2 Expectations After Chapter 9, students will: calculate torques produced by forcesrecognize the condition of complete equilibriumcalculate the location of the center of gravity of a collection of objectsuse the rotational form of Newton’s second law of motion to analyze physical situationscalculate moments of inertia
3 Expectations After Chapter 9, students will: calculate the rotational work done by a torquecalculate rotational kinetic energycalculate angular momentumapply the principle of the conservation of angular momentum in an isolated system
4 Preliminary Definitions TorqueComplete EquilibriumCenter of Gravity
5 Torque Torque: the rotational analog to force Force produces changes in linear motion (linear acceleration). A force is a push or a pull.Torque produces changes in angular motion (angular acceleration). A torque is a twist.
6 Torque Mathematical definition: The lever arm is the line length of lever armMathematical definition:The lever arm is the linethrough the axis ofrotation, perpendicular tothe line of action of theforce.SI units: N·mtorqueforce
7 Torque Torque is a vector quantity. It magnitude is given by and its direction by the right-hand rule:
8 TorqueFor a given force, the torque depends on the location of the force’s application to a rigid object, relative to the location of the axis of rotation.more torqueless torque
9 TorqueFor a given force, the torque depends on the force’s direction.
10 Complete EquilibriumA rigid object is in complete equilibrium if the sum of the forces exerted on it is zero, and the sum of the torques exerted on it is zero.An object in complete equilibrium has zero translational (linear) acceleration, and zero angular acceleration.
11 Center of GravityIn analyzing the equilibrium of an object, we see that where a force is applied to an object influences the torque produced by the force.In particular, we sometimes need to know the location at which an object’s weight force acts on it.Think of the object as a collection of smaller pieces.
12 Center of GravityIn Chapter 7, we calculated the location of the center of mass of this system of pieces:Multiply numerator and denominator by g:
13 Center of Gravity But: Substituting: It is intuitive that the weight force acts at the effective location of the mass of an object.
14 Newton’s Second Law: Rotational Consider an object, mass m, in circular motion with a radius r. We apply a tangential force F:The result is atangential accelerationaccording to Newton’s second law.
15 Newton’s Second Law: Rotational The torque produced by the force isBut the tangential accelerationis related to the angularacceleration:Substituting:
16 Newton’s Second Law: Rotational This is an interesting result.If we define the quantityas the moment of inertia,we havethe rotational form of Newton’s second law.
17 Moment of Inertia The equation gives the moment of inertia of a “particle” (meaning an object whose dimensions are negligible compared with the distance r from the axis of rotation).Scalar quantity; SI units of kg·m2
18 Moment of InertiaNot many real objects can reasonably be approximated as “particles.” But they can be treated as systems of particles …
19 Moment of Inertia The moment of inertia of an object depends on: the object’s total massthe object’s shapethe location of the axis of rotation
20 Rotational Work and Energy By analogy with the corresponding translational quantities:Translational RotationalSI units: N·m = JSI units: (kg·m2) / s2 = N·m = J
21 Total Mechanical Energy We now add a term to our idea of the total mechanical energy of an object:total energygravitationalpotential energytranslationalkinetic energyrotationalkinetic energy
22 Angular Momentum By analogy with linear momentum: Angular momentum is a vector quantity. Its magnitude is given byand its direction is the same as the direction of w.w must be expressed in rad/s.SI units: kg·m2 / s
23 Angular Momentum: Conservation If a system is isolated (no external torque acts on it), its angular momentum remains constant.[If a system is isolated (no external force acts on it), its linear momentum remains constant.]