9.2 The Second Condition for Equilibrium

Rotational equilibrium is crucial for understanding how objects remain stable and balanced. It requires both zero net force and zero net torque, ensuring an object doesn't move or rotate when forces are applied.

Torque, the rotational equivalent of force, plays a key role in rotational equilibrium. Calculated using lever arm, force magnitude, and angle, torque determines an object's tendency to rotate. Understanding torque is essential for analyzing stability in various systems.

Rotational Equilibrium and Torque

Second condition for equilibrium

• States net torque acting on an object must be zero for the object to be in static equilibrium
• Necessary for rotational equilibrium, while first condition (net force equals zero) necessary for translational equilibrium
• Ensures object does not rotate or accelerate angularly when in equilibrium
• Allows analysis of forces and moments acting on a system to determine stability and balance (crane, bridge)

Torque calculation and significance

• Torque ($\tau$) is rotational equivalent of force, causing object to rotate about an axis
• Calculate using formula: $\tau = rF\sin\theta$
  • $r$ is lever arm (distance from axis of rotation to point where force is applied)
  • $F$ is magnitude of force applied
  • $\theta$ is angle between force vector and lever arm
• Responsible for causing angular acceleration and rotational motion
• Magnitude determines rate of angular acceleration
• Direction (clockwise or counterclockwise) determines direction of rotation

Net torque in rotational equilibrium

• For an object to be in rotational equilibrium, net torque acting on it must be zero
  1. Calculate individual torques acting on object using torque formula
  2. Sum torques, considering signs (clockwise typically negative, counterclockwise positive)
• When analyzing objects in rotational equilibrium:
  1. Identify axis of rotation and forces acting on object
  2. Determine lever arm for each force
  3. Calculate individual torques and sum them to ensure net torque is zero
• Applications:
  • Balancing a see-saw or mobile
  • Analyzing stability of a crane or bridge
  • Determining forces required to maintain equilibrium in a system with multiple forces acting at different points (pulleys, gears)

Rotational Dynamics and Equilibrium

• Angular momentum is conserved in a system with no external torques
• Moment of inertia describes an object's resistance to rotational acceleration
• Newton's Second Law for Rotation relates torque to angular acceleration and moment of inertia
• A rigid body in equilibrium experiences no translational or rotational acceleration