Work, Energy, Power and Energy Conservation for SS 1

Work, Energy, Power and Energy Conservation

Performance Objectives

By the end of the lesson, students should be able to:

1. explain the meaning of work, energy, and power, and give example of each;
2. identify the type of energy possessed by a body under given conditions;
3. distinguish between kinetic energy and potential energy;
4. identify energy transformation from one form into another;
5. state the laws of conservation of energy;
6. calculate:
   1. the work done, given a force and displacement it produces in its direction;
   2. the gravitational potential energy at a height h above a given reference plane;
   3. the power in watts, given an applied force and the time it takes to produce a displacement;

Introduction to Work, Energy, and Power

In physics, the terms work, energy, and power have specific meanings that are different from their everyday usage. These concepts are fundamental to understanding how forces cause motion and how systems change. They help to describe interactions between objects and the environment, from simple tasks like lifting a bag to complex processes like power generation.

Understanding Work

Work is done when a force causes a displacement of an object in the direction of the force.

For work to be done, two conditions must be met:

1. A force must be applied to an object.
2. The object must move a certain distance (be displaced) in the direction of the applied force, or at least have a component of its displacement in that direction.

If a force is applied but the object does not move, no work is done. For example, pushing against a solid wall does not result in work being done, even if a lot of effort is exerted, because the wall does not move.

Calculating Work Done

When a constant force acts on an object and causes a displacement in the same direction as the force, the work done (W) is calculated using the formula:

W = F × d

Where:

1. W is the work done (measured in Joules, J).
2. F is the magnitude of the force applied (measured in Newtons, N).
3. d is the magnitude of the displacement (measured in metres, m).

The unit of work is the Joule (J). One Joule is the work done when a force of one Newton moves an object by one metre in the direction of the force (1 J = 1 N·m).

Work Done When Force is at an Angle

Sometimes, the force applied to an object is not exactly in the same direction as its displacement. For example, when pulling a box with a rope, the rope might be held at an angle to the ground. In such cases, only the component of the force that acts in the direction of the displacement does work.

If a force F acts at an angle θ to the direction of displacement d, the work done is given by:

W = Fd cosθ

Here, F cosθ is the component of the force acting in the direction of the displacement.

Examples of work done include:

1. A person pushing a wheelbarrow across a field.
2. A crane lifting a heavy load upwards.
3. A child pulling a toy car along the floor.

Understanding Energy

Energy is the ability to do work.

Any object or system that can do work possesses energy. Energy exists in many forms, such as mechanical energy, heat energy, chemical energy, electrical energy, and nuclear energy. In this unit, the focus is on mechanical energy.

The unit of energy is also the Joule (J), just like work, because energy is the capacity to do work.

Examples of objects or systems possessing energy include:

1. Fuel in a car engine (chemical energy).
2. Food consumed by a person (chemical energy).
3. Water stored at a height in a dam (potential energy).
4. A moving vehicle (kinetic energy).

Interchangeability of Work and Energy

Work and energy are closely related and are often interchangeable. When work is done on an object, its energy changes. Conversely, an object must possess energy to do work.

For instance, when a person lifts a heavy bag, work is done on the bag against gravity. This work increases the bag’s potential energy. If the bag is then allowed to fall, its potential energy is converted into kinetic energy, and it can do work on another object upon impact.

Understanding Power

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

Power tells us how quickly work is performed. A powerful machine can do a large amount of work in a short time, or it can do work at a faster rate than a less powerful machine.

Calculating Power

Power (P) is calculated using the formula:

P = W / t

Where:

1. P is the power (measured in Watts, W).
2. W is the work done (measured in Joules, J).
3. t is the time taken to do the work (measured in seconds, s).

The unit of power is the Watt (W). One Watt is defined as one Joule of work done per second (1 W = 1 J/s).

Examples of power include:

1. A powerful car engine can accelerate quickly, doing work rapidly.
2. A person running up a flight of stairs quickly demonstrates more power than a person walking up the same stairs slowly.
3. An electric motor lifting a load in a short time has higher power than one that lifts the same load over a longer time.

