Phy 201: General Physics I Chapter 9: Rotational Dynamics Lecture Notes

Rigid Objects & Torque Torque is a vector quantity that represents the application of force to a body resulting in rotation (or change in rotational state) Definition: The SI units for torque are N. m (not to be confused with joules) Torque depends on: –The Lever arm (leverage), –The component of force perpendicular to lever arm,

Newton’s 2 nd Law (for Rotational Motion) When there is a net torque exerted on a rigid body its state of rotation ( ) will change depending on: 1.Amount of net torque, 2.Rotational inertia of object, I Note: No net torque means  = 0 and  = constant} Alternatively, the net torque exerted on a body is equal to the product of the rotational inertia and the angular acceleration: This is Newton’s 2 nd Law (for rotation)!!

Rigid Objects in Equilibrium A rigid body is in equilibrium when: –No net force  no change in state of motion (v = constant) –No net torque  no change in state of rotation (  = constant) Both of the above conditions must be met for an object to be in mechanical equilibrium! Note: an object can be both moving and rotating and still be in mechanical equilibrium

Center of Gravity (COG) The COG is the location where all of an object’s weight can be considered to act when calculating torque The COG may or may not actually be on the object (i.e. consider a hollow sphere or a ring) When an object’s weight produces a torque on itself, it acts at its center of mass To calculate center of gravity:

Moment of Inertia The resistance of a rigid body to changes in its state of rotational motion ( ) is called the moment of inertia ( or rotational inertia) The Moment of Inertia (I) depends on: 1.Mass of the object, m 2.The axis of rotation 3.Distribution (position) of mass about the axis of rotation Definition (For a discrete distribution of mass): Where: m i is the mass of a small segment of the object r i is the distance of the mass mi from the axis of rotation The SI units for I are kg. m 2 Note: For a continuous distribution of mass there are more sophisticated techniques for calculating the moment of inertia

Moment of Inertia (common examples) 1.A simple pendulum: 2.A thin ring: 3.A solid cylinder: m dm H R

Moment of Inertia (common examples, cont.) 4.A hollow sphere: 5.A solid sphere: 6.A thin rod:. m m R m R L

The Parallel Axis Theorem The Parallel Axis Theorem is used to determine the moment of inertia for a body rotated about an axis a distance, l, from the center of mass: Example: A 0.2 kg cylinder (r=0.1 m) rotated about an axis located 0.5 m from its center: L

Work & Rotational Kinetic Energy When torque is applied to an object and a rotation is produced, the torque does work: When there is net torque: Since  2 -  o 2 = 2  The rotational kinetic energy (K rot ) for an object is defined as: This is the Work-Energy Theorem (for rotation)!!

Rolling objects & Inclined Planes Consider a solid disc & a hoop rolling down a hill (inclined plane): 1.Apply Newton’s 2 nd Law (force) to each object 2.Apply Newton’s 2 nd Law (torque) to each object 3.What is the acceleration of each object as they roll down the hill? 4.Which one reaches the bottom first?  Solid disc Hoop

Angular Momentum Angular momentum is the rotational analog of linear momentum It represents the “quantity of rotational motion” for an object (or its inertia in rotation) Angular Momentum (a vector we will treat as a scalar) is defined as: L = I.  Note: Angular Momentum is related to Linear Momentum: L = r. p sin  r is distance from the axis of rotation p is the linear momentum  is the angle between r and p SI units of angular momentum are kg. m 2 /s r  p

Conservation of Angular Momentum When there is no net torque acting on an object (or no net torque acts on a system) its angular momentum (L) will remain constant or L 1 = L 2 = … = a constant value Or I 1  1 = I 2  2 = etc… This principle explains why planets move faster the closer the get to the sun and slower the they move when they are farther away Note: this is a re-statement of Newton’s 1 st Law slowest fastest

Conservation of Angular Momentum (Examples) A figure skater A high diver Water flowing down a drain

Archimedes ( BC) Possibly the greatest mathematician in history Invented an early form of calculus Discovered the Principle of Buoyancy (now called Archimedes’ Principle) Introduced the Principle of Leverage (Torque) and built several machines based on it. “Give me a point of support and I will move the Earth”