subject: Principle Of Conservation Of Mechanical Energy [print this page] Definition of mechanical energy : The sum of kinetic energy and potential energy of a body is called its mechanical energy.
The principle of conservation of mechanical energy:
The principle states that if a body or system is in motion under a conservative system of forces, the sum of its kinetic energy and potential energy is a constant. i.e the mechanical energy is constant.
Work done is equal to negative change of potential energy.
Combining the equation (1) and (2),
'rArr E_(k2) - E_(k1) = -(E_(p2) - E_(p1)) '
'"rArr E_(k2) -E_(k1) = E_(p1) - E_(p2) '
'"rArr E_(k2) +E_(p2) = E_(k1) +E_(p1)'
Which means the mechanical energy (i.e sum of the potential energy and kinetic energy) of a system of particles remains constant during the motion under the action of the conservative forces.
Verification of the Principle of Conservation of Mechanical Energy for a Freely Falling Body :
Consider a body of mass m which is at rest , at a height h from the ground as shown in figure.
principle of conservation of mechanical energy
(i) Mechanical energy of the body at A
Potential energy at A = mgh
Kinetic energy at A = '1/2' mv2= '1/2 ' m(0)2 = 0
':.' Mechanical energy = total energy = mgh + 0 = mgh ---------------------------------- (1)
(ii) Mechanical energy of the body at B
Let the body fall freely and strike the ground with a velocity v.
At A , initial velocity =0
At B, final velocity =v
Vertical distance travelled = h
':.' For bodies falling down v2 = u2 +2gs
'rArr v^2 = 0 +2gh or v^2 = 2gh'
':.' Kinetic energy at B = '1/2' mv2 = '1/2' m(2gh) = mgh
Potential energy at B = mg(0) = 0
':.' Mechanical energy at B = 0+mgh=mgh -------------------------- (2)
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(iii) Mechanical energy at C
Let C be an intermediate point at a depth x below A during its travel from A to B
Height at C above B = (h-x)
':.' Potential energy at C = mg(h-x)
At C , the velocity of the body is v1
At A , initial velocity = 0
Vertical distance travelled = x
':.' For bodies falling freely 'v_1^2 = u^2 +2gx'
'rArr v_1^2 = 0^2+2gx rArr v_1^2 = 2gx'
Kinetic energy at C = '1/2mv_1^2 =1/2m(2gx) = mgx'
Mechanical energy at C = mg(h-x) +mgx = mgh -----------------------(3)
From equations 1,2,and 3 it is clear that the mechanical energy of the body remains constant throughout the motion.
The explanation of an easy harmonic motion be just to the speeding up cause the activity a, of the object be proportional also within opponent to its dislocation x as of its balance location. a(t) ? -x(t). Wherever, k is a constant. In simple harmonic motion the energy is proportional toward the square of amplitude. That is E= (1/2) kA2.
Simple Harmonic Motion
Simple harmonic motion is the action of an easy harmonic oscillator, a periodic motion specifically neither determined nor damp. A body within easy harmonic motion experience a particular force which be known through Hookes law; that is to say, the force be straight relative to the displacement x also point within the opposing direction.
It is directly connected toward circular motion since be able to be see but we obtain an object to travel within a round pathway, similar toward a sphere stuck lying on a turntable.
Simple harmonic motion
Energy in Simple Harmonic Motion
Consider an object connect toward a spring display effortless harmonic motion. Permit one ending of the spring exists connect toward a wall also allow the object go parallel lying on a frictionless chart.
Whole energy of the object:
Object's kinetic energy be written as, K = (1/2) mv2
= (1/2) mw2A2sin2 (wt+f),
Potential energy is an expandable possible energy. The expandable potential energy store within a spring dislocate a distance x as of its stability location is U= (1/2) kx2. The object's possible energy therefore is written as,
