Work and Energy
- You’re pushing a box across the floor.
- You apply a force, and the box moves.
- But what exactly are you doing? You’re doing work.
Work is a way to quantify how much energy is transferred when a force causes an object to move.
Definition of Work: The Relationship Between Force and Displacement
Work
Work is defined as the product of the force applied to an object, the displacement of the object, and the cosine of the angle between the force and the displacement.
Mathematically, this is expressed as:
$$
W = Fs \cos \theta
$$
where:
- $W$ is the work done (measured in joules, J).
- $F$ is the magnitude of the force applied (in newtons, N).
- $s$ is the displacement of the object (in meters, m).
- $\theta$ is the angle between the force and the displacement.
- Work is only done when the force has a component in the direction of the displacement.
- If the force is perpendicular to the displacement ($\theta = 90^\circ$), no work is done because $\cos 90^\circ = 0$.
Positive, Negative, and Zero Work
- Positive Work: When the force and displacement are in the same direction ($0^\circ \le \theta< 90^\circ$), work is positive.
- Negative Work: When the force opposes the displacement ($90^\circ < \theta \le 180^\circ$), work is negative.
- Zero Work: When the force is perpendicular to the displacement ($\theta = 90^\circ$), no work is done.
A person pulls a sled with a force of 50 N at an angle of $30^\circ$ to the horizontal. The sled moves 10 m along the ground.
Calculate the work done by the force.
Solution
- Using the formula:$$
W = Fs \cos \theta
$$ - Substitute the values:$$
W = 50 \times 10 \times \cos 30^\circ$$ $$= 50 \times 10 \times 0.866 = 433 \text{ J}
$$ - The work done is 433 joules.
The Work-Energy Theorem: Connecting Work Done to Changes in Kinetic Energy
The work-energy theorem
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy.
In other words, when you do work on an object, you are transferring energy to or from it, changing its motion.
Kinetic Energy
Kinetic energy
Kinetic energy is the energy an object possesses due to its motion.
It is given by the formula:
$$
E_K = \frac{1}{2} m v^2
$$
where:
- $E_K$ is the kinetic energy (in joules, J).
- $m$ is the mass of the object (in kilograms, kg).
- $v$ is the velocity of the object (in meters per second, $\text{m s}^{-1}$).
Work-Energy Theorem
The work-energy theorem can be expressed as:
$$
W_{\text{net}} = \Delta E_K = E_{K,\text{final}} - E_{K,\text{initial}}
$$
This means that the net work done on an object is equal to the change in its kinetic energy.
A 5 kg block is initially at rest. A horizontal force accelerates it to a speed of $4 \ \text{m s}^{-1}$. Calculate the work done on the block.
Solution
- Calculate the initial kinetic energy:$$
E_{K / \text{initial}} = \frac{1}{2} \times 5 \times 0^2 = 0 \text{ J}
$$ - Calculate the final kinetic energy:$$
E_{K / \text{final}} = \frac{1}{2} \times 5 \times 4^2 = 40 \text{ J}
$$ - Calculate the change in kinetic energy:$$
\Delta E_K = E_{K / \text{final}} - E_{K / \text{initial}} $$ $$= 40 \text{ J} - 0 \text{ J} = 40 \text{ J}
$$ - The work done on the block is 40 joules.
Energy Transformations: Analyzing Energy Conversion in Systems
- Energy can change forms but is never lost.
