In physics, forces can be categorized into conservative and non-conservative forces. Conservative forces, such as gravity and electrostatic forces, are path-independent and allow energy to be fully recovered. On the other hand, non-conservative forces, like friction and air resistance, depend on the path taken and cause energy dissipation.
In this topic, we will prove that frictional force is non-conservative by examining its properties, impact on mechanical energy, and how it behaves in different scenarios.
1. Understanding Conservative and Non-Conservative Forces
1.1 What is a Conservative Force?
A conservative force has the following characteristics:
-
The work done by the force does not depend on the path taken but only on the initial and final positions.
-
The total mechanical energy (kinetic + potential) is conserved in a closed system.
-
The work done by the force over a closed loop is zero.
Examples:
-
Gravitational force
-
Electrostatic force
-
Spring force (elastic force)
1.2 What is a Non-Conservative Force?
A non-conservative force has these properties:
-
The work done by the force depends on the path taken.
-
Energy is converted into heat, sound, or other forms, leading to a loss of mechanical energy.
-
The work done by the force over a closed loop is not zero.
Examples:
-
Frictional force
-
Air resistance
-
Viscous force in fluids
Since friction is a non-conservative force, let’s prove why it does not conserve mechanical energy.
2. Proving That Frictional Force is Non-Conservative
2.1 Work Done by Friction Depends on the Path
One of the main characteristics of non-conservative forces is that the work done depends on the path taken.
Let’s consider an object moving from point A to point B on a rough surface:
-
If the object moves in a straight line, friction does negative work by opposing the motion.
-
If the object takes a curved path to reach the same point B, friction still opposes motion but does more work because the path is longer.
Since the work done by friction changes with the path length, friction is a non-conservative force.
2.2 Work Done by Friction in a Closed Loop is Not Zero
A key property of conservative forces is that when an object returns to its starting position, the net work done is zero.
Let’s consider a block moving in a circular track with friction:
-
The block starts at point A and moves around the track.
-
Friction acts opposite to the direction of motion at every point.
-
When the block completes the loop and returns to point A, the total work done by friction is not zero-instead, energy is lost as heat.
Since friction does negative work throughout the motion, the total mechanical energy of the system decreases, proving that friction is non-conservative.
2.3 Friction Converts Mechanical Energy into Heat
Another proof that friction is non-conservative is that it reduces mechanical energy by converting it into heat.
Example:
-
A moving box on a rough surface eventually stops due to friction.
-
The kinetic energy of the box is converted into thermal energy, increasing the temperature of the surface.
-
Unlike conservative forces (which store energy as potential energy), friction permanently removes energy from the system.
This energy conversion makes friction irreversible, reinforcing that it is a non-conservative force.
3. Mathematical Proof: Work Done by Friction
3.1 Work Done by Friction Formula
The work done by friction is given by:
Where:
-
F_{text{friction}} = Force of friction (N)
-
d = Distance moved (m)
-
theta = Angle between friction force and displacement (180°)
Since friction always opposes motion, the angle theta = 180^circ , so:
Thus,
Since work is always negative, friction always removes mechanical energy from the system, proving that it is non-conservative.
4. Experimental Evidence That Friction is Non-Conservative
4.1 Inclined Plane Experiment
A simple experiment demonstrates that friction is non-conservative:
-
Step 1: Place a block on a smooth frictionless inclined plane and let it slide down.
-
The block gains kinetic energy as it descends.
-
If no friction is present, it moves back to its original height when going up.
-
This is an example of conservative motion.
-
-
Step 2: Repeat the experiment with a rough surface (friction present).
-
The block loses some energy as heat while sliding down.
-
When it moves back up, it does not reach the original height because energy was dissipated.
-
This proves that frictional force is non-conservative.
-
5. Applications of Friction as a Non-Conservative Force
Even though friction is non-conservative, it plays an important role in real-world applications:
5.1 Braking Systems in Vehicles
-
When a car brakes, the kinetic energy is converted into heat energy due to friction between the brake pads and wheels.
-
This energy cannot be recovered, proving friction is non-conservative.
5.2 Walking and Running
-
Friction between shoes and the ground allows movement.
-
Each step loses energy as heat, making the process non-conservative.
5.3 Mechanical Systems and Machines
-
In engines and machines, friction causes energy loss in the form of heat.
-
Lubricants are used to reduce friction and improve efficiency.
6. Summary of the Proof
| Property | Conservative Forces | Friction (Non-Conservative) |
|---|---|---|
| Work depends on path? | No | Yes |
| Work done in a closed loop | Zero | Not zero |
| Energy conservation | Total energy is conserved | Energy is dissipated as heat |
| Example forces | Gravity, electrostatic, spring force | Friction, air resistance, viscosity |
From the table, it is clear that frictional force does not meet the criteria of a conservative force.
Frictional force is non-conservative because:
-
The work done by friction depends on the path taken.
-
The work done over a closed loop is not zero.
-
Mechanical energy is lost as heat, making it irreversible.
While friction causes energy loss, it is essential in daily life, enabling movement, braking, and mechanical operations. Understanding friction as a non-conservative force helps in designing efficient machines, vehicles, and energy-saving systems.