🚀 IB Physics Study Notes: Work, Energy, and Power (A.3)

Welcome to Chapter A.3! This chapter is fundamental to understanding how forces and motion relate in the real world. We move beyond just *how* things move (kinematics) and start asking *why* they move and *how much effort* is required to change their motion. This is the heart of the "Space, time and motion" section!

Don't worry if the math looks intimidating; the concepts are based on simple everyday actions like pushing a heavy box or climbing stairs. Let's break it down!

1. Work (W)

1.1 Definition and Conceptual Understanding

In physics, the word Work has a very specific meaning. It is not just effort; it requires movement!

  • Work is done when a force causes a displacement of an object.
  • It measures the energy transferred to or from an object by applying a force.

Key Term: Work is a scalar quantity, meaning it only has magnitude (no direction), even though force and displacement are vectors.

Unit: The standard unit for work is the Joule (J). One Joule is the amount of work done when a force of one Newton moves an object a distance of one meter: \(1 \text{ J} = 1 \text{ N} \cdot 1 \text{ m}\).

1.2 The Work Formula and the Scalar Product

The crucial detail about work is that only the component of the force parallel to the displacement actually contributes to the work done.

The General Formula

$$W = F s \cos \theta$$

Where:

  • \(F\) is the magnitude of the applied force.
  • \(s\) is the magnitude of the displacement (distance moved).
  • \(\theta\) is the angle between the force vector (\(\vec{F}\)) and the displacement vector (\(\vec{s}\)).

💭 The Scalar (Dot) Product: In advanced vector notation (especially useful for HL students), work is defined as the dot product of force and displacement:

$$W = \vec{F} \cdot \vec{s}$$

This notation is a concise way of saying "multiply the parts of F and s that point in the same direction." This ensures that work remains a scalar.

Example/Memory Aid: Imagine pulling a heavy suitcase on wheels. You pull the handle upwards at an angle (\(\theta\)) but the suitcase moves horizontally (\(s\)). Only the horizontal component of your pulling force, \(F \cos \theta\), is actually doing work to move the suitcase forward.

1.3 Types of Work

The angle \(\theta\) determines if the work is positive, negative, or zero:

1. Positive Work (\(0^\circ \le \theta < 90^\circ\)):

  • The force is in the same direction as the displacement.
  • The object speeds up (gains energy).
  • Example: The force applied by an engine accelerating a car.

2. Zero Work (\(\theta = 90^\circ\)):

  • The force is perpendicular to the displacement.
  • No energy is transferred along the direction of motion.
  • Common Mistake to Avoid: A person carrying a box horizontally across a room does zero work on the box (because the lifting force is vertical, while the displacement is horizontal).

3. Negative Work (\(90^\circ < \theta \le 180^\circ\)):

  • The force opposes the displacement.
  • The object slows down (loses energy).
  • Example: The force of friction acting on a sliding object, or the force of gravity acting on a ball thrown upwards.
✅ Quick Review: Work

Work requires a force AND displacement, and they must have parallel components. \(W = F s \cos \theta\).

2. Energy (E)

2.1 Definition and the Conservation Principle

Energy is fundamentally defined as the ability to do work. Energy is also a scalar quantity, measured in Joules (J).

The Law of Conservation of Energy: This is perhaps the most important law in all of physics. It states that energy cannot be created or destroyed; it can only be transformed from one form into another or transferred from one system to another.

2.2 Kinetic Energy (Energy of Motion)

Kinetic Energy (\(E_k\)) is the energy possessed by an object due to its motion. If an object is moving, it has the ability to do work by colliding with something else.

Formula for Kinetic Energy

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

Where:

  • \(m\) is the mass (kg).
  • \(v\) is the speed (m s\(^{-1}\)).

Note that because speed (\(v\)) is squared, doubling the speed quadruples the kinetic energy!

2.3 Gravitational Potential Energy (GPE)

Gravitational Potential Energy (\(E_p\)) is the energy stored in an object due to its position within a gravitational field. It is the work done against gravity to lift the object to that height.

Formula for GPE (Near Earth's Surface)

$$E_p = m g h$$

Where:

  • \(m\) is the mass (kg).
  • \(g\) is the acceleration due to free fall (9.81 m s\(^{-2}\)).
  • \(h\) is the vertical height above a defined zero reference level (m).

Important Note: GPE is relative! You must always choose a reference level (often the ground) where you define \(E_p = 0\). The absolute value of \(E_p\) changes depending on this choice, but the *change* in GPE (\(\Delta E_p\)) remains the same.

2.4 Elastic Potential Energy (EPE)

Elastic Potential Energy (\(E_{PE}\)) is the energy stored in an object that is stretched or compressed, such as a spring or a rubber band.

This concept requires knowledge of Hooke's Law, which states that the force exerted by an ideal spring is proportional to its extension/compression: \(F = k x\).

