Conservative force and nonconservative force

**1. Conservative Force**

1.1 Weight (w)

Observe an object which moves vertically upwards until reaching a maximum height before moving downwards towards its initial position. When moving vertically upwards by h, the weight is opposite in direction from displacement. Thus, the weight does negative work on the object.

W = w h (cos 180^{o}) = – w h = – m g h

After reaching a maximum height, the object moves downwards towards its initial position by h. When moving downwards, the weight is in the same direction as the displacement. Because it is in the same direction as displacement, the weight does positive work.

W = w h (cos 0^{o}) = w h = m g h

The object’s mass (m), gravitational acceleration (g), and height (h) are the same, so the work done by the weight since the object starts moving vertically upwards until it returns to its initial position is zero.

W = m g h – m g h = 0

1.2 Spring Force

Observe a spring placed in a horizontal position. If the right end of the spring is pushed or compressed to the left, the spring exerts a thrust force to the right. You can prove this by compressing a spring. For instance, place an object on the right end, then press the object to the left. After the spring deviates by Δx, remove your hand from the object and the spring. When the hand is no longer in contact with the spring, the spring will push the object back to the right.

When the object moves to the left, the direction of its motion and displacement are opposite to the direction of the spring’s force. Because the directions are opposite to each other, the spring’s force does negative work.

W = – ½ k (Δx)^{2}

When the object moves to the right, the direction of its motion and displacement is the same as the spring’s force. Because the directions are the same, the spring’s force does positive work.

W = ½ k (Δx)^{2}

The spring used is the same, so the spring’s constant (k) is the same. The spring deviation (Δx) is also the same.

Thus, the work done by weight when the object starts to move to the left by Δx and then moves back to the right by Δx is zero.

W = ½ k (Δx)^{2 } – ½ k (Δx)^{2 }= 0

The work done by weight and spring force since the object starts to move from its initial position until it returns there is equal to zero. If the work was done by a force since it moves from its initial position until it returns there is equal to zero, the force is called conservative force. As such, spring force and weight are examples of conservative forces.

**2. Non-Conservative Force**

Observe an object which is pushed to the right and then pushed back to the left. When the object is moving or displaced to the right, the direction of the object’s displacement is the same as the direction of thrust force (F) and opposite to the direction of kinetic frictional force (f_{k}). As it is in the same direction as displacement, the thrust force does positive work on the object.

W = F s

On the contrary, the kinetic frictional force does negative work on the object.

W = – f_{k }s

The work was done by the thrust force and frictional force on the object when the object starts to move from its initial position until it returns there is:

W = 2 F s

W = -2 f_{k }s

The work done by the thrust force and kinetic frictional force since the object starts to move from its initial position until it returns there is not equal to zero. If the work done by a force since the object starts to move from its initial position until it returns there is not equal to zero, the force is called non-conservative force. As such, thrust force and kinetic frictional force are examples of non-conservative forces.

**1. What is a conservative force?** A conservative force is one for which the work done in moving an object between two points is independent of the path taken. Gravitational and electrostatic forces are examples of conservative forces.

**2. How is the work done by a conservative force related to potential energy?** The work done by a conservative force on an object moving from point A to point B is equal to the negative change in potential energy (∆U) between those two points: $W=−∆U$ This means that if an object gains potential energy, the work done by the conservative force is negative, and vice versa.

**3. What is a nonconservative force?** A nonconservative force is one for which the work done in moving an object between two points depends on the path taken. Friction and air resistance are typical examples of nonconservative forces.

**4. How does a nonconservative force affect the mechanical energy of a system?** A nonconservative force can change the total mechanical energy (sum of kinetic and potential energies) of a system. This is because the work done by a nonconservative force does not get stored as potential energy but is usually transformed to other forms of energy, such as heat.

**5. How can one determine whether a force is conservative or nonconservative using a closed-loop path?** For a conservative force, the net work done on an object after it travels a complete closed-loop path is zero. If the net work done over a closed loop is not zero, the force is nonconservative.

**6. What role does the curl of a force field play in determining if a force is conservative?** If the curl of a force field is zero everywhere in space, the force is conservative. If the curl is not zero, then the force is nonconservative.

**7. How is gravitational potential energy related to conservative forces?** Gravitational potential energy arises due to the conservative nature of the gravitational force. An increase in height in a gravitational field leads to an increase in gravitational potential energy, whereas the work done by the gravitational force is negative, implying a decrease in potential energy when an object moves closer to the Earth.

**8. Why doesn’t friction have an associated potential energy like other conservative forces?** Friction is a nonconservative force; therefore, the work done by friction does not get stored as potential energy. Instead, it’s often transformed into other forms, such as heat, and cannot be completely recovered as mechanical energy.

**9. In a scenario where an object slides down an inclined plane, how do conservative and nonconservative forces play a role?** When an object slides down an inclined plane, the gravitational force, a conservative force, acts on the object, converting its potential energy to kinetic energy. If there is friction between the object and the inclined plane (a nonconservative force), it will oppose the motion and reduce the kinetic energy of the object, converting some of it into heat.

**10. Can a system’s total energy be conserved even if nonconservative forces are at play?** Yes, a system’s total energy (including all forms of energy like mechanical, thermal, etc.) remains conserved according to the law of conservation of energy. However, when nonconservative forces act, the mechanical energy (sum of kinetic and potential energies) may not be conserved because some of it is converted to other forms of energy, such as heat or sound.

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