Work, Energy and Power Concept Page - 5

Definition

Conservative and Non-Conservative Forces

If work done by any force in a closed path is zero then the force is conservative else it would be non-conservative. For this defining any arbitrary closed path work done has to be calculated.
For the closed path:
∮F⃗ .d⃗ x=0${\displaystyle \oint \overrightarrow{F}.\overrightarrow{d}x=0}$

Definition

Examples of Conservative and Non-Conservative Forces

Force due to gravity is conservative force as work done from taking an object from height h$h$ to ground is +mgh$+mgh$ whereas it's −mgh$-mgh$ on the other way around. However, friction is an example of non-conservative force. Work done by friction in dragging an object to distance r$r$ is −fr$-fr$ and in retracing the same path work done would be −fr$-fr$ so total work done will be −2fr$-2fr$.

Definition

Conservative forces

The Conservative Force is a Force whose Work done is independent of the Path taken and depends only on the Initial and Final position of the object acted upon.
E.g. Gravitational Force, Electrostatic Force, etc.

Definition

Non-conservative forces:

Non-conservative forces:

A Non-Conservative force is a force whose work done is dependent of the path taken.

E.g. Frictional force, air resitance force, etc.

Result

proof- conservative forces are path independent

We know that the gravitational force is given by the expression. 

F=mg$F=mg$

Let us now see what happens if we throw a ball in two different paths but with the same start and end points. Let us first throw a ball and find the work done by gravity when the ball reaches the height (H).

For the next case, let us say we have thrown the same ball with a higher velocity such that, the ball passes the height (H), and travels further and reaches a height (L), and then comes back to  (H) and finally reaches the ground. And thus, we have two different paths in which the same ball has reached the height (H). Let us compute the work done in each case and see what is the result:

Case 1:

Work done W1$W_1$= ∫H0(−mg)dx${\int }_0^H(-mg)dx$ = −mgH$-mgH$

Case 2:

Work done W2$W_2$ = ∫L0(−mg)dx−∫HL(−mg)dx${\int }_0^L(-mg)dx-{\int }_L^H(-mg)dx$ = −mgH$-mgH$

We observe that W1=W2$W_1=W_2$. And thus, according to the definition of the conservative forces, we can say that the Work is a 'conservative force' because it is path independent.

Definition

Properties of Conservative Forces

In this particular case in order to calculate work done by gravity in the closed path direct formula mgh$mgh$ can be applied owing to conservative nature of gravitational force.

Example

Work Done by ideal springs

Example: A spring obeying the linear law F=−Kx$F=-Kx$ is first compressed by
10$10$ cm and the work done is W1$W_1$. Next it is compressed by
another 10$10$ cm, the work done now is W2$W_2$, then what is the value of W1:W2$W_1:W_2$ ?
Solution:W1=Δ(P.E)$W_1=\mathrm{\Delta }(P.E)$
=12K(x22−x12)$={\displaystyle \frac{1}{2}}K(x{{}_2}^2-x{{}_1}^2)$
=12K(102−02)$={\displaystyle \frac{1}{2}}K({10}^2-0^2)$
Similarly,  W2=12K(202−102)$W_2={\displaystyle \frac{1}{2}}K({20}^2-{10}^2)$
So, W1:W2=100:300$W_1:W_2=100:300$
=1:3$=1:3$

Formula

Force exerted by spring(Hooke’s law)

Spring force:
The force exerted by spring to restore its relaxed state is known as Spring force.

Figure (a) shows a spring in its relaxed state that is, neither compressed nor extended.

If we stretch the spring by pulling the block to the right as in Fig. (b), the spring pulls on the block toward the left. Because a spring force acts to restore the relaxed state.

If we compress the spring by pushing the block to the left as in Fig. (c), the spring now pushes on the block toward the right.

Because a spring force acts to restore the relaxed state, it is sometimes said to be a restoring force.  

Hooke's law: The spring force is proportional to the displacement of free end from its relaxed state.
Fs∝d$F_s\propto d$
Fs=−k×d$F_s=-k\times d$
k = Spring constant
d = displacement of free end from its relaxed position
Negative sign represents the direction of spring force which is always opposite to the direction of displacement of spring.

A spring force is a variable force because it is a function of x (the position of the free end).

Definition

Conservation of total mechanical energy

Mechanical energy is the sum of the potential and kinetic energies in a system. The principle of the conservation of mechanical energy states that the total mechanical energy in a system (i.e., the sum of the potential plus kinetic energies) remains constant as long as the only forces acting are conservative forces.
Example: Consider a person on a sled sliding down a 100 m$100m$ long hill on a 30$30$ 0${}^0$ incline. The mass is 20 kg$20kg$, and the person has a velocity of 2 m/s$2m/s$ down the hill when they're at the top. How fast is the person traveling at the bottom of the hill?
 
Solution:At the top: P.E=mgh=(20)(9.8)(100sin30$P.E=mgh=(20)(9.8)(100sin30$^0)=9800 J$)=9800J$
KE=1/2mv2=1/2×(20)×(2)2=40J$KE=1/2m{v}^2=1/2\times (20)\times (2{)}^2=40J$
Total mechanical energy at the top =9800+40=9840 J$=9800+40=9840J$
At the bottom: PE=0,K.E.=1/2mv2$PE=0,K.E.=1/2m{v}^2$
Total mechanical energy at the bottom = 1/2mv2$1/2m{v}^2$

If we conserve mechanical energy, then the mechanical energy at the top must equal what we have at the bottom. This gives:

1/2mv2=9840,v=31.3m/s.$1/2m{v}^2=9840,v=31.3m/s.$

Example

Conservation of energy when one of the forces is non conservative

Example:
A coconut of mass m$m$ falls from the tree through a vertical distance of s$s$ and could reach ground with a velocity of v$v$ ms−1${}^{-1}$ due to air resistance. Find the work done by air resistance.
Solution:
Mechanical energy before falling = Mechanical energy after falling + Energy loss due to air resistanceEnergy loss due to air resistance , Ea=mgs−12mv2$E_a=mgs-{\displaystyle \frac{1}{2}}m{v}^2$ 
Work done by air resistance is, W=−Ea=−mgs+12mv2

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