Newton's second law, which you're familiar with as F = ma, can be extended. Similarly, there is an equation for rotational dynamics. Understanding these concepts helps in real-world situations, like figure skating or spinning wheels.

• Momentum and Impulse
• Newton's Second Law Extended
• Simple: F = ma
• Complex: FΔt = Δ(mv)
  • This change in equation connects momentum change and impulse.
• Newton’s Second Law in Rotational Terms
• Equation: τ = I Δω Δt + ω ΔI Δt
  • This means torque (τ) is needed to
  • a. Change the angular speed of an object with a steady moment of inertia. Example - Accelerating a spinning toy top.
  • b. Keep a constant speed for an object with changing moment of inertia. Example - Ever watched figure skating during the Olympics? When a skater pulls in their arms, they spin faster. This is because of the change in their moment of inertia.
• Worked Example: The Sandy Disc Dilemma 🏖️
• Scenario: Sand's pouring on a spinning disc. Let's visualize it like adding toppings on a rotating pizza.
• Given
  • Rate of sand = 8.0gs^-1
  • Distance from center = 10cm
  • Moment of inertia of disc = 0.040kgm²
  • Initial angular speed = 5.0rads^-1

No external forces messing with our spinning pizza. (I mean, disc!)

Calculations

• a. Rate of Change of Moment of Inertia: ΔI/Δt = Δm/Δt * R² = 0.0080kgs^-1 * (0.10m)² = 8.0 × 10^-5 kgm² s^-1
• b. Initial Angular Acceleration: Using the equation: α = Δω/Δt = -ω/I * ΔI/Δt = -5.0/0.040 × 8.0 × 10^-5 = -0.010rads^-2 This means our disc (or pizza) slows down as more sand is added.
• c. Constant Acceleration?: Nope! As the disc gathers more sand (more toppings!), it slows down, and its moment of inertia goes up. So, the rate at which it's slowing down (angular acceleration) also changes.

The moment of inertia is like the "resistance to change in rotation" – the more it is, the harder it's to change the object's spinning speed.