Premed · Premed · Physics 1
Lecture 11: Rotational Dynamics and Torque
Physics I — Mechanics & Thermodynamics
Learning Objectives
By the end of this lecture, students will be able to:
- Define torque and calculate it using the cross product and the lever arm method
- Apply Newton's second law for rotation (net torque = I alpha)
- Solve problems involving pulleys with mass and rotating systems
- Calculate rotational kinetic energy
- Apply energy conservation to problems involving both translation and rotation
Lecture Content
I. Torque — Definition
Torque (tau) is the rotational analog of force: it is the quantity that causes angular acceleration. Formally, torque is defined as the cross product tau = r x F, where r is the position vector from the axis of rotation to the point where the force is applied, and F is the applied force. The magnitude is tau = r F sin(theta) = F d_perp, where theta is the angle between r and F, and d_perp = r sin(theta) is the lever arm (or moment arm), defined as the perpendicular distance from the axis to the line of action of the force. Equivalently, tau = r F_perp, where F_perp = F sin(theta) is the component of force perpendicular to r.
The SI unit of torque is the Newton-meter (N m). Although dimensionally the same as Joules, torque is not energy. By convention, counterclockwise torque is positive and clockwise torque is negative, consistent with the right-hand rule.
<image>A wrench tightening a bolt. The force F is applied at the end of the wrench at angle theta to the wrench handle (vector r). The lever arm d_perp = r sin(theta) is shown as the perpendicular distance from the bolt (axis) to the line of action of F. Three cases are illustrated: (1) F perpendicular to r gives maximum torque, (2) F at an angle gives intermediate torque, (3) F along r gives zero torque. Each case shows the lever arm and the resulting torque value.</image>
II. Newton's Second Law for Rotation
The rotational analog of F_net = ma is tau_net = I alpha, where tau_net is the sum of all torques about the chosen axis, I is the moment of inertia about that same axis, and alpha is the angular acceleration in rad/s^2. This equation governs the rotational motion of rigid bodies in exactly the way that F = ma governs translational motion.
For a system undergoing both rotation and translation, two equations apply simultaneously: F_net = m a_cm for the translational motion of the center of mass, and tau_net = I_cm alpha for the rotation about the center of mass.
III. Pulleys with Mass
A real pulley has mass and a moment of inertia, typically modeled as a disk with I = (1/2)MR^2. Unlike an ideal massless pulley, a pulley with mass causes the string tensions on each side to differ: T_1 and T_2 are not equal. For a pulley of radius R and moment of inertia I, the torque equation is tau_net = (T_1 - T_2) R = I alpha, and the string constraint (assuming no slipping) gives a = R alpha.
To solve such problems, write F = ma for each hanging mass, tau = I alpha for the pulley, and use the constraint a = R alpha. The pulley's inertia effectively adds to the system's total inertia, reducing the overall acceleration compared to the massless-pulley case.
IV. Rotational Kinetic Energy
A rotating body possesses kinetic energy due to its rotation: K_rot = (1/2) I omega^2. This is the direct rotational analog of the translational kinetic energy K_trans = (1/2) m v^2. For a body that is simultaneously translating and rotating, the total kinetic energy is K_total = K_trans + K_rot = (1/2) m v_cm^2 + (1/2) I_cm omega^2.
For rolling without slipping, where v_cm = R omega, this becomes K_total = (1/2)(m + I_cm/R^2) v_cm^2. The fraction of kinetic energy in rotation versus translation depends on the object's shape through its moment of inertia.
<image>Panel A: A solid disk, a hollow cylinder, and a solid sphere all at the top of the same incline, about to roll down without slipping from the same height h. Panel B: At the bottom of the incline, bar charts for each object show the split between translational KE and rotational KE. The total energy (mgh) is the same for all, but the solid sphere has the most translational KE (fastest), and the hollow cylinder has the least (slowest). A ranking of speeds at the bottom is shown: v_sphere > v_disk > v_hoop.</image>
V. Energy Conservation with Rotation
When rolling objects are involved, rotational kinetic energy must be included in the energy equation: (1/2)mv_i^2 + (1/2)I omega_i^2 + mgy_i = (1/2)mv_f^2 + (1/2)I omega_f^2 + mgy_f, plus W_nc if non-conservative forces do work. For rolling without slipping, the constraint v = R omega allows one variable to be eliminated.
A key insight is that objects with larger I/MR^2 ratios reach the bottom of an incline more slowly. The ranking for rolling down an incline is: solid sphere (fastest) > solid cylinder > hollow sphere > hollow cylinder (slowest). Remarkably, this ranking is independent of mass and radius and depends only on the shape (mass distribution). It is also worth noting that static friction provides the torque needed for rolling but does no work, because the contact point has zero velocity.
VI. Work and Power in Rotation
The work done by a torque is W = integral of tau d theta, which reduces to W = tau Delta theta for constant torque. The rotational work-energy theorem states W_net = Delta K_rot = (1/2)I omega_f^2 - (1/2)I omega_i^2. Power in rotational systems is P = tau omega, analogous to P = Fv in linear motion. These relationships are useful for analyzing engines, motors, and biological systems involving joint torques.
VII. Summary of Linear-Rotational Analogies
The correspondence between translational and rotational quantities runs deep. Position x corresponds to angle theta, velocity v to angular velocity omega, acceleration a to angular acceleration alpha, mass m to moment of inertia I, force F to torque tau, momentum p = mv to angular momentum L = I omega (explored in the next lecture), kinetic energy (1/2)mv^2 to (1/2)I omega^2, Newton's second law F = ma to tau = I alpha, work W = Fd to W = tau theta, and power P = Fv to P = tau omega. This systematic analogy makes it possible to transfer intuition built from translational mechanics directly to rotational problems.
<image>A comprehensive two-column analogy table with "Translational" on the left and "Rotational" on the right. Each row pairs a translational quantity with its rotational analog: position/angle, velocity/angular velocity, acceleration/angular acceleration, mass/moment of inertia, force/torque, momentum/angular momentum, kinetic energy formulas, Newton's second law, work, and power. Matching colors connect each pair across the columns.</image>
