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CISCEISC (Science) ISC Class 11
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Conservation of Mechanical Energy

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Topics

  • Conservation of mechanical energy
  • Principle of conservation of Energy
  • Conservative forces
  • Non-conservative forces

Notes

Conservative & Non-Conservative Forces

  • Conservative forces are those for which work done depends only on initial and final points.
    Example- Gravitational force, Electrostatic force.

  • Non-Conservative forces are those where the work done or the kinetic energy did depend on other factors such as the velocity or the particular path taken by the object.
    Example- Frictional force.


The Conservation of Mechanical Energy

For conservative forces the sum of kinetic and potential energies of any object remains constant throughout the motion.
According to the quantum physics, mass and energy are not conserved separately but are conserved as a single entity called ‘mass-energy’.

  • Mechanical Energy is the energy associated with the motion and position of an object.

  • The quantity K +V(x), is called the total mechanical energy of the system.

  • For a conservative force, ΔK = ΔW = F(x) Δx
    Also, - V(x) = F(x) Δx

  • This employs Δ(K+V) =0 for a conservative force.

  • Individually the kinetic energy K and the potential energy V(x) may vary from point to point, but the sum is a constant.

    Conservative Force:

  • A force F(x) is conservative if it can be derived from a scalar quantity V(x) by the relation  : F(x) = - dv/dx

  • The work done by the conservative force depends only on the end points.

  • A third definition states that the work done by this force in a closed path is zero.

  • The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.


The above discussion can be made more concrete by considering the example of the gravitational force once again and that of the spring force in the next section. Fig. below depicts a ball of mass m being dropped from a cliff of height H.


The total mechanical energies E0, Eh, and EH of the ball at the indicated heights zero (ground level), h and H, are
`E_H = mgH`
`E_h = mgh + 1/2 mv_h^2`
`E_0 = (1/2) mv_f^2`

The constant force is a special case of a spatially dependent force F(x). Hence, the mechanical energy is conserved. Thus
`E_H = E_0`
`mgH = 1/2 mv_f^2`
`v_f= sqrt(2gH)`
a result for a freely falling body.

Further,
`E_H=E_h`
which implies,
`v_h^2 = 2g (H-h)`
and is a familiar result from kinematics.

At the height H, the energy is purely potential. It is partially converted to kinetic at height h and is fully kinetic at ground level. This illustrates the conservation of mechanical energy.

Related QuestionsVIEW ALL [26]

A heavy stone is thrown in from a cliff of height h in a given direction. The speed with which it hits the ground
(a) must depend on the speed of projection
(b) must be larger than the speed of projection
(c) must be independent of the speed of projection
(d) may be smaller than the speed of projection.

You lift a suitcase from the floor and keep it on a table. The work done by you on the suitcase does not depend on

(a) the path taken by the suitcase
(b) the time taken by you in doing so
(c) the weight of the suitcase
(d) your weight

A body falls towards earth in air. Will its total mechanical energy be conserved during the fall? Justify.

A small heavy block is attached to the lower end of a light rod of length l which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle?

In the following figure shows two blocks A and B, each of mass of 320 g connected by a light string passing over a smooth light pulley. The horizontal surface on which the block Acan slide is smooth. Block A is attached to a spring of spring constant 40 N/m whose other end is fixed to a support 40 cm above the horizontal surface. Initially, the spring is vertical and unstretched when the system is released to move. Find the velocity of the block A at the instant it breaks off the surface below it. Take g = 10 m/s2.

A heavy stone is thrown from a cliff of height h with a speed v. The stoen will hit the ground with maximum speed if it is thrown 

A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated.

  1. How much work does she do against the gravitational force?
  2. Fat supplies 3.8 x 107J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?

One end of a spring of natural length h and spring constant k is fixed at the ground and the other is fitted with a smooth ring of mass m which is allowed to slide on a horizontal rod fixed at a height h (following figure). Initially, the spring makes an angle of 37° with the vertical when the system is released from rest. Find the speed of the ring when the spring becomes vertical.

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