Kinetic Energy and Work
- Kinetic Energy (K): The energy an object has due to its motion. Formula: \[ K = \frac{1}{2}mv^2 \] where \( m \) = mass and \( v \) = speed.
- Work (W): Done by a force on an object when it causes displacement. Formula for constant force: \[ W = \mathbf{F} \cdot \mathbf{d} \] (Dot product of force \( \mathbf{F} \) and displacement \( \mathbf{d} \)).
The Work-Energy Theorem
The change in kinetic energy of an object equals the total work done by all forces acting on it:
\[ K_{\text{final}} – K_{\text{initial}} = W_{\text{net}} \]Example: If a force speeds up or slows down an object, the work done directly affects its kinetic energy.
Example: Raindrop Falling
A raindrop (mass = 1.00 g) falls from 1.00 km height and hits the ground at 50.0 m/s.
- Work done by gravity (\( W_g \)): \[ W_g = mgh = (10^{-3} \, \text{kg})(10 \, \text{m/s}^2)(10^3 \, \text{m}) = 10.0 \, \text{J} \] (Gravity does positive work, speeding up the drop).
- Work done by resistive force (\( W_r \)):
Using the work-energy theorem: \[ \Delta K = W_g + W_r \] \[ 1.25 \, \text{J} = 10.0 \, \text{J} + W_r \] \[ W_r = -8.75 \, \text{J} \] (Negative work means the resistive force opposes motion).
Key Points About Work
- Work depends on the force component in the direction of displacement.
- If force and displacement are perpendicular, no work is done.
- Work can be positive (force aids motion) or negative (force resists motion).
Important Concepts for NEET
- Work-Energy Theorem: \( \Delta K = W_{\text{net}} \) is a core concept for solving energy-related problems.
- Work Done by Gravity: \( W_g = mgh \) frequently appears in free-fall or inclined plane problems.
- Resistive Forces: Calculating work done by forces like air resistance (often negative).
- Kinetic Energy Change: Relating speed changes to work done by forces.
