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WORK ENERGY AND POWER

P. H. GAJBHIYE

PGT PHYSICS JNV GONDIA

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WORK

  • In physics, the definition of work is the application of a force through a distance

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W = F·d

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  • W is the work done
  • F is the force applied
  • d is the distance through which the force acts
  • Only the force that acts in the direction of motion counts towards work

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WORK AS SCALAR PRODUCT

  • WORK IS SCALAR PRODUCT OF TWO VECTORS
  • If under a constant force F the object displaced through a distance d, then work done by the force is given by
  • W = F * d = F d cos θ

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Work as a dot product of force F and displacement d

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�Positive, Negative, and Zero Work

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�Graphical Representation of the Work Done �by a Constant Force

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Work done by variable force

  • Work done by a variable force is given by
  • W = ∫ F * ds
  • It is equal to the area under the force-displacement graph along with proper sign.
  • Work done = Area ABCDA

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�Work Done by a Non-Constant Force

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�Work Done by a Continuously Varying Force

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Work done by system of forces

  • Work done in displacing any body under the action of a number of forces is equal to the work done by the resultant force.
  • In equilibrium (static or dynamic), the resultant force is zero therefore resultant work done is zero.

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Conservative Non-conservative forces

  • If work done by a force during a rough trip of a system is zero, then the force is conservative, otherwise it is called non-conservative force.
  • Gravitational force, electrostatic force, magnetic force, etc are conservative forces.
  • All the central forces are conservative forces.
  • Frictional force, viscous force, etc are non-conservative forces.

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ENERGY

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ENERGY & ITS UNIT

  • Energy of a body is its capacity of doing work.
  • It is a scalar quantity.
  • Its SI unit is joule and CGS unit is erg. Its dimensional formula is [ML2T-2].

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FORMS OF ENERGY

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MECHANICAL ENERGY

KINETIC ENERGY

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POTENTIAL ENERGY

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Kinetic Energy

  • Kinetic Energy: the energy of motion
  • Moving things carry energy in the amount:

K.E. = ½mv2

  • Note the v2 dependence—this is why:
    • a car at 60 m/s is 4 times more dangerous than a car at 30 m/s
    • hurricane-force winds at 100 m/s are much more destructive (4 times) than 50 m/s gale-force winds
    • a bullet shot from a gun is at least 100 times as destructive as a thrown bullet, even if you can throw it a tenth as fast as you could shoot it

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UCSD: Physics 8; 2005

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Spring 2005

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Numerical examples of kinetic energy

  • A baseball (mass is 0.145 kg = 145 g) moving at 30 m/s has kinetic energy:

K.E. = ½×(0.145 kg)×(30 m/s)2

= 65.25 kg·m2/s2 ≈ 65 J

  • A quarter (mass = 0.00567 kg = 5.67 g) flipped about four feet into the air has a speed on reaching your hand of about 5 m/s. The kinetic energy is:

K.E. = ½×(0.00567 kg)×(5 m/s)2

= 0.07 kg·m2/s2 = 0.07 J

UCSD: Physics 8; 2005

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Spring 2005

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POTENTIAL ENERGY

GRAVITATIONAL PE

ELASTIC PE

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PE OF SPRING MASS SYSTEM�

F = -K Δx

How much work is done to move the block?

Be careful with the signs of the forces!

The force to move the block must be equal and opposite to the force of the spring on the block.

F = K Δx

Force

Distance

X

F=Kx

W = ½ F X

W = ½ K X2

Work done to move a mass on a spring a distance “x”:

This work is stored in the form of potential energy of spring

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WORK ENERGY THEOREM FOR CONSTANT FORCE

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WORK ENERGY THEOREM FOR VARIABLE FORCE

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MASS ENERGY EQUIVALENCE

  • ACCORDING TO EINSTEINS MASS ENERGY RELATION ENERGY AN BE CONVERTED TO MASS AND MASS CAN BE CONVERTED TO ENERGY.

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POWER

  • The time rate of work done by a body is called its power.
  • Power = Rate of doing work = Work done / Time taken
  • If under a constant force F a body is displaced through a distance s in time t, the power
  • p = W / t = F * s / t
  • But s / t = v ; uniform velocity with which body is displaced.
  • ∴ P = F * v = F v cos θ
  • where θ is the smaller angle between F and v.
  • power is a scalar quantity. Its S1 unit is watt and its dimensional formula is [ML2T-3].
  • Its other units are kilowatt and horse power,
  • 1 kilowatt = 1000 watt
  • 1 horse power = 746 watt

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COLLISIONS

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TYPES OF COLLISION

  • Collision between two or more particles is the interaction for a short interval of time in which they apply relatively strong forces on each other.
  • In a collision physical contact of two bodies is not necessary. There are two types of collisions:

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TYPES OF COLLISION

  • Elastic collision
  • The collision in which both the momentum and the kinetic energy of the system remains conserved are called elastic collisions.
  • In an elastic collision all the involved forces are conservative forces.
  • Total energy remains conserved.
  • Inelastic collision
  • The collision in which only the momentum remains conserved but kinetic energy does not remain conserved are called inelastic collisions.
  • In an inelastic collision some or all the involved forces are non-conservative forces.
  • Total energy of the system remains conserved.
  • If after the collision two bodies stick to each other, then the collision is said to be perfectly inelastic

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Applying Newton’s experimental law, we have

Velocities after collision

v1 = (m1 – m2) u1 + 2m2u2 / (m1 + m2)

v2 = (m2 – m1) u2 + 2m1u1 / (m1 + m2)

When masses of two colliding bodies are equal, then after the collision, the bodies exchange

their velocities.

v1 = u2 and v2 = u1

If second body of same mass (m1 = m2) is at rest, then after collision first body comes to rest

and second body starts moving with the

initial velocity of first body.

v1 = 0 and v2 = u1

If a light body of mass m1 collides with a very heavy body of mass m2 at rest, then after collision.

v1 = – u1 and v2 = 0

It means light body will rebound with its own velocity and heavy body will continue to be at

rest.

If a very heavy body of mass m1 collides with a light body of mass m2(m1 > > m21) at rest, then

after collision

v1 = u1 and v2 = 2u1

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Inelastic One Dimensional Collision

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Elastic collision in one dimension

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In horizontal direction,

m1u1 cos α1 + m2u2 cos α2= m1v1 cos β1 + m2v2 cos β2

In vertical direction.

m1u1 sin α1 – m2u2 sin α2 = m1u1 sin β1 – m2u2 sin β2

If m1 = m2 and α1 + α2 = 90°

then β1 + β2 = 90°

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If the initial and final velocities of colliding bodies do not lie along the same line, then the collision is called two dimensional or oblique Collision.

Two Dimensional or Oblique Collision