Newton’s Second Law of Motion
Key Idea
Force is directly related to how fast an object’s momentum changes. The bigger the force, the faster the change in momentum.
Momentum
- Momentum (\( \mathbf{p} \)) = mass (\( m \)) × velocity (\( \mathbf{v} \)):
\( \mathbf{p} = m \mathbf{v} \) - Examples:
- A heavy truck needs more force to stop than a light car moving at the same speed.
- A faster bullet causes more damage because it has greater momentum.
Newton’s Second Law Formula
- Force (\( \mathbf{F} \)) equals the rate of change of momentum:
\( \mathbf{F} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} \) - For constant mass, this simplifies to:
\( \mathbf{F} = m \mathbf{a} \) (where \( \mathbf{a} \) = acceleration)
Important Notes
- Force is a vector: It has direction. For example, gravity changes the vertical velocity of a ball, but not its horizontal velocity.
- Force depends on “here and now”: Acceleration at any moment depends only on the force acting at that moment.
- Internal forces cancel out: When considering a system (like two colliding balls), internal forces don’t affect the system’s overall motion.
Impulse
- Impulse = Force × Time = Change in momentum.
- Example: A cricket ball caught by pulling hands backward reduces force by increasing time of contact.
- Formula:
\( \text{Impulse} = \mathbf{F} \Delta t = \Delta \mathbf{p} \)
Examples
- Bullet Stopping in a Block:
- Mass = 0.04 kg, initial speed = 90 m/s, distance = 0.6 m.
- Retardation: \( a = \frac{-90^2}{2 \times 0.6} = -6750 \, \text{m/s}^2 \).
- Force: \( F = 0.04 \times 6750 = 270 \, \text{N} \).
- Impulse in Cricket Ball:
- Ball mass = 0.15 kg, speed = 12 m/s.
- Impulse = \( 0.15 \times 12 – (-0.15 \times 12) = 3.6 \, \text{N s} \).
Important Concepts for NEET
- \( \mathbf{F} = m \mathbf{a} \): Force-mass-acceleration relationship.
- Impulse-momentum theorem: Calculating force or time using momentum change.
- Vector nature of force: Direction matters in force equations.
- Action-reaction pairs (Third Law): Forces always occur in equal, opposite pairs.
