Work and Energy

Work

Work is a scalar quantity that has only magnitude but not direction. Work is said to be done when energy is expended.

Work is the product of the force and displacement. Work has unit Joule.

Energy

Energy is the capability of an object to do work. The types of Energy are:


The unit of Energy is the Joule

The order of vectors is then given by: O = Vector(A) × Vector(B) x cos(Angle between them)
O = Vector(A) × Vector(B) × cos(θ) Work done = f × cosθ

Work done by an external force on a body is the product of the force and the displacement x, provided that the force is in the direction of the displacement.
w = f.s

If the force acts at an angle θ to the displacement, the work done is the resolved component of the force along the direction of the displacement multiplied by the displacement.

Fcosθ

The graph of the force against the displacement

Hence, the workdone is defined as the scalar product of force F that moves its point of application through a displacement S.

Examples

  1. A body moves through a distance S = 5i + 2j - 3k from the origin (0, 0, 0) under the influence of a constant force F = 4i + 3j - 2k. Determine

    (a) the work done
    (b) the angle between F and S

    Solution

    (a) Workdone = FS
    Displacement = Final Reaction - Initial Position
    = 5i + 2j - 3k - [0,0,0]
    = 5i + 2j - 3k

    (b)

    `abs(F) = sqrt(4^2 + 3^2 + -2^2) = sqrt(29) = 5.4N`

    `abs(S) = sqrt(5^2 + 2^2 + -3^2) = sqrt(38) = 6.16m`

    `W = abs(F) abs(S) cos theta`

    `cos theta = W / (abs(F) abs(S))`

    `theta = cos^-1 (W /(abs(F) abs(S)))`

    `theta = cos^-1 (32 / (5.4 times 6.16)) = 15.8^O`

  2. A force F = (6i - 2j)N acts on a particle that undergoes a displacement Δr = (3i + j)m. Find
    (a) the work done by the force on the particle
    (b) the angle between F and Δr 

    Solution

    (a)

    `W.D = Fxx + Fyy`

    `W.D = (6 × 3) + (-2 × 1)`

    `W.D = 18 + (-2)`

    `W.D = 18 - 2`

    `W.D = 16J`

    (b)

    `θ = cos^-1 ((F Delta r)/(abs(F)abs(Delta r)))`

    `θ = cos^-1(16 / sqrt((6^2 + 2^2) + (3^2 + 1^2)))`

    `θ = cos^-1 (16 / (40 + 10))`

    `θ = cos^-1 (16 / 50)`

    `θ = cos^-1 0.32`

    `θ = 71.34^o`

Mechanical Energy

Mechanical Energy is the energy possessed by a body due to its virtue of position or its velocity.

The types of Mechanical Energy are:

  1. Kinetic Energy

    The kinetic energy of a body is the energy possessed by the body by virtue of its motion. A solid body, a bullet fired from the rifle, and a swinging pendulum all possess kinetic energy

    A body is capable of doing work, but in the process of doing this work, its velocity gradually decreases. The amount of work that can be done depends both on the magnitude of the velocity and the mass of the body.

    A heavy bullet will penetrate a wooden plank deeper than a light bullet of equal size moving with equal velocity.

    Let us consider a body of mass m moving at a velocity B in a straight line. Suppose that it is acted upon by a resistant force F resisting its motion, which produces a retardation A, then force F is the product of the mass and the retardation.

    `F = -ma`eq 5

    Let dx be the displacement of the body before it comes to rest. But retardation a is expressed as `a = (dv)/(dt)`. This can be simplified as `(dv)/(dx) times (dx) / (dt)`

    velocity = `(dx)/(dt)` acceleration = `(dv) / (dx) + (dv) / x`eq 6

    If we substitute equation 6 from equation 5, `F = -mv times (dv) / (dx)`eq 7 hence, the work done in bringing the body to rest is given by: `W = int F dx = -int mv (dv)/(dx)` `= int -mv dv`eq 8 `w = -F((v^2)/(2))` `lim_(v->0) = wo - wv`` = 1/2 mv^2`eq 9 `= 0 - (-v)`

    This workdone is equal to the kinetic energy of the body. `Ek` `= 1/2 mv^2`eq 10

  2. Potential Energy

    This is the energy stored in a body by virtue of its position, its state of strain.
    Examples of bodies possessing potential energy are

    • Water stored in a reservoir
    • A warmed spring
    • A stretched rubber cord
    Potential energy is given by the amount of work done by the force acting on the body when the body moves from its given position to another position.

    Expression of Potential Energy

    Let us consider a body of mass m which is at rest at a height H above the ground. The work done in raising the body from the ground to height H is stored in the body as its potential energy. When the body falls to the ground, the same amount of work can be got back from it. Therefore in order to lift the body vertically up, a force mg = W equal to the weight of the body should be applied. When the body istaken vertically up through thr height H, then work done W = Force × Displacement W = mg × h This work done is stored as the potential energy of the body which is measured in Joules. i.e Ep = mgh eq 4