Force

Conservative and Non-Conservative Forces

While analysing the problems using the principle of conservation of energy, it is important to distinguish between two types of forces:

Conservative forces: There are two ways in which we can characterise a conservative force. A force is conservative if:

The net work done against the force in moving a mass between two points depends only on the location of two points and not on the path followed

The net work done against the force in moving a mass through any closed path is zero.

These two criteria are equivalent. A conservative force follows both properties.

Examples of conservative forces are-gravitational forces, ideal spring forces, electrostatic forces.

We can/always define an associated potential energy for every conservative force. Corresponding to these conservative forces we have gravitational potential energy, elastic potential energy and electrostatic potential energy.

Non-conservative forces:Those forces which do not satisfy the above mentioned criteria are non conservative forces. Friction and viscous forces are the most common examples of non-conservative forces.

Conservative forces and potential energy:For every conservative force, there is a corresponding potential energy function.

In each case, the potential energy expression depends only on position.

For every conservative force F_x, that depends only on the position x, there is an associated potential energy function U(x). When conservative force does positive work, the potential energy of the system decreases.

Work done by conservative force is

which, in the limit, becomes

Integrating both sides for a displacement from to we have:

Motion in a Vertical Circle

Illustration

A mass m is tied to a string of length l and is rotated in a vertical circle with centre at the other end of the string.

Find the minimum velocity of the mass at the top of the circle so that it is able to complete the circle.
Find the minimum velocity at the bottom of the circle.

At all positions, there are two forces acting on the mass:

its own weight and the tension in the string.

Let the radius of the circle = l

(a) At the top: Let v_t = velocity at the top

Net force towards centre

For the movement in the circle, the string should remain tight i.e. the tension must be positive at all positions.

As the tension is minimum at the top

Minimum or critical velocity at the top =

(b) Let v_b be the velocity at the bottom. As the particle goes up, its KE decreases and GPE (Gravitational

Potential Energy increases.

Loss in KE = gain in GPE

Note : When a particle moves in a vertical circle, its speed decreases as it goes up and its speed increases as it comes down. Hence it is an example of non-uniform circular motion.

Illustration:

A particle of mass m is suspended from a string of length l fixed to the point o. What velocity should be imparted to the particle in its lowermost position so that the string is just able to reach the horizontal diameter of the circle?

Let v be the required velocity imparted to the particle. The particle just stops at B.

Loss in Ke = Gain in GPE

1/2mv² _ 0 = mgl

Brief analysis of motion in a vertical circle

If v_b : velocity of particle at bottom

l: radius of the circle ABCD

1. If the particle will move in the circle ABCD.

2. If the particle will oscillate right and left around point A.

3. If it will oscillate along the semicircle BAD.

4. If

The particle will cross the diameter DOB but will not be able to complete the circle. It will leave the circular path somewhere between B and C at a point where tension (or normal reaction) becomes zero.