Formula for Elastic Potential Energy (HL/SL Required)

$$E_{PE} = \frac{1}{2} k x^2$$

Where:

  • \(k\) is the spring constant (N m\(^{-1}\))—a measure of the stiffness of the spring.
  • \(x\) is the extension or compression (m) from the equilibrium position.
🏹 Did You Know? Conservative Forces

Gravity and the spring force are conservative forces. This means the work done by these forces on an object depends only on the starting and ending points, not the path taken. Friction is a non-conservative force; the work done by friction depends entirely on the path.

3. The Work-Energy Theorem

The Work-Energy Theorem connects the concepts of Work and Kinetic Energy, giving us a powerful tool for solving motion problems.

3.1 What the Theorem States

The net work done (\(W_{net}\)) on an object by all forces is equal to the change in the object's kinetic energy (\(\Delta E_k\)).

$$W_{net} = \Delta E_k = E_{k, \text{final}} - E_{k, \text{initial}}$$

or

$$W_{net} = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2$$

3.2 Applying the Theorem: A Step-by-Step Approach

If you push a stationary car, the work you do directly contributes to its motion (Kinetic Energy).

  1. Identify All Forces: List all forces acting on the system (e.g., applied force, friction, gravity, normal force).
  2. Calculate Net Work: Find the work done by each force, remembering that friction does negative work and forces perpendicular to motion do zero work. Sum them up to find \(W_{net}\).
  3. Relate to Change in Speed: Set \(W_{net}\) equal to the final kinetic energy minus the initial kinetic energy.

If \(W_{net}\) is positive, the object speeds up. If \(W_{net}\) is negative, the object slows down.

4. Conservation of Mechanical Energy

In many ideal situations (where we ignore air resistance and friction), the sum of potential and kinetic energy remains constant. This is known as the conservation of Mechanical Energy.

$$E_{\text{Mechanical}} = E_k + E_p = \text{Constant}$$

4.1 The Conservation Equation

In an isolated system where only conservative forces (like gravity and spring force) are acting, the total initial mechanical energy equals the total final mechanical energy.

$$E_{k, \text{initial}} + E_{p, \text{initial}} = E_{k, \text{final}} + E_{p, \text{final}}$$

Analogy: Think of a roller coaster. At the top of the first hill (maximum height), all the energy is GPE (\(E_k = 0\)). As it drops, GPE converts into KE. At the bottom (maximum speed), nearly all the energy is KE (\(E_p = 0\)). The total amount of energy (GPE + KE) stays the same throughout the ride.

4.2 Dealing with Non-Conservative Forces (Friction)

In the real world, non-conservative forces like friction or drag always reduce mechanical energy, converting it into unusable thermal energy (heat).

When non-conservative work ($W_{nc}$) is done, the conservation equation must include this energy loss:

$$E_{k, \text{initial}} + E_{p, \text{initial}} + W_{\text{other}} = E_{k, \text{final}} + E_{p, \text{final}}$$

If $W_{\text{other}}$ is negative (like friction), the final mechanical energy is less than the initial energy.

✅ Key Takeaway: Energy Conservation

Total energy is always conserved. Mechanical energy is conserved ONLY when friction and air resistance are negligible.

5. Power (P) and Efficiency

Work and energy tell us *how much* is done. Power tells us *how quickly* it is done.

5.1 Definition of Power

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

Unit: The standard unit for power is the Watt (W). One Watt equals one Joule of work done per second: \(1 \text{ W} = 1 \text{ J s}^{-1}\).

The Formula for Power

$$P = \frac{W}{t}$$

Where:

  • \(W\) is the work done (J).
  • \(t\) is the time taken (s).

Analogy: Two people might do the same amount of work lifting boxes onto a truck. The person who finishes faster has generated more power.

5.2 Power as Force times Velocity

A very useful formula links power, force, and instantaneous velocity. We know \(W = F s \cos \theta\) and \(P = W/t\).

If the force is applied parallel to the motion (\(\cos \theta = 1\)):

$$P = \frac{F s}{t}$$

Since \(v = s/t\), we get the instantaneous power formula:

$$P = F v$$

This formula is vital for analyzing things like cars or motors operating at constant velocity against resisting forces. To maintain high speed (\(v\)), a powerful engine must be able to apply a large force (\(F\)).

5.3 Efficiency

No machine is 100% efficient. Efficiency is a measure of how much of the energy input is converted into useful work output, rather than wasted (usually as heat or sound).

Formula for Efficiency (\(\eta\))

$$\eta = \frac{\text{Useful Power Output}}{\text{Total Power Input}} \times 100\%$$

OR

$$\eta = \frac{\text{Useful Energy Output}}{\text{Total Energy Input}} \times 100\%$$

Step-by-step calculation:

  1. Identify the total energy (or power) supplied to the system (e.g., fuel burned).
  2. Identify the useful work (or power) the system was designed to do (e.g., lifting a mass).
  3. Divide output by input. The result is always less than 1 (or less than 100%).

Congratulations! You have covered the core concepts of Work, Energy, and Power. Remember, these concepts are deeply interconnected: work is the mechanism for energy transfer, and power is simply how quickly that transfer happens!

Keep practicing those conservation problems—they are the key to mastering this chapter!