Kinetics of Rigid Bodies: Newton's Second Law

Why it matters

Understanding the kinetics of rigid bodies using Newton's Second Law is crucial for designing and analyzing structures and mechanical systems. It helps engineers predict how structures will respond to various forces, ensuring safety and functionality in real-world applications.

Key ideas

• Rigid Body: An idealization where deformation is neglected, and the body does not change shape under applied forces.
• Newton's Second Law for Rigid Bodies: Extends the concept from particles to rigid bodies, considering both translational and rotational motion.
• Equations of Motion: For a rigid body, the equations of motion include both linear and angular components.
  • Translational Motion: Governed by F = m·a, where F is the net force, m is the mass, and a is the acceleration.
  • Rotational Motion: Governed by τ = I·α, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration.
• Free Body Diagram (FBD): A crucial tool for visualizing forces and moments acting on a body.

Formulas

For planar motion use ΣF = m a_G and ΣM_G = I_G α about the center of mass G. About a fixed rotation axis O, use ΣM_O = I_O α. The scalar equation is not a general arbitrary-axis three-dimensional relation.

• F = m·a
  • F: Net force (N)
  • m: Mass (kg)
  • a: Acceleration (m/s²)
• τ = I·α
  • τ: Net torque (N·m)
  • I: Moment of inertia (kg·m²)
  • α: Angular acceleration (rad/s²)

Worked example

Given: A body rotates about a fixed axis O with moment of inertia I_O = 5 kg·m². A 50 N force acts with perpendicular moment arm 2 m about O, and no other torque acts about that axis. Find angular acceleration.

1. Calculate the torque using τ = F·r.
   • τ = 50 N · 2 m = 100 N·m
2. Use the rotational motion formula τ = I·α to find α.
   • 100 N·m = 5 kg·m² · α
   • α = 100 N·m / 5 kg·m² = 20 rad/s²

Final Answer: 20 rad/s²

Common mistakes

• Confusing mass with weight; remember weight is a force (W = m·g).
• Neglecting rotational effects when analyzing rigid bodies.
• Incorrectly drawing or interpreting Free Body Diagrams.

For GATE CE

• Focus on problems involving both translational and rotational motion.
• Practice drawing Free Body Diagrams and applying equations of motion.
• Be prepared for questions that integrate concepts from other topics like friction and equilibrium.

Quick check

1. What is the formula for torque in terms of force and distance?
2. How does Newton's Second Law apply to rotational motion?
3. What is the unit of moment of inertia?

Answers: 1. τ = F·r when r is the perpendicular moment arm 2. τ = I·α 3. kg·m²