Mechanics: Work, Energy, and Power
Work, Energy, and Power
What Is Work?
In Science, work is done when a force causes an object to move in the direction of the force. The formula for work is:
\[ W = F \cdot d \cdot \cos \theta \]
Where:
- \( W \) is the work done (in joules, J),
- \( F \) is the force applied (in newtons, N),
- \( d \) is the distance moved by the object (in metres, m),
- \( \theta \) is the angle between the direction of the force and the direction of motion.
Work is a scalar quantity, meaning it only has magnitude, not direction.
What Is Energy?
Energy is the ability to do work. The SI unit of energy is the joule (J), which is the same as the unit of work. There are different forms of energy, including:
- Kinetic Energy: The energy possessed by an object due to its motion.
- Potential Energy: The energy stored in an object due to its position or configuration.
- Chemical Energy: The energy stored in the bonds of chemical compounds.
- Thermal Energy: The energy associated with the temperature of an object.
Kinetic Energy
The kinetic energy \( KE \) of an object is given by the equation:
\[ KE = \frac{1}{2} m v^2 \]
Where:
- \( m \) is the mass of the object (in kilograms, kg),
- \( v \) is the velocity of the object (in metres per second, m/s).
Potential Energy
The potential energy \( PE \) of an object due to its height above the ground is given by:
\[ PE = mgh \]
Where:
- \( m \) is the mass (in kilograms, kg),
- \( g \) is the gravitational acceleration (\( 9.8 \, \text{m/s}^2 \)),
- \( h \) is the height above the ground (in metres, m).
What Is Power?
Power is the rate at which work is done or energy is transferred. It is given by the formula:
\[ P = \frac{W}{t} \]
Where:
- \( P \) is the power (in watts, W),
- \( W \) is the work done (in joules, J),
- \( t \) is the time taken (in seconds, s).
Power can also be expressed as the rate of change of energy:
\[ P = \frac{E}{t} \]
Where \( E \) is energy and \( t \) is time.
The Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form to another. For example, in a rollercoaster, potential energy at the top is converted into kinetic energy as the coaster descends.
Applications of Work, Energy, and Power
Work in Everyday Life
Work is involved in many everyday activities. For instance, when you lift a book, you apply a force against gravity over a distance. The work you do is equal to the force (your weight) multiplied by the height you lift the book.
Energy in Machines
Machines convert energy from one form to another. For example, a car engine converts chemical energy from fuel into mechanical energy, allowing the car to move.
Power in Appliances
In appliances like kettles, the electrical energy is converted into heat (thermal energy) to boil water. The power rating of an appliance indicates how much energy it uses per unit of time. For example, a 1000W kettle uses 1000 joules of energy per second.
Example Problem
Problem: A car with a mass of \( 1500 \, \text{kg} \) accelerates from rest to a speed of \( 20 \, \text{m/s} \). How much kinetic energy does the car gain?
Solution:
The kinetic energy is given by:
\[ KE = \frac{1}{2} m v^2 \]
Substitute the known values:
\[ KE = \frac{1}{2} \times 1500 \times (20)^2 = 0.5 \times 1500 \times 400 = 300,000 \, \text{J} \]
So, the car gains \( 300,000 \, \text{J} \) of kinetic energy.
Common Mistakes in Work, Energy, and Power
- Forgetting Units: Always check the units used for force, distance, time, and energy.
- Incorrect Application of Formulas: Make sure to use the correct formula for the specific problem. For example, potential energy applies to height, not to motion.
- Ignoring Friction: In real-world applications, friction reduces the efficiency of energy conversion. Always account for friction in energy problems.
Practice Questions
- A car with a mass of \( 1200 \, \text{kg} \) is travelling at a speed of \( 15 \, \text{m/s} \). Calculate its kinetic energy.
- A person lifts a box weighing \( 50 \, \text{N} \) to a height of \( 2 \, \text{m} \). How much work is done?
- An electric heater operates at \( 2000 \, \text{W} \). How much energy does it use in \( 5 \, \text{minutes} \)?
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