🎡AP Physics 1 Unit 7 – Torque and Rotational Motion

Key Concepts and Definitions

• Torque ($\tau$) is the rotational equivalent of force, causing an object to rotate about an axis, calculated as the cross product of the force and the moment arm ($\tau = r \times F$)
• Moment of inertia ($I$) represents an object's resistance to rotational motion, determined by its mass distribution relative to the axis of rotation
• Angular displacement ($\theta$) measures the angle through which an object rotates, expressed in radians or degrees
• Angular velocity ($\omega$) is the rate of change of angular displacement with respect to time, typically measured in radians per second
• Angular acceleration ($\alpha$) represents the rate of change of angular velocity with respect to time, expressed in radians per second squared
• Rotational equilibrium occurs when the net torque acting on an object is zero, resulting in no angular acceleration
• Center of mass is the point where the entire mass of an object can be considered to be concentrated, acting as the average position of the object's mass

Angular Motion vs. Linear Motion

• Angular motion involves rotation about an axis, while linear motion refers to movement along a straight line
• Rotational variables have angular counterparts to linear variables (displacement vs. angular displacement, velocity vs. angular velocity, acceleration vs. angular acceleration)
• Tangential velocity ($v_t$) is the linear velocity of a point on a rotating object, perpendicular to the radius, and can be calculated using the formula $v_t = r \omega$
• Centripetal acceleration ($a_c$) is the acceleration directed towards the center of the circular path, causing an object to move in a circular motion, calculated as $a_c = \frac{v_t^2}{r} = r \omega^2$
  • Objects undergoing circular motion experience a centripetal force, which is always directed towards the center of the circle
• Tangential acceleration ($a_t$) is the linear acceleration tangent to the circular path, caused by the angular acceleration, and can be calculated using the formula $a_t = r \alpha$
• Relationships between angular and linear variables:
  • $v_t = r \omega$
  • $a_t = r \alpha$
  • $a_c = \frac{v_t^2}{r} = r \omega^2$

Torque and Rotational Equilibrium

• Torque is the rotational equivalent of force, causing an object to rotate about an axis
• The magnitude of torque depends on the applied force and the moment arm (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, with the thumb pointing in the direction of the angular displacement
• Net torque is the sum of all torques acting on an object, considering both magnitude and direction
• Rotational equilibrium occurs when the net torque acting on an object is zero, resulting in no angular acceleration
  • In rotational equilibrium, the sum of clockwise torques equals the sum of counterclockwise torques
• The torque due to gravity (weight) on an object can be calculated using the formula $\tau = r \times F_g$, where $F_g$ is the force of gravity (weight) acting at the center of mass
• A couple is a pair of equal and opposite forces that produce a net torque on an object without causing linear acceleration

Moment of Inertia

• Moment of inertia is a measure of an object's resistance to rotational motion, determined by its mass distribution relative to the axis of rotation
• The formula for moment of inertia depends on the object's shape and mass distribution (point mass, rod, disk, sphere, etc.)
  • For a point mass: $I = mr^2$
  • For a thin rod (about an axis perpendicular to its length): $I = \frac{1}{12}mL^2$
  • For a thin rectangular plate (about an axis perpendicular to its surface): $I = \frac{1}{12}m(a^2 + b^2)$
• 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 perpendicular distance between the axes
• Radius of gyration ($k$) is the distance from the axis of rotation at which the entire mass of an object can be concentrated without changing its moment of inertia, calculated as $k = \sqrt{\frac{I}{m}}$
• Rotational kinetic energy ($KE_{rot}$) is the energy associated with an object's rotational motion, calculated as $KE_{rot} = \frac{1}{2}I\omega^2$
• The work-energy theorem for rotational motion states that the net work done on an object equals the change in its rotational kinetic energy

