Chapter 11 - Rotational Dynamics Physics Intro

11.1 Torque Two factors that determine the amount of torque. Torque: A force that causes or opposes rotation. The force is applied perpendicularly to the line from the axis of rotation and the location where the force is applied. Two factors that determine the amount of torque. How much force F is exerted. The distance r between the axis of rotation and the location where the force is applied.

11.1 Torque cont… A net torque can cause an object to start rotating clockwise or counterclockwise. A torque that causes counterclockwise rotation is a positive torque. A negative torque causes clockwise rotation. The unit for torque is the newton-meter (N·m). work and energy are also measured using newton-meters, or, equivalently, joules. Torque

11.2 Torque, angle and lever arm F and r are vectors, so we can describe the angle between them. The position vector r points from the axis of rotation to the point where the force is applied. The sine of the angle θ between r and F. determines the amount of torque applied. Θ is the angle made by the two vectors when they are placed tail-to-tail. The angle used is the smaller angle made between the two vectors. Seesaw

11.3 Cross product of vectors Cross product: A vector whose magnitude equals the product of the magnitudes of two vectors and the sine of the smaller angle between them. Its direction is determined by the right-hand rule. First, make sure the vectors are placed tail-to-tail. Then, point the fingers of your right hand along the first vector, and curl them toward the second vector. (Curl your fingers through the smaller of the two angles between the vectors.) The direction of your thumb is the direction of the resulting vector: It is perpendicular to both of the two vectors you are multiplying. Cross-product

11.4 Torque, moment of inertia and angular acceleration Rotational inertia (I) is the resistance of an object to changes in its angular velocity. It is the rotational equivalent of mass. Units - Kg•m2 It is directly proportional to torque and inversely proportional to angular acceleration. An object with a greater moment of inertia requires more net torque to angularly accelerate at a given rate than an object with a lesser moment of inertia. The net torque equals the moment of inertia times the angular acceleration. This equation, Στ = Iα, resembles Newton’s second law, ΣF = ma. Like mass, the moment of inertia is always a positive quantity. F=ma then τ=Iα or τ =Fr Torque and moment of inertia

11.5 Calculating the moment of inertia More than the amount of mass is required to determine the moment of inertia; the distribution of the mass also matters. When a rigid object or system of particles rotates about a fixed axis, each particle in the object contributes to its moment of inertia. The moment equals the sum of each particle’s mass times the square of its distance from the axis of rotation. I=Σmr2 A single object often has a different moment of inertia when its axis of rotation changes. Moment of inertia

11.6 Table of moments of inertia 11.7 Sample problem: a seesaw 11.8 Interactive checkpoint: moment of inertia 11.9 Sample problem: an Atwood machine 11.10 Interactive problem: close the bridge

11.11 Parallel axis theorem Parallel axis theorem The parallel axis theorem is a tool for calculating the moment of inertia of an object. You use it when you know the moment of inertia for an object rotating about an axis that passes through its center of mass and want to know the moment when it rotates around a different but parallel axis of rotation. The theorem states that the moment of inertia when the disk rotates about the axis on its edge will be the sum of two values: the moment of inertia when the disk rotates about its center of mass, and the product of the disk’s mass and the square of the distance between the two axes. Parallel axis theorem

11.12 Rotational work With rotational motion, the work equals the product of the angular displacement and the torque. Rotational work

Power in Rotary Motion P = TΔθ/Δt P=Tω Work in rotary motion is torque times angular displacement so power is torque times angular displacement divided by time. P = TΔθ/Δt or P=Tω

11.13 Rotational kinetic energy Kinetic energy is the energy of a moving object so objects which are rotating also have KE So and τ=Iα or Rotational kinetic energy

11.14 Physics at work: flywheels Flywheels are rotating objects used to store energy as rotational kinetic energy. Flywheels can be powered by “waste” energy or can receive power from more traditional sources. Flywheels

11.15 Rolling objects and kinetic energy The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies. Rotational kinetic energy equals ½ Iω2 Linear kinetic energy equals ½ mvCM2 The “CM” subscript indicates that the point used in calculating the linear speed is the coin's center of mass Kinetic energy of rolling object

11.16 Sample problem: rolling down a ramp 11.17 Sample problem: rolling cylinders 11.18 Interactive checkpoint: rolling to a stop 11.19 Physics at play: a yo-yo 11.20 Sample problem: acceleration of a yo-yo

11.21 Angular momentum of a particle in circular motion Angular momentum is the rotational analog of linear momentum. Angular momentum is proportional to mass and velocity. However, with rotational motion, the distance of the particle from the origin must be taken into account, as well. Angular momentum equals the product of mass, speed and the radius of the circle: mvr.The amount of angular momentum equals the linear momentum mv times the radius r. Angular momentum is a vector. When the motion is counterclockwise, by convention, the vector is positive. The angular momentum of clockwise motion is negative. The units for angular momentum are kilogram-meter2 per second (kg·m2/s). Angular momentum

11.22 Angular momentum of a rigid body Rigid means the particles all rotate with the same angular velocity, and each remains at a constant radial distance from the axis. Angular momentum can be determined by summing the angular momenta of all the particles that make it up. The magnitude of the angular momentum of the CD equals the product of its moment of inertia, I, and its angular velocity, ω. Angular momentum of a rigid body

11.23 Angular momentum: general motion The angular momentum of a particle with respect to the origin is the cross product of the particle’s position vector and its linear momentum. The angular momentum equals the mass times the cross product of the radius and velocity. Its direction is calculated with the right-hand rule for cross products. Angular momentum: general case

11.24 Sample problem: object moving in a straight line 11.25 Comparison of rotational and linear motion

11.26 Torque and angular momentum A net torque changes a rotating object’s angular velocity, and this changes its angular momentum. The product of torque and an interval of time equals the change in angular momentum (impulse). Torque and angular momentum

11.27 Conservation of angular momentum Angular momentum is conserved when there is no net external torque. Since angular momentum equals the product of the moment of inertia and angular velocity, if one of these properties changes, the other must as well for the angular momentum to stay the same. Conservation of angular momentum 11.28 Interactive checkpoint: a rotating disk 11.29 Interactive summary problem: dynamics of skating