Overview
This lecture covered the fundamentals of work, energy, and power in physics, including key formulas, problem-solving techniques, and conceptual understanding relevant for competitive exams.
Work, Force, and Displacement
- Work is the dot product of force and displacement: ( W = \vec{F} \cdot \vec{S} ).
- Work depends on the displacement at the point where force acts.
- Work is frame-dependent, scalar, and can be positive, negative, or zero.
- Positive work: kinetic energy increases; zero work: kinetic energy constant; negative work: kinetic energy decreases.
- The angle between force and displacement determines sign of work (0–90°: positive, 90°: zero, 90–180°: negative).
Kinetic Energy, Momentum, Work-Energy Theorem
- Change in momentum (( \Delta p )) = force × time.
- ( KE = \frac{p^2}{2m} ).
- Change in kinetic energy equals work done (( \Delta KE = W )).
- For constant force and displacement: ( W = F \cdot S ).
- Work done is required to change kinetic energy.
Problem Solving and Examples
- For variable forces, integrate ( F ) with respect to displacement.
- Work by gravity: ( W_{grav} = mgh ).
- Work-energy theorem: total work by all forces = change in kinetic energy.
- Work by conservative force: ( W_{cons} = -\Delta U ); by non-conservative force: ( W_{non-cons} = \Delta U ) (only when ( \Delta KE = 0 )).
- Conservative force work is path-independent; non-conservative is path-dependent.
Conservation of Energy and Potential Energy
- Mechanical energy conserved if no non-conservative work is done: ( KE_i + PE_i = KE_f + PE_f ).
- Potential energy defined only for conservative forces.
- For springs, potential energy ( U = \frac{1}{2}kx^2 ).
- When a spring is cut, spring constant is inversely proportional to the new length.
Momentum and Kinetic Energy Links
- With equal kinetic energies, momentum ratio of masses (( m_1, m_2 )): ratio is ( \sqrt{\frac{m_1}{m_2}} ).
- Kinetic energy increased by ( n)%: momentum increases by ( \sqrt{n+1}-1 ) times original percentage.
Vertical Circular Motion
- Conservation of energy applies: ( V^2 = V_0^2 + 2gL(\cos\theta - 1) ).
- Tension at any angle involves gravitational and centripetal terms.
- Minimum velocity to complete loop: ( V_{min} = \sqrt{5gR} ) for string, ( \sqrt{4gR} ) for rod.
Power
- Power is the rate of doing work: ( P = \frac{dW}{dt} ).
- Instantaneous power: ( P = \vec{F} \cdot \vec{v} ).
- Average power: total work divided by total time.
- Power in engines/pumps proportional to ( v^3 ); force to ( v^2 ).
- Power can be positive (energy supplied) or negative (energy removed).
Key Terms & Definitions
- Work — Transfer of energy via force acting over displacement (( W = \vec{F} \cdot \vec{S} )).
- Kinetic Energy (KE) — Energy due to motion (( KE = \frac{1}{2}mv^2 )).
- Momentum (p) — Product of mass and velocity (( p = mv )).
- Potential Energy (PE/U) — Stored energy due to position, for conservative forces.
- Conservative Force — Force whose work is path-independent (e.g., gravity).
- Non-Conservative Force — Force whose work depends on path (e.g., friction).
- Power (P) — Rate of energy transfer or work done over time.
Action Items / Next Steps
- Practice solving NEET/JEE previous year problems on work, energy, and power.
- Review and memorize key formulas and special case derivations.
- Complete assigned textbook problems on energy conservation and vertical circular motion.