Principle Of Conservation Of Mechanical Energy
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.
Explanation of Conservation of Mechanical Energy:
From principle of work and energy
Work done = change in its kinetic energy
'rArr W_(1rarr2) = E_(k2) - E_(k1)' ---------------------------- (1)
If a body moves under the action of a conservative force, work done is stored as potential energy.
'rArr W_(1rarr2) = - (E_(p2) - E_(p1))' --------------------------- (2)
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)
Check this awesome easy math word problems - http://www.mathwordproblems.org/easy-math-word-problems.html i recently used.
(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,
U = (1/2) kx2 = (1/2) mw2x2 = (1/2) mw2A2cos2 (wt+f).
Total automatic energy of the object can be written,
E = K+U = (1/2) mw2A2 (sin2 (wt+f) +cos2 (wt+f)) = (1/2) mw2A2.
Energy E within the classification is proportional toward the square of amplitude.
E = (1/2) kA2.
It be an always altering combination of kinetic energy with potential energy.
Used for a few object perform easy harmonic motion through angular frequency w, the reinstate force F = -mw2x obey Hooke's law.
Therefore it is a conventional force. We be able to describe a potential energy U = (1/2) mw2x 2.
Also the whole energy of the object be known through E = (1/2) mw2A2.
by: mathqa
Prestige Ferns Residency Bangalore : New Launch At Harlur Road Off Sarjapur Road Outline Milford Sound Trips Provide You With Trips Around New Zealand Use Study Pills For Boosting Energy Before Exams Energy Efficient Ways To Heat Water Places Near Mumbai To Celebrate New Year Party Mortgage Net Branch: An Alternative For New Entrants To The Market Customize Your Website Through Magento Extensions To Reach New Heights Outfits And New Inclinations Why You Need A Business Plan For Your New Or Existing Business Visit Terranea Resort And Experience The Cuisine Of Their Wonderful New Chefs December 2012 Moving Out Of An Old Paradigm And Into A New One Upgrading To A Classy Look With New Kitchen Tile Countertops Fiberglass Insulation Saves Energy