Torque Equations

Torque is all about how forces cause rotation. These equations help us understand the relationship between force, distance, and rotation, making them essential for solving problems in AP Physics 1. Mastering torque is key to grasping rotational motion concepts.

1. τ = r × F (torque equals the cross product of position vector and force)

   • Torque (τ) is a vector quantity that measures the tendency of a force to rotate an object about an axis.
   • The position vector (r) is the distance from the axis of rotation to the point where the force is applied.
   • The direction of torque is determined by the right-hand rule, indicating the axis of rotation.

2. τ = r F sin θ (scalar form of torque equation)

   • This equation calculates the magnitude of torque using the angle (θ) between the position vector and the force vector.
   • The sine function accounts for the effective component of the force that contributes to rotation.
   • Maximum torque occurs when the force is applied perpendicular to the position vector (θ = 90°).

3. τ = I α (torque equals moment of inertia times angular acceleration)

   • Moment of inertia (I) represents an object's resistance to changes in its rotational motion.
   • Angular acceleration (α) is the rate of change of angular velocity.
   • This equation shows the relationship between torque and the rotational motion of an object.

4. Στ = 0 (sum of torques equals zero for rotational equilibrium)

   • In a system at rotational equilibrium, the total torque acting on the object is zero.
   • This condition implies that the object is either at rest or moving with constant angular velocity.
   • Analyzing torques helps determine unknown forces or distances in equilibrium problems.

5. τ = F d (torque equals force times perpendicular distance)

   • The perpendicular distance (d) is the shortest distance from the axis of rotation to the line of action of the force.
   • This equation emphasizes that torque is maximized when the force is applied at a right angle to the lever arm.
   • Understanding this relationship is crucial for solving problems involving levers and rotational systems.

6. W = τ θ (work done by torque equals torque times angular displacement)

   • Work (W) done by torque is the product of torque and the angular displacement (θ) in radians.
   • This equation highlights the energy transfer involved in rotational motion.
   • It is essential for understanding how torque contributes to the work done in rotating systems.

7. P = τ ω (power equals torque times angular velocity)

   • Power (P) is the rate at which work is done or energy is transferred in a rotational system.
   • Angular velocity (ω) measures how fast an object rotates and is expressed in radians per second.
   • This equation shows how torque influences the power output of rotating machinery.

8. τnet = Iα (net torque equals moment of inertia times angular acceleration)

   • This equation is a restatement of Newton's second law for rotation, relating net torque to the resulting angular acceleration.
   • It is crucial for analyzing dynamic rotational systems and predicting motion.
   • Understanding net torque helps in solving complex problems involving multiple forces and torques.