Tuesday, March 3, 2015

Potential and Kinetic Energy

Potential and Kinetic Energy

Introduction

Mechanical engineering is concerned principally with two kinds of energy, potential energy and kinetic energy.

Potential Energy

Potential energy is energy due to the position of the body. The force exerted on a mass of m kg is mg N (where g D 9.81 m/s2, the acceleration due to gravity). When the mass is lifted vertically through a height h m above some datum level, the work done is given by: force ð distance D (mg)(h) J. This work done is stored as potential energy in the mass.

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(the potential energy at the datum level being taken as zero).

For example, if a car of mass 800 kg is climbing an incline at 10° to the horizontal, the increase in potential energy of the car as it moves a distance of 50 m up the incline is determined as follows:

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Kinetic Energy

Kinetic energy is the energy due to the motion of a body. Suppose a force F acts on an object of mass m originally at rest (i.e. u D 0) and accelerates it to a velocity v in a distance s:

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Principle of Conservation of Energy

Energy may be converted from one form to another. The principle of conservation of energy states that the total amount of energy remains the same in such conversions, i.e. energy cannot be created or destroyed.

In mechanics, the potential energy possessed by a body is frequently converted into kinetic energy, and vice versa. When a mass is falling freely, its potential energy decreases as it loses height, and its kinetic energy increases as its velocity increases. Ignoring air frictional losses, at all times:

potential energy + kinetic energy = a constant

If friction is present, then work is done overcoming the resistance due to friction and this is dissipated as heat. Then,

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Kinetic energy is not always conserved in collisions. Collisions in which kinetic energy is conserved (i.e. stays the same) are called elastic collisions, and those in which it is not conserved are termed inelastic collisions.

Kinetic Energy of Rotation

The tangential velocity v of a particle of mass m moving at an angular velocity ω rad/s at a radius r metres (see Figure 21.2) is given by:

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If all the masses were concentrated at the radius of gyration it would give the same moment of inertia as the actual system.

For example, a system consists of three small masses rotating at the same speed about a fixed axis; the masses and their radii of rotation are: 16 g at 256 mm, 23 g at 192 mm and 31 g at 176 mm.

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Flywheels

The function of a flywheel is to restrict fluctuations of speed by absorbing and releasing large quantities of kinetic energy for small speed variations.

To do this they require large moments of inertia and to avoid excessive mass they need to have radii of gyration as large as possible. Most of the mass of a flywheel is usually in its rim.

For example, a cast iron flywheel is required to release 2.10 kJ of kinetic energy when its speed falls from 3020 rev/min to 3010 rev/min. The moment of inertia of the flywheel is assumed to be concentrated in its rim which is to be of rectangular section, the external and internal diameters being 670 mm and 600 mm. The radius of gyration of the rim may be assumed to be its mean radius. Taking the density of cast iron as 7800 kg/m3 , the required width for the flywheel is determined as follows:

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