Types of Mechanical Energy

Mechanical energy is the energy associated with the motion and position of an object. It exists in two main forms: potential energy and kinetic energy.

Potential Energy (PE)

Potential energy is the energy stored in an object due to its position, condition, or state.

This stored energy has the potential to be converted into other forms of energy or to do work. The most common type of potential energy discussed in SS1 Physics is gravitational potential energy.

Gravitational Potential Energy (GPE)

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, specifically its height above a reference level.

The higher an object is lifted, the more gravitational potential energy it gains. This energy is stored as a result of the work done against gravity to lift the object.

Calculating Gravitational Potential Energy

The formula for gravitational potential energy (GPE) is:

GPE = mgh

Where:

1. GPE is the gravitational potential energy (measured in Joules, J).
2. m is the mass of the object (measured in kilograms, kg).
3. g is the acceleration due to gravity (approximately 9.8 m/s² or 10 m/s² on Earth).
4. h is the height of the object above the reference level (measured in metres, m).

Examples of objects with gravitational potential energy include:

1. Water held behind a dam.
2. A stone held high above the ground.
3. A book placed on a high shelf.

Kinetic Energy (KE)

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

Any object that is moving has kinetic energy. The faster an object moves and the greater its mass, the more kinetic energy it possesses.

Calculating Kinetic Energy

The formula for kinetic energy (KE) is:

KE = ½mv²

Where:

1. KE is the kinetic energy (measured in Joules, J).
2. m is the mass of the object (measured in kilograms, kg).
3. v is the speed (or velocity) of the object (measured in metres per second, m/s).

Examples of objects with kinetic energy include:

1. A moving car.
2. A flying bird.
3. Flowing water in a river.
4. A thrown ball.

Distinguishing Between Kinetic Energy and Potential Energy

Kinetic energy and potential energy are both forms of mechanical energy, but they differ in their origin:

1. Kinetic Energy: This is the energy of motion. An object must be moving to possess kinetic energy. Its value depends on the object’s mass and speed.
2. Potential Energy: This is stored energy due to an object’s position or state. An object can have potential energy even when it is at rest, as long as it is in a position where it can potentially do work (e.g., at a height). Its value depends on the object’s mass, height (for GPE), and the acceleration due to gravity.

Consider a ball thrown upwards:

1. As the ball leaves the hand and moves upwards, it has both kinetic and potential energy.
2. At its highest point, just before it starts to fall, its speed is momentarily zero, so its kinetic energy is zero, but its potential energy is maximum.
3. As it falls back down, its potential energy decreases, and its kinetic energy increases.

Work Done in a Force Field (Gravitational)

When an object is moved within a force field, work is done. For the gravitational force field, work is done when an object’s height changes.

1. Work Done Against Gravity: When an object is lifted upwards, work is done against the force of gravity. The work done is equal to the gain in the object’s gravitational potential energy. If an object of mass ‘m’ is lifted through a height ‘h’, the work done (W) is W = mgh.
2. Work Done By Gravity: When an object falls downwards, the force of gravity does work on the object. The work done by gravity is equal to the loss in the object’s gravitational potential energy, which is converted into kinetic energy. If an object of mass ‘m’ falls through a height ‘h’, the work done by gravity is W = mgh.

Conservation of Mechanical Energy

The Law of Conservation of Energy states that energy cannot be created or destroyed, but it can be transformed from one form to another.

In systems where only conservative forces (like gravity) are doing work, the total mechanical energy (the sum of kinetic and potential energy) remains constant. This is known as the principle of conservation of mechanical energy.

Total Mechanical Energy = Kinetic Energy + Potential Energy = Constant

This means that as an object moves, its kinetic energy can be converted into potential energy, and potential energy can be converted into kinetic energy, but their sum remains the same, assuming no energy is lost to non-conservative forces like air resistance or friction.

