Lecture on Torque

Introduction to Torque

• Torque is the rotational equivalent of force.
• It is not exactly a force but a rotational kind of equivalent.
• Torque Equation: Torque ( \tau = F \times R \times \sin(\theta) )
  • ( F ): Applied force
  • ( R ): Distance from the axis of rotation
  • ( \theta ): Angle between the applied force and R (measured in degrees)

Key Concepts

• Center of Mass: Often used for solving torque problems, consider the center of mass as a point.
• 90 Degree Angle: If ( \theta = 90° ), then ( \sin(90°) = 1 ), simplifying ( \tau = F \times R ).

Newton's Laws and Torque

• Newton's First Law (Rotational Version): An object in rotational motion stays in motion unless acted upon by a net torque.
• Rotational Equilibrium: No net torque means no angular acceleration.
• Newton's Second Law (Rotational): ( \tau = I \times \alpha )
  • ( I ): Moment of inertia
  • ( \alpha ): Angular acceleration

Units and Measurements

• Torque Units: Newton-meters (N·m)
• Moment of Inertia: ( \text{kg} \times \text{m}^2 )
• Angular Acceleration Units: Radians per second squared (rad/s²)

Right-Hand Rule for Torque

• Use right-hand rule to determine the direction of torque.
• Right-Hand Rule: Fingers point in direction of R, curl towards F, thumb points in direction of torque.

Practical Application: Example Problem

1. Example Setup: Uniform metal rod pivoted, supported frictionlessly.

   • Objective (Part A): Find the force exerted by support.
   • Objective (Part B): Calculate initial angular acceleration after support removal.

2. Solution Steps (Part A):

   • Consider torques due to gravity and support.
   • Use center of mass to calculate torque: ( \tau = F \times R ) (for equilibrium)
   • Equilibrium implies ( F_1 \times R_1 = F_2 \times R_2 )
   • Calculate force exerted by support: 24 Newtons.

3. Solution Steps (Part B):

   • Remove support, calculate torque as ( \tau = F \times R ) using center of mass.
   • Use ( \tau = I \times \alpha ) to find angular acceleration.
   • Calculation yields angular acceleration: ( \alpha = 4.0 \text{ rad/s}^2 ).

Conclusion

• Understanding torque involves recognizing its non-linear nature and applying Newton's laws to rotational systems.
• Problems often require calculations using moment of inertia and recognizing equilibrium conditions.