Chapter 7: Work and Energy – Physics Study Notes

Work and Energy

Introduction to Work and Energy

Work and energy are fundamental concepts in physics that describe how forces cause changes in motion and how energy is transferred or transformed within systems. This chapter covers the work-energy theorem, kinetic energy, work done by constant and variable forces, and the distinction between conservative and non-conservative forces.

• Work-Energy Theorem: Relates the net work done on an object to its change in kinetic energy.

• Kinetic Energy: The energy associated with the motion of an object.

• Work: The process of energy transfer via force acting over a distance.

• Conservative vs. Non-Conservative Forces: Determines whether energy is conserved or dissipated in a system.

Different Forms of Energy

Energy exists in various forms and can be transformed from one type to another. The photograph of wind turbines illustrates several forms of energy:

• Kinetic Energy: The motion of the wind and rotating blades.

• Mechanical Energy: The sum of kinetic and potential energy in the moving parts.

• Electrical Energy: Generated by the turbines from mechanical motion.

• Solar Energy: The sunlight visible in the image.

Example: Wind turbines convert the kinetic energy of wind into electrical energy, which can be used to power homes and industries.

Work

Scientific Definition of Work

In physics, work is defined as the product of the component of force in the direction of displacement and the magnitude of the displacement.

• Formula: $W = F \cdot d \cdot \cos\theta$

• F: Magnitude of the force

• d: Displacement of the object

• $\theta$: Angle between the force and displacement vectors

Work is positive when the force acts in the direction of displacement, negative when opposite, and zero when perpendicular.

Work Done by a Constant Force

When a constant force acts on an object, the work done is straightforward to calculate:

• Formula: $W = F \cdot d$ (if force and displacement are in the same direction)

• Example: Pushing a crate up a ramp with a constant force.

Work Done by Multiple Forces

When several forces act on an object, the total work done is the sum of the work done by each force.

• Formula: $W_{\text{total}} = W_1 + W_2 + ... + W_n$

• Example: Moving a box up an incline with applied force, gravity, and friction.

Kinetic Energy

Definition and Formula

Kinetic energy is the energy an object possesses due to its motion.

• Formula: $K = \frac{1}{2}mv^2$

• m: Mass of the object

• v: Speed of the object

Kinetic energy is a scalar quantity and always non-negative.

Work-Energy Theorem

The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy.

• Formula: $W_{\text{net}} = K_f - K_i$

• $K_f$: Final kinetic energy

• $K_i$: Initial kinetic energy

Example: If a cart starts at rest and a force accelerates it, the work done by the force equals the increase in the cart's kinetic energy.

Potential Energy

Gravitational Potential Energy

Potential energy is the energy stored due to an object's position or configuration. Gravitational potential energy depends on height above a reference point.

• Formula: $U_g = mgy$

• m: Mass of the object

• g: Acceleration due to gravity

• y: Height above reference level

The change in gravitational potential energy is $\Delta U_g = mg(y_f - y_i)$.

Spring Potential Energy

Energy stored in a stretched or compressed spring is called spring potential energy.

• Formula: $U_{\text{spring}} = \frac{1}{2}kx^2$

• k: Spring constant (N/m)

• x: Displacement from equilibrium (positive for stretch, negative for compression)

Example: Stretching a spring by 0.47 m with $k = 860$ N/m stores $U_{\text{spring}} = \frac{1}{2} \times 860 \times (0.47)^2$ J of energy.

Conservative and Non-Conservative Forces

Conservative Forces

A conservative force is one for which the work done depends only on the initial and final positions, not the path taken. Energy lost to conservative forces can be fully recovered.

• Examples: Gravity, spring force

• Potential Energy: Can be defined for conservative forces

Non-Conservative Forces

Non-conservative forces depend on the path taken and dissipate energy, usually as heat or sound.

• Examples: Friction, air resistance

• Energy: Cannot be fully recovered; mechanical energy is not conserved

Total Mechanical Energy

Definition and Conservation

Total mechanical energy is the sum of kinetic and potential energy in a system.

• Formula: $E = K + U$

• Conservation: If only conservative forces act, $E$ remains constant.

• Non-Conservative Work: $E_f = E_i + W_{\text{non-conservative}}$

Example: A block launched by a spring up a ramp converts spring potential energy into kinetic and gravitational potential energy.

Power

Definition and Formula

Power is the rate at which work is done or energy is transferred.

• Formula: $P = \frac{W}{t}$

• Instantaneous Power: $P = Fv \cos\theta$

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

Example: Calculating the power required for a person to run up stairs or for an elevator to lift passengers.

Summary Table: Conservative vs. Non-Conservative Forces

Type of Force	Path Dependence	Energy Recovery	Examples
Conservative	No (depends only on endpoints)	Fully recoverable	Gravity, spring force
Non-Conservative	Yes (depends on path)	Not fully recoverable	Friction, air resistance

Key Equations

• $W = Fd \cos\theta$

• $K = \frac{1}{2}mv^2$

• $U_g = mgy$

• $U_{\text{spring}} = \frac{1}{2}kx^2$

• $W_{\text{net}} = K_f - K_i$

• $E = K + U$

• $P = \frac{W}{t}$

Additional info: These notes expand on the provided slides and images, filling in academic context and definitions for clarity and completeness.