Key Concepts in Mechanics
1. Banked Roads and Circular Motion
When a car moves on a banked road, the normal force (N) and friction help provide the centripetal force needed for turning. The maximum speed (v_{\text{max}}) to avoid slipping is:
\[ v_{\text{max}} = \sqrt{ Rg \frac{\mu_s + \tan \theta}{1 – \mu_s \tan \theta} } \]
The optimum speed (v_o), where no friction is needed, is:
\[ v_o = \sqrt{ Rg \tan \theta } \]
Example: For a banked road with R = 300 \, \text{m} and \theta = 15^\circ, v_o = 28.1 \, \text{m/s}.
2. Frictional Force in Circular Motion
Friction provides the centripetal force on flat roads. The condition to avoid slipping is:
\[ v^2 \leq \mu_s R g \]
Example: A cyclist moving at 18 \, \text{km/h} (5 \, \text{m/s}) on a turn of radius 3 \, \text{m} will slip because v^2 = 25 \, \text{m}^2/\text{s}^2 exceeds \mu_s R g = 2.94 \, \text{m}^2/\text{s}^2.
3. Free-Body Diagrams (FBDs)
FBDs help visualize forces acting on a system. Steps to draw them:
- Draw all external forces (e.g., gravity, normal force, friction).
- Ignore internal forces between parts of the system.
- Apply Newton’s laws to solve unknowns.
Example: A block + cylinder system accelerating downward has two forces in its FBD: gravity (270 \, \text{N}) and normal force (R’).
4. Newton’s Laws of Motion
- First Law: An object remains at rest or in uniform motion unless acted upon by a net force.
- Second Law: F = ma (force equals mass × acceleration).
- Third Law: Every action has an equal and opposite reaction. Action-reaction pairs act on different bodies.
Example: The block exerts a 20 \, \text{N} force on the floor (action), and the floor exerts a 20 \, \text{N} normal force (reaction).
5. Problem-Solving Steps
- Draw a diagram of the system.
- Choose a part of the system to analyze.
- Draw an FBD showing all external forces.
- Apply Newton’s laws and solve equations.
- Use action-reaction pairs when switching between systems.
Important Concepts for NEET
- Banked Roads: Calculate v_{\text{max}} and v_o using friction and banking angle.
- Friction in Circular Motion: Condition v^2 \leq \mu_s Rg for no slipping.
- Free-Body Diagrams: Critical for visualizing forces and applying Newton’s laws.
- Newton’s Third Law: Identify action-reaction pairs (e.g., block-floor interactions).
- Problem-Solving Framework: Systematic approach to analyze forces and motion.