Rotational Kinematics

• Rotational kinematics describes the motion of an object undergoing rotation without considering the forces causing the motion
• Angular displacement ($\Delta \theta$) is the angle through which an object rotates, measured in radians or degrees
• Angular velocity ($\omega$) is the rate of change of angular displacement with respect to time, typically measured in radians per second
  • Average angular velocity: $\omega_{avg} = \frac{\Delta \theta}{\Delta t}$
  • Instantaneous angular velocity: $\omega = \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$
• Angular acceleration ($\alpha$) is the rate of change of angular velocity with respect to time, measured in radians per second squared
  • Average angular acceleration: $\alpha_{avg} = \frac{\Delta \omega}{\Delta t}$
  • Instantaneous angular acceleration: $\alpha = \lim_{\Delta t \to 0} \frac{\Delta \omega}{\Delta t} = \frac{d\omega}{dt}$
• Rotational kinematic equations (for constant angular acceleration):
  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
  • $\omega = \omega_0 + \alpha t$
  • $\theta = \theta_0 + \frac{1}{2}(\omega_0 + \omega)t$
  • $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

Rotational Dynamics and Newton's Second Law

• Rotational dynamics deals with the forces causing rotational motion and the resulting motion of objects
• Newton's second law for rotational motion states that the net torque acting on an object equals the product of its moment of inertia and angular acceleration ($\sum \tau = I \alpha$)
• The rotational analog of Newton's first law states that an object at rest stays at rest, and an object rotating at a constant angular velocity continues to rotate at that velocity unless acted upon by a net external torque
• The rotational analog of Newton's third law states that when two objects interact, they apply torques to each other that are equal in magnitude and opposite in direction
• Rotational inertia is an object's resistance to changes in its rotational motion, determined by its moment of inertia
• Rolling motion occurs when an object rotates and translates simultaneously, with the condition that the object rolls without slipping
  • For rolling without slipping: $v_{CM} = R \omega$, where $v_{CM}$ is the linear velocity of the center of mass, $R$ is the radius of the object, and $\omega$ is the angular velocity

Angular Momentum and Conservation

• Angular momentum ($L$) is the rotational analog of linear momentum, defined as the product of an object's moment of inertia and its angular velocity ($L = I \omega$)
• The vector direction of angular momentum is determined by the right-hand rule, with the thumb pointing in the direction of the angular velocity
• The law of conservation of angular momentum states that the total angular momentum of a system remains constant if no external net torque acts on the system
  • $L_{initial} = L_{final}$ or $I_{initial} \omega_{initial} = I_{final} \omega_{final}$
• Angular impulse ($\Delta L$) is the change in angular momentum, equal to the product of the net torque and the time interval over which it acts ($\Delta L = \sum \tau \Delta t$)
• The angular momentum of a system is conserved in the absence of external torques, even if internal torques are present (such as in collisions or explosions)
• The conservation of angular momentum explains various phenomena, such as the increase in angular velocity when a spinning figure skater pulls their arms inward, reducing their moment of inertia

Real-World Applications and Examples

• Torque wrenches are used to precisely apply a specific torque to fasteners (nuts and bolts) to ensure proper tightness without overtightening
• Balancing acts (tightrope walkers, acrobats) rely on maintaining rotational equilibrium by adjusting their center of mass and applying counterbalancing torques
• Flywheels are used in engines and machines to store rotational kinetic energy and smooth out power fluctuations
• Gyroscopes, which have a high moment of inertia and angular momentum, are used in navigation systems (aircraft, spacecraft) and stabilization devices (Segways, camera gimbals)
• The conservation of angular momentum is demonstrated by a figure skater spinning on ice – as they pull their arms inward, their moment of inertia decreases, and their angular velocity increases to conserve angular momentum
• Planetary motion and the formation of solar systems can be explained by the conservation of angular momentum, as a collapsing cloud of gas and dust begins to rotate faster as it contracts
• The precession of a spinning top or gyroscope occurs due to the torque applied by gravity, causing the axis of rotation to trace out a circular path
• The design of car and bicycle wheels involves optimizing the moment of inertia to balance stability, acceleration, and maneuverability