Work done by different forces
Work Done by Different Forces
In physics, work is defined as the measure of energy transfer that occurs when an object is moved over a distance by an external force. The concept of work done by different forces is crucial in understanding how energy is transferred in various physical systems. In this article, we will explore the work done by various types of forces, including gravitational, frictional, elastic, and applied forces.
Definition of Work
Work is defined as the product of the force applied to an object and the displacement of the object in the direction of the force. Mathematically, work (W) is given by:
$$ W = \vec{F} \cdot \vec{d} = Fd \cos(\theta) $$
where:
- $\vec{F}$ is the force vector
- $\vec{d}$ is the displacement vector
- $F$ is the magnitude of the force
- $d$ is the magnitude of the displacement
- $\theta$ is the angle between the force vector and the displacement vector
Work is measured in joules (J) in the International System of Units (SI).
Types of Forces and Work Done
| Force Type | Description | Formula for Work Done | Positive or Negative Work |
|---|---|---|---|
| Gravitational Force | The force due to gravity acting on an object with mass. | $W_g = mgd \cos(\theta)$ | Positive when object moves against gravity, negative when it moves with gravity. |
| Frictional Force | The force that opposes the motion of an object across a surface. | $W_f = -f_k d$ | Always negative since friction opposes motion. |
| Elastic Force | The force exerted by a spring or any elastic material returning to its original shape. | $W_e = \frac{1}{2}kx^2$ | Positive when work is done against the elastic force, negative when work is done by the elastic force. |
| Applied Force | The force applied by an external agent to move an object. | $W_a = Fd \cos(\theta)$ | Positive when force and displacement are in the same direction, negative otherwise. |
Gravitational Force
The work done by gravitational force is significant when an object is lifted against gravity or falls under the influence of gravity. The formula for work done by gravity is:
$$ W_g = mgd \cos(\theta) $$
where $m$ is the mass of the object, $g$ is the acceleration due to gravity, and $\theta$ is the angle between the gravitational force and the displacement.
Example: Lifting a 10 kg object vertically upwards for 5 meters would do work against gravity:
$$ W_g = (10 \, \text{kg})(9.8 \, \text{m/s}^2)(5 \, \text{m})\cos(0^\circ) = 490 \, \text{J} $$
Frictional Force
Frictional force does negative work because it always opposes the direction of displacement. The work done by friction is given by:
$$ W_f = -f_k d $$
where $f_k$ is the kinetic frictional force and $d$ is the displacement.
Example: If a box is pushed across a floor with a kinetic frictional force of 20 N for a distance of 3 meters, the work done by friction is:
$$ W_f = -(20 \, \text{N})(3 \, \text{m}) = -60 \, \text{J} $$
Elastic Force
The work done by or against an elastic force, such as a spring, can be calculated using Hooke's Law:
$$ W_e = \frac{1}{2}kx^2 $$
where $k$ is the spring constant and $x$ is the displacement from the spring's equilibrium position.
Example: Compressing a spring with a spring constant of 200 N/m by 0.1 m would require work:
$$ W_e = \frac{1}{2}(200 \, \text{N/m})(0.1 \, \text{m})^2 = 1 \, \text{J} $$
Applied Force
The work done by an applied force depends on the direction of the force relative to the displacement. If the force is in the same direction as the displacement, the work is positive. If the force is opposite to the displacement, the work is negative.
Example: Pushing a box with a force of 50 N for a distance of 4 meters along the direction of the force:
$$ W_a = (50 \, \text{N})(4 \, \text{m})\cos(0^\circ) = 200 \, \text{J} $$
Conclusion
Understanding the work done by different forces is essential in analyzing the energy transfer in physical systems. Each type of force contributes to the total work done on an object, and the direction of the force relative to the displacement determines whether the work is positive or negative. By applying the formulas and concepts discussed, one can calculate the work done by various forces in a wide range of physical scenarios.