not conserved, but momentum is.

• In inelastic collisions, some mechanical energy is converted to thermal energy.

Power

Definition and Calculation of Power

Power is the rate at which energy is transferred or transformed.

• $P = \frac{\Delta E}{\Delta t}$

• $P = \frac{W}{\Delta t}$

• Unit: watt (W), where $1\,\text{W} = 1\,\text{J}/\text{s}$

• For a force acting on a moving object: $P = Fv$

Example: Output Power for a Weightlifter

• Weightlifter raises a 190 kg bar 1.9 m in 1.8 s.

• Change in potential energy: $\Delta U_g = mg\Delta y = 190\,\text{kg} \times 9.8\,\text{m/s}^2 \times 1.9\,\text{m} = 3540\,\text{J}$

• Power: $P = \frac{3540\,\text{J}}{1.8\,\text{s}} \approx 1970\,\text{W}$

Summary Table: Forms of Energy

Form of Energy	Definition	Formula
Kinetic Energy	Energy of motion	$K = \frac{1}{2}mv^2$
Gravitational Potential Energy	Energy due to height	$U_g = mgy$
Elastic Potential Energy	Energy stored in spring	$U_s = \frac{1}{2}kx^2$
Thermal Energy	Internal energy of molecules	N/A
Chemical Energy	Energy in molecular bonds	N/A
Nuclear Energy	Energy in atomic nuclei	N/A