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**Law of Conservation of Mechanical Energy**

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**Law of Conservation of Mechanical Energy**

Principle of Conservation of Mechanical Energy Conservation of Mechanical Energy Equation Law of Conservation of Mechanical Energy Conservation of Mechanical Energy: Mathematical Problem

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Principle of Conservation of Mechanical Energy According to the law of conservation of mechanical energy, in an isolated system, that is, in the absence of non-conservative forces like friction, the initial total energy of the system equals to the total energy of the system. Simply stated, the total mechanical energy of a system is always constant (in case of absence of non-conservative forces). For instance, if a ball is rolled down a frictionless roller coaster, the initial and final energies remain constant. Conservative forces are those that don't depend on the path taken by an object. For example, gravity, spring and electrical forces are examples of mechanical energy. Law of Conservation of Mechanical Energy

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Conservation of Mechanical Energy Equation The quantitative relationship between work and energy is stated by the mechanical energy equation. UT = Ki + Pi + Wext = Kf + Pf, where, UT = Total mechanical energy Ki = Initial kinetic energy Kf = Final kinetic energy Pi = Initial potential energy Pf = Final potential energy Wext = External work done Law of Conservation of Mechanical Energy

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**This is a general equation for mechanical energy conservation**

This is a general equation for mechanical energy conservation. In case, there are some external or internal forces acting on the object, that is the forces are non-conservative like friction, air resistance, etc, then only Wext is considered. In absence of such forces, Wext = 0 and so the mechanical energy conservation equation takes the form: UT = Ki + Pi = Kf + Pf Law of Conservation of Mechanical Energy JoJaRiRheCha’s “Munchkins” Caraga Regional Science high School Surigao City

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Conservation of Mechanical Energy: Mathematical Problem Let us consider a mathematical problem that involves the use of law of conservation of mechanical energy in finding the values of unknown quantities. Question: A 20 g stone is put in a sling shot with a spring constant of 100 N/m and it is stretched back to 0.7 m. Determine the maximum velocity that the stone will acquire and the speed of stone when it is shot straight up? Law of Conservation of Mechanical Energy JoJaRiRheCha’s “Munchkins” Caraga Regional Science high School Surigao City

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Solution: In this problem, we ignore the air resistance and heat effects that are present while operating the sling shot. This makes external work done zero, that means we can easily apply the law of conservation of mechanical energy formula. Total energy in the beginning of the event Ei = Ki + Gravitational potential energy (mgh) + spring force (½ kx2). Here, Ki = (0.5 mv2) = (0.5)m (0)2 = 0 (Since v = 0 initially) Gravitational potential energy = mg(0) = 0 (since h = 0 initially) Spring force = ½ kx2 = (0.5)(100)(0.7)2 = 24.5 J = Ei Once out of the sling shot, the stone gains some maximum velocity before it reaches some altitude. Law of Conservation of Mechanical Energy JoJaRiRheCha’s “Munchkins” Caraga Regional Science high School Surigao City

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**Ef = 0. 5 mv2 + mgh + ½ kx2 = (0. 5)(0. 02)(v)2 + mg(0) + (0**

Ef = 0.5 mv2 + mgh + ½ kx2 = (0.5)(0.02)(v)2 + mg(0) + (0.5)k(0)2 = 0.001v2 Since Ei = Ef Therefore, 24.5 J = 0.001v2 = 24,500 = v2. Therefore, v = m/s (approximate value) At the highest point, the velocity of stone is zero. Therefore, Ef = 24.5 J = 0.5 mv2 + mgh + ½ kx J = 0.5mv(0)2 + mgh + 1/2k(0)2 = 24.5 J = (0.02)(9.8 N/Kg)h = 125 m. Answer: Velocity attained = m/s and height attained = 125 m Law of Conservation of Mechanical Energy JoJaRiRheCha’s “Munchkins” Caraga Regional Science high School Surigao City

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