Lutfi Wiandani

Lutfi 1 Nama : Lutfi Listia Wiandani Email :- MECHANICAL ENERGY Conseervation of mechanical energy is a direct consequence of Newton’s second law. Consider the scalar case (3.1.2). where the force 𝑓 can be a function of discplacement 𝑥. 𝑚𝑣̇ = 𝑓(𝑥) Multiply both sides by 𝑣 𝑑𝑡 and use the fact that 𝑣 = 𝑚𝑣 𝑑𝑣 = 𝑣𝑓(𝑥 )𝑑𝑡 = 𝑑𝑥 𝑑𝑡 𝑑𝑥 𝑑𝑡 𝑓 (𝑥 )𝑑𝑡 = 𝑓(𝑥 )𝑑𝑥 Integrate both sides. ∫ 𝑚𝑣 𝑑𝑣 = 𝑚𝑣 2 2 = ∫ 𝑓 (𝑥 )𝑑𝑥 + 𝑐 (3.1.4) Where C is a constant of intergration. Work is force times displacement, so the integral on the right represents the total work done on the mass by the force 𝑓 (𝑥 ) is independt of the path and depends only on the end ponts, then the force 𝑓(𝑥 ) is derivable from a function 𝑉 (𝑥 ) as follow. 𝑑𝑉 𝑓 (𝑥 ) = − 𝑑𝑥 (3.1.5) Then, in this case, 𝑓(𝑥) is called a conservattive force. If we intergrate both sides of the last equation, we obtain 𝑉(𝑥) = ∫ 𝑑𝑉 = − ∫ 𝑓 (𝑥 )𝑑𝑥 Or from (3.1.4), 𝑚𝑣 2 2 + 𝑉 (𝑥 ) = C (3.1.6) Lutfi 2 The equation shows that 𝑉(𝑥) has the same units as kinetic energy. 𝑉(𝑥) is called the potential energy (PE) function. Equation (3.1.6) states that the sum of the kinetic and potential energics must be constant, if No. force other than the conservative force is applied. If 𝑣 and 𝑥 have the values 𝑣0 and 𝑥0 at the time 𝑡0 , then 𝑚𝑣02 + 𝑉(𝑥0 ) = C 2 Comparing this with (3.1.6) gives 𝑚𝑣 2 2 - 𝑚𝑣02 2 + 𝑉(𝑥 ) − 𝑉 (𝑥0 ) = 0 (3.1.7) Which can be expressed as (3.1.8) ∆𝐾𝐸 + ∆𝑃𝐸 = 0 Where the change in kinetic energi is ∆KE = 𝑚(𝑣 2 − 𝑣02 ) 2 and the change in potential energy is ∆PE = 𝑉(𝑥 ) − V(𝑥0 ). For some problems, the following from of the principle is more convenient to use: 𝑚𝑣02 2 + 𝑉(𝑥0 ) = 𝑚𝑣 2 2 + 𝑉(𝑥) (3.1.9) In the form (3.1.8), we see that conservation of mechanical energy states that the change in kinetic energy plus the change in potential energy is zero. Note that the potential energy has a relative value only. The choice of reference point for measuring 𝑥 determines only the value of C, which (3.1.7) shows to be irrelevent. Gravity is an examples of a conservative force, for which 𝑓 = −𝑚g. The gravity force is conservative because the work done lifting an object depends only on the change in height and not on the path taken. Thus, if 𝑥 represents vertical displacement, 𝑉(𝑥) = 𝑚g𝑥 And 𝑚𝑣 2 2 + 𝑚g𝑥 = C (3.1.10) Lutfi 3 𝑚𝑣 2 2 − 𝑚𝑣02 2 + 𝑚g(𝑥 − 𝑥0 ) = 0 (3.1.11)