Physics 1901 (Advanced)

Physics 1901 (Advanced) Prof Geraint F. Lewis Rm 560, A29 gfl@physics.usyd.edu.au www.physics.usyd.edu.au/~gfl/Lecture 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. 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! 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 http://www.physics.usyd.edu.au/~gfl/Lecture

Angular Variables Angular acceleration is the change of  with time Tangential acceleration is given by 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: 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? 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; 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. 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. 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. http://www.physics.usyd.edu.au/~gfl/Lecture

Energy in Rotation For a mass at point P Total kinetic energy 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? 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! 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. 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 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? 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 http://www.physics.usyd.edu.au/~gfl/Lecture

Torque and Acceleration At point P, the tangential force gives a tangential acceleration of This becomes 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) 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. http://www.physics.usyd.edu.au/~gfl/Lecture

Rolling without Slipping The total kinetic energy is and WhereC is the constant of the Moment of Inertia http://www.physics.usyd.edu.au/~gfl/Lecture

Rolling without Slipping • Independent of mass & size • Any sphere beats any hoop! • What is the source of torque? Conservation of energy 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). 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. 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 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) 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. 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? 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”. http://www.physics.usyd.edu.au/~gfl/Lecture

Angular Momentum 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 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. 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. http://www.physics.usyd.edu.au/~gfl/Lecture

Physics 1901 (Advanced)