Energy Transformations

Energy transformations are common in everyday life:

1. Falling Object: As an object falls from a height, its gravitational potential energy is converted into kinetic energy. At the moment it hits the ground, most of its initial potential energy has been transformed into kinetic energy.
2. Swinging Pendulum: When a pendulum swings, its energy continuously transforms. At the highest points of its swing, its speed is momentarily zero, so its kinetic energy is zero, and its potential energy is maximum. As it swings downwards, potential energy is converted into kinetic energy, reaching maximum kinetic energy (and minimum potential energy) at the lowest point of its swing. This process repeats as it swings upwards again.
3. Hydroelectric Power Plant: Water stored at a height in a dam has high potential energy. As it flows downwards, this potential energy is converted into kinetic energy, which then turns turbines to generate electrical energy.

Calculations of Work, Energy, and Power

Here are some worked examples to illustrate the calculation of work, energy, and power.

Example 1: Calculating Work Done (Horizontal Force)

A boy pushes a trolley with a force of 50 N over a distance of 15 m on a flat surface. Calculate the work done by the boy.

Solution:

1. Identify the given values:
   1. Force (F) = 50 N
   2. Displacement (d) = 15 m
2. State the formula for work done:

   W = F × d

3. Substitute the values and calculate:

   W = 50 N × 15 m

   W = 750 J

The work done by the boy is 750 J.

Example 2: Calculating Work Done (Force at an Angle)

A girl pulls a sled with a rope. The rope makes an angle of 30° with the horizontal. If the tension in the rope is 40 N and the sled is pulled a distance of 20 m, calculate the work done.

Solution:

1. Identify the given values:
   1. Force (F) = 40 N
   2. Displacement (d) = 20 m
   3. Angle (θ) = 30°
2. State the formula for work done at an angle:

   W = Fd cosθ

3. Substitute the values and calculate (cos 30° ≈ 0.866):

   W = 40 N × 20 m × cos 30°

   W = 40 N × 20 m × 0.866

   W = 692.8 J

The work done by the girl is approximately 692.8 J.

Example 3: Calculating Gravitational Potential Energy

A 5 kg bag of rice is lifted to a height of 3 m above the ground. Calculate its gravitational potential energy. (Take g = 10 m/s²)

Solution:

1. Identify the given values:
   1. Mass (m) = 5 kg
   2. Height (h) = 3 m
   3. Acceleration due to gravity (g) = 10 m/s²
2. State the formula for gravitational potential energy:

   GPE = mgh

3. Substitute the values and calculate:

   GPE = 5 kg × 10 m/s² × 3 m

   GPE = 150 J

The gravitational potential energy of the bag of rice is 150 J.

Example 4: Calculating Kinetic Energy

A car with a mass of 1200 kg is moving at a speed of 15 m/s. Calculate its kinetic energy.

Solution:

1. Identify the given values:
   1. Mass (m) = 1200 kg
   2. Speed (v) = 15 m/s
2. State the formula for kinetic energy:

   KE = ½mv²

3. Substitute the values and calculate:

   KE = ½ × 1200 kg × (15 m/s)²

   KE = 600 kg × 225 m²/s²

   KE = 135,000 J

The kinetic energy of the car is 135,000 J.

Example 5: Calculating Power

A student of mass 60 kg climbs a flight of stairs to a height of 4 m in 8 seconds. Calculate the power developed by the student. (Take g = 10 m/s²)

Solution:

1. First, calculate the work done by the student against gravity (which is equal to the gain in GPE):
   1. Mass (m) = 60 kg
   2. Height (h) = 4 m
   3. Acceleration due to gravity (g) = 10 m/s²
   4. Work done (W) = mgh = 60 kg × 10 m/s² × 4 m = 2400 J
2. Now, calculate the power developed:
   1. Work done (W) = 2400 J
   2. Time (t) = 8 s
3. State the formula for power:

   P = W / t

4. Substitute the values and calculate:

   P = 2400 J / 8 s

   P = 300 W

The power developed by the student is 300 W.