Work done
Understanding Work Done
Work done is a fundamental concept in physics and engineering that relates to the amount of energy transferred by a force to move an object. It is a scalar quantity, which means it has magnitude but no direction. The basic formula for work done (W) when a force (F) moves an object through a displacement (s) is:
[ W = F \cdot s \cdot \cos(\theta) ]
where:
- ( F ) is the magnitude of the force applied,
- ( s ) is the magnitude of the displacement,
- ( \theta ) is the angle between the force vector and the displacement vector.
Table of Differences and Important Points
| Aspect | Description |
|---|---|
| Scalar vs. Vector | Work is a scalar quantity, whereas force and displacement are vector quantities. |
| Units | The SI unit of work is the joule (J), which is equivalent to a newton-meter (N·m). |
| Positive vs. Negative Work | Work is positive when the force has a component in the direction of displacement, and negative when the force has a component opposite to the displacement. |
| Zero Work | Work is zero when the force is perpendicular to the displacement or when there is no displacement. |
| Energy Transfer | Work done on an object results in a transfer of energy to or from the object. |
Formulas for Work Done
Constant Force: When a constant force acts on an object along a straight line, the work done is given by the formula mentioned above.
Variable Force: If the force varies along the path, the work done is calculated using the integral form:
[ W = \int_{s_1}^{s_2} F(s) \cdot ds ]
where ( F(s) ) is the force as a function of position and ( s_1 ) and ( s_2 ) are the initial and final positions, respectively.
- Work Done by Gravity: When an object falls freely under gravity, the work done by the gravitational force is:
[ W = m \cdot g \cdot h ]
where:
- ( m ) is the mass of the object,
- ( g ) is the acceleration due to gravity,
- ( h ) is the height through which the object falls.
Work Done in Lifting: When lifting an object against gravity, the work done is the same as the work done by gravity but with a positive sign.
Work Done by Spring Force: For a spring that obeys Hooke's law, the work done in stretching or compressing the spring is:
[ W = \frac{1}{2} k \cdot x^2 ]
where:
- ( k ) is the spring constant,
- ( x ) is the displacement from the equilibrium position.
Examples to Explain Important Points
Example 1: Positive Work
A person applies a force of 100 N to push a box for a distance of 5 meters along a frictionless surface. The force is applied in the direction of the movement.
[ W = F \cdot s \cdot \cos(\theta) ] [ W = 100 \, \text{N} \cdot 5 \, \text{m} \cdot \cos(0^\circ) ] [ W = 500 \, \text{J} ]
The work done is positive because the force and displacement are in the same direction.
Example 2: Negative Work
Now, if the same person pulls the box towards themselves with a force of 100 N for a distance of 5 meters, the work done by the person is negative because the force is opposite to the direction of displacement.
[ W = 100 \, \text{N} \cdot 5 \, \text{m} \cdot \cos(180^\circ) ] [ W = -500 \, \text{J} ]
Example 3: Zero Work
If the person applies a force of 100 N to the box but the box does not move, the work done is zero because there is no displacement.
[ W = 100 \, \text{N} \cdot 0 \, \text{m} \cdot \cos(\theta) ] [ W = 0 \, \text{J} ]
Example 4: Work Done by Gravity
An object with a mass of 2 kg is dropped from a height of 10 meters.
[ W = m \cdot g \cdot h ] [ W = 2 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 10 \, \text{m} ] [ W = 196 \, \text{J} ]
The work done by gravity is 196 J, which is also the kinetic energy gained by the object just before it hits the ground.
Understanding work done is crucial for solving problems in mechanics, energy conservation, and many other areas of physics and engineering. It is important to consider the direction of the force relative to the displacement to determine whether the work done is positive, negative, or zero.