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Conservative & Non-conservative Force
When the work done by a force is independent of path taken by the object and only depends on it’s initial and final position, only then the force is called as conservative force.
In such cases, work done by conservative force will be same for all paths. i.e. w1 = w2 = w3.
In case of conservative force, work done in any close loop is zero. Or in other words if initial and final positions are same work done will be zero in case of conservative force.
Gravitational force, spring force, Electrostatic force etc. are examples of conservative forces. Any constant or variable force whose work can be calculated without the help of path travelled is conservative force.
Non conservative force
A force, whose work cannot be calculated without the help of path travelled, is called as non-conservative force.
- In case of non-conservative force, work done in different paths may or may not be zero
- In case of non-conservative force work done in close loop will be non-zero
Kinetic friction, air resistance, viscous force etc are example of non-conservative force.
Relation between Conservative force & Potential energy
We know that, from previous discussion,
Potential energy is equal to negative of work done by conservative force.
Let’s take, Force vector and displacement vector,
Let’s do partial differentiation w.r.t. x,
Can be also written as, if partial differentiation is done with respect to radius,
Note: Above equations only holds when force is conservative.
Now we have,
is conservative force, is also known as nabla.
We know that, if we do cross product of a vector with itself, result will be zero.
Now we have,
Identify whether the force is conservative or not.
We know that, for conservative force.
Hence, give force is conservative.
Equilibrium of a particle
A particle is said to be in equilibrium if the acceleration of the particle is zero with respect to the observer.
- Equilibrium is frame dependent
- At equilibrium position, net force acting on the particle is zero, net force includes pseudo force also
In case 1, Block is in equilibrium for both A & B, as acceleration of block is zero.
In case 2, Block is in equilibrium w.r.t. A (as acceleration is zero wrt to A), but not w.r.t. B. For B, block is accelerating with acceleration a.
In case 3, though speed of moon is constant but it’s still not in equilibrium because moon is changing it’s direction at every moment and hence it’s speed is constant but velocity is not. So moon is accelerating at each point of its trajectory, so not in equilibrium.
Circular motion is always accelerated motion so there will be no equilibrium.
When velocity and acceleration both are zero, equilibrium is called static equilibrium.
Acceleration is zero, but velocity is not zero, i.e. constant non zero velocity.
When particle is displaced slightly from its equilibrium point in either direction, it returns back to the equilibrium point, this type of equilibrium is called stable equilibrium at this point.
Above examples are in stable equilibrium. If we displace object from equilibrium point, it will try to go at equilibrium point again. Notice that when object is at equilibrium point, it’s acceleration is zero.
When the particle is displaced from equilibrium point, In either direction, it moves away from equilibrium point, such type of equilibrium is called unstable equilibrium at that point.
When particle is displaced from it’s equilibrium slightly in either direction, if it neither moves away nor returns back then the equilibrium will be neutral.
- In case of stable equilibrium force is always towards equilibrium point
- In case of unstable equilibrium force is away from equilibrium point