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Physics 1901 (Advanced) Prof Geraint F. Lewis Rm 560, A29 [email protected] www.physics.usyd.edu.au/~gfl/Lecture Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Motion So far we have examined linear motion; Newton’s laws Energy conservation Momentum Rotational motion seems quite different, but is actually familiar. Remember: We are looking at rotation in fixed coordinates, not rotating coordinate systems. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Variables Rotation is naturally described in polar coordinates, where we can talk about an angular displacement with respect to a particular axis. For a circle of radius r, an angular displacement of corresponds to an arc length of Remember: use radians! Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Variables Angular velocity is the change of angle with time There is a simple relation between angular velocity and velocity Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Variables Angular acceleration is the change of with time Tangential acceleration is given by Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Kinematics Notice that the form of rotational relations is the same as the linear variables. Hence, we can derive identical kinematic equations: Linear Rotational a=constant =constant v=u+at =o+ t s=so+ut+½at2 =o+ot+½t2 Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Net Acceleration Remember, for circular motion, there is always centripetal acceleration The total acceleration is the vector sum of arad and atan. What is the source of arad? Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Dynamics As with rotational kinematics, we will see that the framework is familiar, but we need some new concepts; Semester 1 2009 Linear Rotational Mass Moment of Inertia Force Torque http://www.physics.usyd.edu.au/~gfl/Lecture Moment of Inertia This quantity depends upon the distribution of the mass and the location of the axis of rotation. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Moment of Inertia Luckily, the moment of inertia is typically; where c is a constant and is <1. Object Solid sphere Hollow sphere Rod (centre) Rod (end) I 2/5 M R2 2/3 M R2 1/12 M L2 1/3 M L2 Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Energy in Rotation To get something moving, you do work on it, the result being kinetic energy. To get objects spinning also takes work, but what is the rotational equivalent of kinetic energy? Problem: in a rotating object, each bit of mass has the same angular speed , but different linear speed v. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Energy in Rotation For a mass at point P Total kinetic energy Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Parallel Axis Theorem The moment of inertia depends upon the mass distribution of an object and the axis of rotation. For an object, there are an infinite number of moments of inertia! Surely you don’t have to do an infinite number of integrations when dealing with objects? Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Parallel Axis Theorem If we know the moment of inertia through the centre of mass, the moment of inertia along a parallel axis d is; The axis does not have to be through the body! Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Torque Opening a door requires not only an application of a force, but also how the force is applied; It is ‘easier’ pushing a door further away from the hinge. Pulling or pushing away from the hinge does not work! From this we get the concept of torque. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Torque Torque causes angular acceleration Only the component of force tangential to the direction of motion has an effect Torque is Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Torque Like force, torque is a vector quantity (in fact, the other angular quantities are also vectors). The formal definition of torque is where the x is the vector cross product. In which direction does this vector point? Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Vector Cross Product The magnitude of the resultant vector is and is perpendicular to the plane containing vectors A and B. Right hand grip rule defines the direction Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Torque and Acceleration At point P, the tangential force gives a tangential acceleration of This becomes Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Torque and Acceleration For an arbitrarily shaped object We have the rotational equivalent of Newton’s second law! Torque produces an angular acceleration. Notice the vector quantities. All rotational variables point along the axis of rotation. (Read torques & equilibrium 11.0-11.3 in textbook) Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rolling without Slipping For a rolling wheel which does not slide, then the distance it travels is related to how much it turns. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rolling without Slipping The total kinetic energy is and Where C is the constant of the Moment of Inertia Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rolling without Slipping Conservation of energy Independent of mass & size Any sphere beats any hoop! What is the source of torque? Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rolling without Slipping Torque is provided by friction acting at the surface (otherwise the ball would just slide). Note that the normal force does not produce a torque (although it can with deformable surfaces and rolling friction). Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Work In linear mechanics, the work-kinetic energy theorem can be used to solve problems. In rotational mechanics, we note that a force, Ftan, applied to a point on a wheel always points along the direction of motion. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Rotational Work If the torque is constant Hence, we now have a rotational work-kinetic energy theorem, except Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum In linear dynamics, complex interaction (collisions) can be examined using the conservation of momentum. In rotational dynamics, the concept of angular momentum similarly eases complex interactions. (Derivation similar to all other rotational quantities) Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum In linear dynamics: In rotational dynamics: Hence, the net torque is equal to the rate of change of angular momentum. Hence, if there is no net torque, angular momentum is conserved. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum We can change the angular velocity by modifying the moment of inertia. Angular momentum is conserved, but where has the extra energy come from? Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum I have to apply a force on the mass to change its linear velocity. Through NIII, the mass applies a force on me. For every torque there is an equal and opposite “retorque”. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum Consider a lecturer on a rotating stool holding a spinning wheel, with the axis of the wheel pointing towards the ceiling. What happens when the wheel is turned over? http://www.physics.lsa.umich.edu/demolab/demo.asp?id=696 Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Angular Momentum As with linear momentum, we can use conservation of angular momentum without having to worry about the various (internal) torques in action. External torques will change the value of the total angular momentum. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture Linear & Angular Momentum What is the angular momentum of an object moving along a straight line? Objects moving linearly have constant angular momentum. Rotational mechanics is linear mechanics in a different coordinate system. Semester 1 2009 http://www.physics.usyd.edu.au/~gfl/Lecture