Showing posts with label ENERGY. Show all posts
Showing posts with label ENERGY. Show all posts

Potential Energy of Spring

The spring force is an example of a variable force which is conservative. Fig. one shows a block attached to a spring and resting on a smooth horizontal surface. The other end of the spring is attached to a rigid wall. The spring is light and may be treated as mass less. In an ideal spring, the spring force Fs is proportional to x where x is the displacement of the block from the equilibrium position. The displacement could be either positive [Fig. 1(b)] or negative [Fig. 1(c)]. This force law for the spring is called Hooke’s law and is mathematically stated as

Fs = − kx 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.


Suppose that we pull the block outwards as in Fig. 1(b). If the extension is xm, the work done by the spring force is
This expression may also be obtained by considering the area of the triangle as in Fig. 1(d). The work done by the external pulling force F is positive since it overcomes the spring force.

The equation is valid true even when the spring is compressed.(129)

Related posts :

Motion of a body in vertical circle
Work
work done by variable force
Work Energy Theorem
Potential energy
Law of conservation of energy
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Law of Conservation of Energy

The total energy of any system is always constant. It neither increases nor decreases and it remains constant.

The mechanical energy of the system is of two forms and they are potential and kinetic energies.

Let that a body undergoes displacement Δx under the action of a conservative force F. Then from the WE theorem we have, ΔK = F(x) Δx .

If the force is conservative, the potential energy function V(x) can be defined such that

− ΔV = F(x) Δx

The above equations imply that

ΔK + ΔV = 0

Δ(K + V ) = 0

which means that K + V, the sum of the kinetic and potential energies of the body is a constant. Over the whole path, xi to xf, this means that Ki + V(xi) = Kf + V(xf ) .

The quantity K +V(x), is called the total mechanical energy of the system. Individually the kinetic energy K and the potential energy V(x) may vary from point to point, but the sum is a constant.

The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.

Related posts :

Work
work done by variable force
Work Energy Theorem
Potential energy
Friction introduction
Rolling Friction





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Potential Energy

The energy possessed by the body by virtue of its state is called potential energy.The term potential energy means ‘stored’ energy. A stretched bow-string possesses potential energy. When it is released, the arrow flies off at a great speed.

The earth’s crust is not uniform, but has discontinuities and dislocations that are called fault lines. These fault lines in the earth’s crust are like ‘compressed springs’. They possess a large amount of potential energy. An earthquake results when these fault lines readjust. Thus, potential energy is the ‘stored energy’ by virtue of the position or configuration of a body. The body left to itself releases this stored energy in the form of kinetic energy .

The gravitational force on a ball of mass m is mg . g may be treated as a constant near the earth surface. That means that height h of the ball above the earth’s surface is very small compared to the earth’s radius so that we can ignore the variation of g near the earth’s surface.

Let us take the upward direction to be positive. Let us raise the ball up to a height h. The work done by the external agency against the gravitational force is mgh. This work gets stored as potential energy. Gravitational potential energy of an object, as a function of the height h, is denoted by V(h) and it is the negative of work done by the gravitational force in raising the object to that height.

V (h) = mgh

If h is taken as a variable, it is easily seen that the gravitational force F equals the negative of the derivative of V(h) with respect to h. Thus,The negative sign indicates that the gravitational force is downward. When released, the ball comes down with an increasing speed. There fore the potential energy is there converted into kinetic energy.

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Common Forces in mechanics
Free body diagram
Circular motion part one and two


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