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Mechanical Energy



(4) Mechanical Energy

  • we already have an idea that energy is associated closely with work and we have defined energy of a body as the capacity of the body to do work
  • In dynamics body can do work either due to its motion ,due to its position or both due to its motion and position
  • Ability of a body to do work due to its motion is called Kinetic energy for example piston of a locomotive is capable of doing of work
  • Ability of a body to do work due to its position or shape is called Potential Energy For example workdone by a body due to gravity above surface of earth
  • Sum of kinetic energy and Potential energy of body is known as its mechanical energy
    Thus
    Mechanical Energy=Kinetic Energy+Potential Energy

(4) Principal of Conservation of Mechanical Energy

  • From work energy theorem , we know that
    ΔK=Wnet
  • Now for conservative forces , we know that
    Δ=F.dx
    or
    W=ΔU
  • If only conservative forces acting on the system ,then ΔK=Wnet
    ΔK=ΔU
    ΔK+ΔU=0
    or
    K2K1+U2U1=0
    K2+U2=K1+U1
    or
    K+U=constant
  • We already know that above quantity is called the mechanical energy of the system
  • So we see that if only conservative forces are acting on the system, the total mechanical energy of the system remains constant.It does not increase or decrease and it is conserved . This is called the Principal of Conservation of Mechanical Energy
  • If non-conservative forces are also present in the system such as friction, the Work Energy Theorem is given as
    Wnet=Wc+Wnc
    Now ,ΔK=Wnet
    Therefore,
    ΔK=Wc+Wnc
    ΔKWc=WNC
    ΔK+ΔU=WNC
  • So, we see that total mechanical energy is not conserved if the non-conservative forces are present

Example-1
A man throws an ball of mass .10 kg from the top of the building of height 10 m with speed of 20 m/s. Find kinetic energy and speed of the ball when it reaches the ground? (take g=10 m/s2)
Solution
From law of conservation of mechanical energy
(Kinetic Energy + Potential energy) at top = ( Kinetic energy ) at ground
KEground=12mv2+mgH=12.1202+.1×10×10=30J
Now
12mvg2=30
or
v=24.5 m/s

Example-2
A object of mass 10 kg moving with a speed 5 m/s on a smooth surface and it collide with a horizontally mounted spring of spring constant 1000 N/m. What is the maximum compression of the spring ?
Solution
At maximum compression , Kinetic energy of the object converts into potential energy of the spring
From law of conservation of mechanical energy
(Kinetic Energy) initially = ( Potential Energy ) at maximum compression
12mv2=12kx2
or
x=vmk
or x=.5 m

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