Kinetics of Rigid Bodies: Impulse and Momentum

Why it matters

Understanding the impulse and momentum of rigid bodies is essential in civil engineering for analyzing the effects of dynamic loads, such as wind or seismic forces, on structures. This knowledge helps in designing structures that can withstand sudden forces and prevent catastrophic failures.

Key ideas

• Impulse: Impulse is the time integral of force, reducing to FΔt for constant force. It changes the momentum of a body.
• Momentum: Momentum is the product of mass and velocity of a body. It is a vector quantity and is conserved in isolated systems.
• Impulse-Momentum Principle: This principle states that the change in momentum of a body is equal to the impulse applied to it.
• Conservation of Momentum: In the absence of external forces, the total momentum of a system remains constant.

Formulas

• Impulse: I = F·Δt
  • I = Impulse (N·s)
  • F = Force (N)
  • Δt = Time duration (s)
• Momentum: p = m·v
  • p = Momentum (kg·m/s)
  • m = Mass (kg)
  • v = Velocity (m/s)
• Impulse-Momentum Principle: I = Δp
  • Δp = Change in momentum (kg·m/s)

For planar rigid-body rotation, ∫ΣM_G dt = I_G(ω₂-ω₁), with constant I_G. Use center-of-mass velocity in linear momentum p = m v_G. A body may conserve linear momentum while an external couple changes its angular momentum.

Worked example

Given: A 10 kg block is moving with a velocity of 5 m/s. A constant net force of 20 N is applied through its center of mass in its initial direction of motion for 3 seconds.

1. Calculate the initial momentum.
   • p_initial = m·v = 10 kg · 5 m/s = 50 kg·m/s
2. Calculate the impulse.
   • I = F·Δt = 20 N · 3 s = 60 N·s
3. Apply the impulse-momentum principle to find the final momentum.
   • I = Δp = p_final - p_initial
   • 60 N·s = p_final - 50 kg·m/s
   • p_final = 110 kg·m/s
4. Calculate the final velocity.
   • p_final = m·v_final
   • 110 kg·m/s = 10 kg · v_final
   • v_final = 11 m/s

Final Answer: The final velocity of the block is 11 m/s.

Common mistakes

• Confusing impulse with force; impulse is force applied over time.
• Forgetting that momentum is a vector and has direction.
• Ignoring external forces when applying conservation of momentum.

For GATE CE

Questions often involve calculating the final velocity or momentum of a body after an impulse is applied. Practice problems involving both linear and angular momentum, and understand how to apply the impulse-momentum principle in different scenarios.

Quick check

1. What is the unit of impulse?
2. How does impulse affect momentum?
3. What principle relates impulse and momentum?

Answers: 1. N·s 2. Impulse changes momentum 3. Impulse-Momentum Principle