Conservative & Non-Conservative Forces
- Conservative forces are those for which work done depends only on initial and final points.
Example- Gravitational force, Electrostatic force.
- Non-Conservative forces are those where the work done or the kinetic energy did depend on other factors such as the velocity or the particular path taken by the object.
Example- Frictional force.
The Conservation of Mechanical Energy
- Mechanical Energy is the energy associated with the motion and position of an object.
- The quantity K +V(x), is called the total mechanical energy of the system.
- For a conservative force, ΔK = ΔW = F(x) Δx
Also, – V(x) = F(x) Δx
- This employs Δ(K+V) =0 for a conservative force.
- Individually the kinetic energy K and the potential energy V(x) may vary from point to point, but the sum is a constant.
- Conservative Force:
- A force F(x) is conservative if it can be derived from a scalar quantity V(x) by the relation : F(x) = – dv/dx
- The work done by the conservative force depends only on the end points.
- A third definition states that the work done by this force in a closed path is zero.
- The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.
Potential energy of spring
- The spring force is an example of a variable force, which is conservative.
- In an ideal spring, Fs = − kx , this force law for the spring is called Hooke’s law.
- The constant k is called the spring constant. Its unit is N m-1.
- The spring is said to be stiff if k is large and soft if k is small.
- Spring force is position dependent as first stated by Hooke, (Fs = − kx)
- Work done by spring force only depends on the initial and final positions. Thus, the spring force is a conservative force.
- We define the potential energy V(x) of the spring to be zero when block and spring system is in the equilibrium position.
- For an extension (or compression) x, V(x) = kx2/2
If the block of mass m is extended to xm and released from rest, then its total mechanical energy at any arbitrary point x (where x lies between – xm and + xm) will be given by: