11. Torque

Professor Shankar begins by reviewing the concept of torque, represented by τ = Iα, where I is the moment of inertia and α is the angular acceleration. He differentiates between scenarios with and without external torques and introduces the special case of no torque or angular velocity, which is pertinent to stability in everyday objects like ladders. The lecture then transitions into a discussion on the conditions for a body to remain at rest, emphasizing the need for both force and torque to sum up to zero.

The professor elaborates on the conditions for a body to be in static equilibrium, where not only must the net force be zero, but also the net torque. He uses the seesaw as an example to illustrate how to calculate the forces needed to maintain balance. The seesaw problem involves determining the force F_2 that, when applied at a certain distance from the pivot, balances the weight of a child (F_1) at a different distance.

Shankar highlights the strategic choice of pivot point when calculating torque. He explains that while one can calculate torque around any point, choosing the point of application of an unknown force makes the problem simpler. This is because the unknown force does not appear in the torque equation when taken through the point of its application, simplifying the solution for the unknown force.

The lecture continues with a more complex problem involving a rod supported by a pivot on a wall. The professor discusses how to calculate the force required to keep the rod from falling. He explains the concept of torque due to gravity and how it can be simplified by considering the center of mass of the rod, assuming a uniform distribution of mass and a constant gravitational force.

Shankar explores the dynamics of a rod with a force applied to it, considering both the vertical and horizontal components of force. He demonstrates how to calculate the force at the pivot and how the horizontal force does not contribute to the torque equation. The discussion then extends to a scenario where a rod is supported by a wire, emphasizing the importance of the angle θ in determining the tension in the wire.

The professor delves into the calculation of torque in more complex systems, such as a rod supported by a wire at an angle θ to the wall. He shows how to find the tension in the wire and the forces at the pivot, highlighting the importance of understanding the direction of forces and torques in achieving equilibrium. The lecture concludes with a discussion on the torque equation's utility in simplifying the analysis of these forces.

Shankar addresses the stability of a ladder leaning against a wall, focusing on the minimum angle θ required for the ladder to remain stable. He introduces the concept of static friction μ and how it affects the ladder's stability. The discussion involves writing down the forces acting on the ladder and setting up the torque equation to find the critical angle θ, considering the coefficient of friction and the ladder's length.

The lecture shifts to three-dimensional dynamics, where the professor introduces the cross product to define torque and angular momentum in 3D space. He explains the right-hand rule for determining the direction of the torque vector and how the cross product is a unique feature of three-dimensional space. The professor also discusses the properties of the cross product and its implications for the dynamics of rotating bodies.

Shankar defines angular momentum for a single point mass as the cross product of the position vector r and the momentum vector p. He shows that the rate of change of angular momentum is equal to the torque, as defined by the cross product of r and the force F. The lecture emphasizes that angular momentum is a vector that can be conserved in a system, provided that the forces between bodies act along the line joining them.

The professor concludes the lecture with a discussion on the gyroscope, a spinning disk on a massless rod. He explains the concept of precession, which is the rotation of the gyroscope's axis when an external torque is applied. The lecture covers how the gyroscope's behavior can be understood through the principles of torque and angular momentum, and how the precessional frequency is determined by the torque and the angular momentum of the gyroscope.