MECHANICAL ENERGY....

- Mechanical energy is the sum of potential & kinetic energy.
- Potential energy can be due to gravitation or in other forms
like an extended or compressed spring.
- The formula, that the potential energy of a body at a height
h = mgh where g is the acc. due to gravity which is treated as constant,is valid only when h is very small compared to
the radius of the earth.
- The potential energy at a distance r from the centre of
earth is
U(earth) = - GM(earth)*m(particle) / r
The negative sign shows that the particle is attracted by
the earth.
- The potential is - GM/(R+h) = -GM/r
As r increases,the potential & potential energy decreases.
It is zero at infinity.
- When h = 0,the potential energy = -GMm/R
( In the approx. calculations, P.E. = 0 at h = 0)
- For a spring, the potential energy = +1/2kx^2
where k is the spring constant & x is the extension
within the elastic limit.
- The P.E. of a spring or a mass oscillating in the simple
harmonic mode is always positive.
- For a mass oscillating in the simple harmonic mode, the
potential energy is +1/2mw^2x^2.
- Maximum P.E. = +1/2mw^2A^2.
- The sum of kinetic + potential energy = max K.E = max P.E.
because of conservation of energy.
Total energy = potential + kinetic
1/2mw^2A^2 = 1/2m(w)^2(x)^2 + 1/2mw^2(A^2 - x^2)
- Velocity of the particle making simple harmonic motion
= w√(A^2 - x^2)
At the equilibrium position, K.E. is max as x = 0,
P.E is max at the max displacement.
- In rotation, one does not discuss P.E.. The P.E is only due to
the position of the centre of mass.

- Work done :
For translation, W = F.dx
By work energy theorm (for all types of motion) =
= final K.E. - initial K.E.
- When the P.E + K.E is constant as in the caser of a body
falling freely from a height h,
work done = initial P.E - final P.E.
- For rotation, work done =τθ where τ is the torque applied
τ = αI where α is the angular acc.
- If rotation or circular motion is uniform, angular velocity
is a constant. The torque is not applied (neglecting friction) α = 0.
- Once rotation starts, in the case of uniform circular motion,
it continues without the necessity of a torque. The principle
of conservation of angular momentum is applicable. If there
is no external torque, rotation, rotation will continue in the
same way because angular momentum is conserved.
- In the same way, if there os no external force, a body in
uniform linear motion continues to move with the same
velocity because of the conservation of linear momentum.