Work done by conservative and non-conservative forces

When you study work done by forces in physics, it is essential to understand the difference between conservative and non-conservative forces. Broadly speaking, work done can be defined as an effect that causes an object to move or change its position under the influence of a force. In this lesson, we will explore these concepts in depth, explain what distinguishes conservative forces from non-conservative forces, and consider how each type contributes to work.

Understanding the work

In physics, work is done when a force is applied to an object and the object moves in the direction of the force. It is expressed by the formula:

W = F × d × cos(θ)

Where:

• W is the work done by the force.
• F is the magnitude of the force.
• d is the displacement of the object.
• θ is the angle between the force and displacement direction.

Work is measured in joules (J), force in newtons (N) and distance in meters (m). The concept of work is important for understanding energy transfer within a system.

Conservative forces

A conservative force is a type of force in which the work done in moving an object between two points is independent of the path taken by it. The work done depends only on the initial and final position of the object. Prominent examples of conservative forces include gravitational force and elastic spring force.

Example of gravitational force

Suppose an object of mass m is raised to a height h from the ground. The work done by the gravitational force in bringing the object down is the same irrespective of the path it takes. It depends only on the difference in height.

In such cases the work done, W_g, is given by:

W_g = m × g × h

Here, g represents the acceleration due to gravity. Note that the work done by gravity depends only on the vertical distance h.

Elastic force example (Hooke's law)

Consider a spring being compressed or stretched. The force exerted by the spring (which is conservative) obeys Hooke's law:

F_s = -k × x

Where:

• F_s is the spring force.
• k is the spring constant (N/m).
• x is the displacement from the equilibrium position.

When the spring is stretched or compressed from initial displacement x_1 to final displacement x_2 the work done by this force is given by:

W_s = 1/2 k (x_2^2 - x_1^2)

Since this type of work depends only on the initial and final conditions, it is clear that the spring force is a conservative force.

Non-conservative forces

Now, let's take a look at non-conservative forces. Non-conservative forces are forces where the work done depends on the path taken. This means that moving an object from one point to another may require different amounts of work depending on how the object is moved. Friction and air resistance are good examples of non-conservative forces.

Example of friction force

Friction is a general non-conservative force. Consider a block sliding on a flat surface. The work done against friction depends on the path taken by the block.

The work done by friction is calculated as follows:

W_f = -f × d

Where:

• W_f is the work done by friction.
• f is the friction force.
• d is the distance over which the force acts.

Since friction opposes motion, its work is often negative. The longer the path, the more work is done against friction.

Implications of non-conservative forces

Unlike conservative forces, non-conservative forces, such as friction, convert mechanical energy into other forms, such as thermal energy, which cannot be recovered as mechanical energy in that system. This is why energy is often "lost" in systems with non-conservative forces. However, energy is conserved throughout the universe due to the principle of conservation of energy.

Total mechanical energy and conservation

In any mechanical system with only conservative forces, the total mechanical energy is conserved. Total mechanical energy is the sum of potential energy and kinetic energy, expressed as:

E_total = K + U

Where:

• E_total is the total mechanical energy.
• K is the kinetic energy of the object.
• U is the potential energy of the object.

Examples of energy conservation include a pendulum swinging in the absence of air resistance or friction. As the pendulum swings, its energy converts between kinetic and potential forms, but the total mechanical energy remains constant.

Real-life examples and applications

The difference between conservative and non-conservative forces has practical implications in engineering, physics research, and understanding natural phenomena.

Roller coaster

Amusement parks operate on these concepts. As the roller coaster climbs up, potential energy increases; as it falls down, potential energy is converted into kinetic energy. Friction plays a role in controlling motion, exemplifying the effects of non-conservative forces.

Automobile

In cars, brakes apply a non-conservative force to stop the vehicle. A car's engine also works to overcome friction and drag, which is another example of a non-conservative force. The efficiency of a car engine is often improved by reducing these losses.

Astronomical events

Astronomers consider gravitational forces when studying the motion of planets, which shows that the work depends only on the relative positions of the celestial bodies, thus it is conservative.

Conclusion

In short, understanding the subtle difference between conservative and non-conservative forces gives us insight into how energy is conserved or transformed in different systems. Through conservative forces, we learn that energy can be completely exchanged between potential and kinetic forms without any loss, while non-conservative forces reflect energy lost in other forms such as heat. This knowledge is basically applied in many fields of science and technology to optimize machinery, vehicles, and even natural processes, helping us design more efficient and sustainable systems.

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