# 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
$\Delta K= W_{net}$
• Now for conservative forces , we know that
$\Delta = - F.dx$
or
$W = -\Delta U$
• If only conservative forces acting on the system ,then $\Delta K= W_{net}$
$\Delta K = - \Delta U$
$\Delta K + \Delta U=0$
or
$K_2 - K_1 + U_2 - U_1=0$
$K_2 + U_2 = K_1 + U_1$
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
$W_{net}= W_{c} + W_{nc}$
Now ,$\Delta K= W_{net}$
Therefore,
$\Delta K=W_{c} + W_{nc}$
$\Delta K -W_{c}= W_{NC}$
$\Delta K + \Delta U = W_{NC}$
• 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
$KE_{ground}= \frac {1}{2} mv^2 + mgH = \frac {1}{2} .1 20^2 + .1 \times 10 \times 10= 30 J$
Now
$\frac {1}{2} mv_g^2 = 30$
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
$\frac {1}{2}mv^2 = \frac {1}{2}kx^2$
or
$x= v \sqrt {\frac {m}{k}}$
or x=.5 m

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