WPE

UMMEED NEET: Work, Energy and Power Study Notes

1. PhD on Work

• Definition of Work: Work is defined as the product of force and displacement in the direction of the force.

  $W = F imes s imes ext{cos}( heta)$
  where,

  • $W$ = Work done,

  • $F$ = Magnitude of the force applied,

  • $s$ = Displacement of the object,

  • $ heta$ = Angle between the force and displacement vector.

• Types of Work done by Forces:

  • Positive Work: Occurs when the displacement is in the direction of the force ($0 < heta < 90°$).

  • Negative Work: Occurs when the displacement is opposite to the direction of the force ($90° < heta < 180°$).

  • Zero Work: Occurs when the force does not cause any displacement ($ heta = 90°$).

2. Work-Energy Theorem

• The work-energy theorem states that the work done by the forces acting on an object results in a change in its kinetic energy.

  $W = rac{1}{2}mv^2_{final} - rac{1}{2}mv^2_{initial}$
  where:

  • $m$ = mass of the object,

  • $v$ = velocity of the object.

3. Potential Energy

• Definition: It is the energy stored in an object due to its position in a force field (e.g., gravitational field).

  • Gravitational Potential Energy:
    $PE = mgh$
    where:

  • $m$ = mass,

  • $g$ = acceleration due to gravity,

  • $h$ = height above a reference point.

4. Conservation of Mechanical Energy

• The mechanical energy (sum of kinetic and potential energy) of an object in the absence of non-conservative forces (like friction) remains constant.

  $E_{total} = KE + PE$

5. Work Done by Non-Conservative Forces

• Non-conservative forces (like friction) do work that depends on the path taken, and they change the total mechanical energy of the system.

6. Work Done by Gravitational Forces

• Work done by gravitational forces is path-independent. The work done depends on the height change and is calculated as:

  $W = mgh$
  (when lifting or lowering an object).

7. Work Done by Springs

• Work done on a spring is calculated by the spring constant and the displacement from its relaxed position:

  $W = rac{1}{2}k x^2$
  where:

  • $k$ = spring constant,

  • $x$ = displacement.

8. Power

• Definition: Power is the rate at which work is done.

  $P = rac{W}{t}$
  where:

  • $P$ = Power,

  • $W$ = Work done,

  • $t$ = Time taken to do the work.

• Power can also be calculated using the instantaneous power formula:

  $P = F imes v$
  where:

  • $F$ = Force,

  • $v$ = Velocity.

9. Numerical Problems

• Sample Question: A particle moves from a point $(-2 extbf{i} + 5 extbf{j})$ to $(4 extbf{i} + 3 extbf{j})$. The force applied is $(4 extbf{i} + 3 extbf{j})$ N. Calculate work done.

  • Solution:
    $W = F imes s = (4 extbf{i} + 3 extbf{j}) imes [(4 + 2) extbf{i} + (3 - 5) extbf{j}]$
    Simplifying gives the work done.

• Sample Question: A force of 10 N displaces an object by 10 m. If work done is 50 J, find the angle between the force and displacement.

  $W = F imes d imes ext{cos}( heta)$
  Rearranging gives you $ heta$.

Additional Questions and Practical Applications

1. Understanding of concepts like friction, normal force, and their role in work calculations.

2. Exploration of how energy conservation principles apply to real-world scenarios like roller coasters, oscillations, and pendulums.

3. Addressing real-world applications: e.g., calculating work done in lifting objects, running engines or pumps.

10. Important Concepts in Circular Motion

• In vertical circular motion, the forces and energies change as the object moves due to gravity, tension in the string, etc.

• Understanding tension changes and forces exerted at different points in the circle, especially at the highest and lowest points.

11. Final Remarks

• Review and practice problems often. Relevant numerical problems illustrate the theoretical concepts thoroughly.