🎡AP Physics 1 (2025) Unit 5 – Torque and Rotational Dynamics

Key Concepts and Definitions

• Torque ($\tau$) is the rotational equivalent of force, causing an object to rotate about an axis
• Moment of inertia ($I$) represents an object's resistance to rotational motion, dependent on its mass distribution
• Angular displacement ($\theta$) measures the angle through which an object rotates
• Angular velocity ($\omega$) is the rate of change of angular displacement with respect to time
• Angular acceleration ($\alpha$) is the rate of change of angular velocity with respect to time
• Rotational kinetic energy ($K_r$) is the energy associated with an object's rotational motion
• Angular momentum ($L$) is the rotational equivalent of linear momentum, representing the product of an object's moment of inertia and angular velocity

Torque Fundamentals

• Torque is calculated using the formula $\tau = r \times F$, where $r$ is the lever arm and $F$ is the force applied
  • The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force
• The direction of torque is determined by the right-hand rule
  • Curl your fingers in the direction of rotation, and your thumb points in the direction of the torque vector
• Net torque is the sum of all torques acting on an object
• When net torque is zero, the object is in rotational equilibrium
• Torque can cause an object to rotate clockwise or counterclockwise, depending on the direction of the force applied
• The SI unit for torque is newton-meter (N⋅m)

Rotational Motion Basics

• Rotational motion occurs when an object rotates about an axis
• Angular displacement is measured in radians (rad) or degrees (°)
  • One full rotation equals 2π radians or 360°
• Angular velocity is measured in radians per second (rad/s) or revolutions per minute (rpm)
• Angular acceleration is measured in radians per second squared (rad/s²)
• Tangential velocity ($v_t$) is the linear velocity of a point on a rotating object, perpendicular to the radius
  • $v_t = r \omega$, where $r$ is the distance from the axis of rotation
• Centripetal acceleration ($a_c$) is the acceleration directed towards the center of the circular path
  • $a_c = \frac{v_t^2}{r} = r \omega^2$

Moment of Inertia

• Moment of inertia depends on the object's mass and its distribution relative to the axis of rotation
• For a point mass, $I = mr^2$, where $m$ is the mass and $r$ is the distance from the axis of rotation
• The parallel-axis theorem states that the moment of inertia about any axis parallel to an axis through the center of mass is equal to the moment of inertia about the axis through the center of mass plus the product of the mass and the square of the distance between the axes
• Radius of gyration ($k$) is the distance from the axis of rotation at which all the mass can be concentrated without changing the moment of inertia
  • $I = mk^2$
• Moment of inertia for common shapes (rod, disk, sphere) can be calculated using specific formulas

Rotational Dynamics Equations

• Newton's second law for rotational motion states that the net torque on an object equals its moment of inertia times its angular acceleration
  • $\sum \tau = I \alpha$
• Rotational work ($W_r$) is the product of torque and angular displacement
  • $W_r = \tau \theta$
• Rotational kinetic energy is given by $K_r = \frac{1}{2} I \omega^2$
• Power in rotational motion ($P_r$) is the rate of doing work or the product of torque and angular velocity
  • $P_r = \tau \omega$
• Rolling motion without slipping occurs when the tangential velocity of the bottom point on the object is zero relative to the surface
  • $v_{cm} = r \omega$, where $v_{cm}$ is the velocity of the center of mass

Angular Momentum and Conservation

• Angular momentum is calculated using $L = I \omega$
• The law of conservation of angular momentum states that if the net external torque on a system is zero, the total angular momentum of the system remains constant
  • $\sum \tau_{ext} = \frac{dL}{dt}$, so if $\sum \tau_{ext} = 0$, then $\frac{dL}{dt} = 0$ and $L$ is conserved
• Angular momentum is conserved in the absence of external torques (friction, air resistance)
• When a system's moment of inertia changes, its angular velocity must change to conserve angular momentum
  • $I_1 \omega_1 = I_2 \omega_2$
• Precession is the gradual change in the orientation of a rotating object's axis (gyroscope, spinning top)

Real-World Applications

• Torque wrenches are used to apply a specific torque to fasteners (bolts, nuts)
• Flywheels store rotational kinetic energy and help maintain a constant angular velocity in engines
• Centripetal force keeps objects moving in a circular path (satellites, amusement park rides)
• Ice skaters and divers change their moment of inertia to control their angular velocity during spins and twists
• Gyroscopes maintain their orientation and are used in navigation systems and stabilization devices

Problem-Solving Strategies

• Identify the key variables given in the problem (mass, radius, force, angular velocity)
• Determine the appropriate equation to use based on the given information and the quantity you're asked to find
• If the problem involves multiple steps, break it down into smaller sub-problems
• Pay attention to units and convert them if necessary
• Use the right-hand rule to determine the direction of torque or angular velocity vectors
• Check if the problem involves conservation of angular momentum and apply the principle accordingly
• Double-check your answer to ensure it makes sense in the context of the problem