Lecture Notes: Calculating Torque on a Rotating Sphere

Problem Statement

• Calculate the torque acting on a system.
• Given:
  • Mass (m): 1.85 kg
  • Diameter: 45 cm
  • Angular Velocity (ω): 2.4 revolutions/second
  • Total Angle: 18.2 revolutions

Key Concepts

Conversion and Calculation

• Radius Calculation:

  • Diameter = 45 cm
  • Radius (r) = Diameter / 2 = 22.5 cm = 0.225 m (converted to meters)

• Angular Velocity Conversion:

  • Given ω = 2.4 revolutions/second
  • 1 revolution = 2π radians
  • ω in radians/second = 2.4 × 2π = 15.1 radians/second

• Total Angle Calculation:

  • Given 18.2 revolutions
  • Total angle in radians = 18.2 × 2π = 114.3 radians

Rotational Dynamics

• Equation for Rotational Motion:

  • Similar to linear motion (v² = u² + 2as), but adapted for rotation:
    • ω² = ω₀² + 2αθ
    • Where:
      • ω is final angular velocity
      • ω₀ is initial angular velocity (given)
      • α is angular acceleration
      • θ is the total angle

• Given:

  • Final velocity (ω) = 0 as the system comes to a stop
  • Initial angular velocity (ω₀) = 15.1 radians/second
  • Total angle (θ) = 114.3 radians

• Solving for Angular Acceleration (α):

  • From ω² = ω₀² + 2αθ, rearrange to solve for α
  • α = -1 radian/second² (negative indicates deceleration)

Moment of Inertia

• For a Solid Sphere:
  • Moment of Inertia (I) = (2/5)mr²
  • Mass (m) = 1.85 kg
  • Radius (r) = 0.225 m
  • Calculated I = 0.037 kg·m²

Calculating Torque

• Torque (τ) Formula:
  • τ = Iα
  • Plug in values: I = 0.037 kg·m², α = -1 radian/second²
  • Torque magnitude |τ| = 0.037 N·m

Conclusion

• Negative sign in α and τ indicates the system is slowing down (decelerating). This means the angular velocity is decreasing as the sphere comes to rest.