Rotational Dynamics Angular Momentum. Collisions. o Rotational Dynamics  Basic Concepts Rotation Angular speed Torque Angular Acceleration  Nature of Angular Momentum (& Energy), origins  Angular Momentum Conservation  Angular Momentum for a system of bodies o Parallel Axis Theorem

o Aim of the lecture  Concepts in Rotational Dynamics Angular speed, torque, acceleration Dependence on mass Dependence on radius of mass, Moment of Inertia Dependence on rotation speed  Newton’s Second Law Conservation of angular momentum o Main learning outcomes  familiarity with  , , d  /dt, I, L, including vector forms  Use of energy and angular momentum conservation  Calculation of moment of inertia Lecture 8

Basic Concepts – Angular Position  r Centre of rotation (axis pointing out of page) m o Rotational Dynamics  Systems rotating about fixed axis  Use r and  to describe position  Natural quantities for rotation o For Rigid Body Rotation  r is fixed  m moves in a circle The position of the mass is (r,  ) or (x,y) they both give same information x y

Basic Concepts – Angular Rotation  r Centre of rotation (axis pointing out of page) m o Rotation – ‘massless’ rod with mass  Mass moving with speed v  r is fixed   changes with time o This is Rigid Body Rotation  m moves in a circle  angular speed is called   see next slide  The position of the mass (r,  ) depends on time (r,  ) = (r,  t) x y v

Basic Concepts – Angular Speed  r Centre of rotation (axis pointing out of page) m o The mass is rotation round  say f revolutions per second  f is the frequency of rotation  f is measured in Hertz, Hz  the time for one revolution is 1/f   depends on f and time o There are 2  radians in a circle, so  the number of radians per second is  2  f =  the angular speed   is measured in radians per second   = (change in angle)/(time taken) (r,  ) depends on time (r,  ) = (r,  t) x y v

For a 98 TVR cebera car the maximum Engine rpm is 6250 (same for all colours) 6250rpm = 6250/60 revs per second = 104 Hz = f = 104 x 2  = 655 rads/sec = 

For dancer, the maximum rpm is much lower, about 60 rpm = 1 Hz= f = 2  rads/sec= 

oRotation is extremely common, it is measured in rpm, Hertz (frequency of rotation) or radians per second (angular speed)

Look here Then here Its not really rotating, so its angular speed is  = 0

Basic Concepts – Angular Speed  r Centre of rotation (axis pointing out of page) m o The mass is moving  at speed v  at radius r  with angular speed  (r,  ) depends on time (r,  ) = (r,  t) x y v In time  t, m moves v  t Which changes the angle (in radians) by v  t/r So  = angle/time = v  t/r  t = v/r  = v/r

In fact whilst  is the angular speed, there is also a vector form, called the Angular velocity. As above. Its direction is along the axis of rotation, such that the object is rotating clockwise looking along the vector 

Basic Concepts - Torque o To make an object rotate about an axis o Must apply a torque, o A force perpendicular to the radius A torque in rotation is like a force in linear motion A torque is a perpendicular force times a radius  A = F A d A  B = F B d B dAdA dBdB

Only the perpendicular component matters so here the torque is  = rFcos(  ) Centre of rotation

Torque is a ‘twisting force’ Distance matters: A high torque is needed for car wheel bolts This is achieved with a long lever A short lever would not work, wheel would fall off UNLESS the force was much bigger The same torque can be achieved with a long lever and small force a short lever and a large force

Basic Concepts – Angular Acceleration Linear:F = mdv/dt = ma Rotation:  = Id  dt = I  oYou can accelerate, ‘spin up’ a rotating object  by applying a torque,   the rate of angular acceleration =  /I where I is the moment of inertia (see later)  Moment of inertia is like mass in the linear case

o Torque is actually a vector  Direction is perpendicular to: Force being applied  It is parallel to: The axis of rotation F Torque is pointing INTO the page [the direction a screw would be driven]

 = I d  /dt = I  This is the vector form for the relationship between torque and angular acceleration ( In advanced work, I is a tensor, but in this course we will just use a constant )

Angular Momentum o Conserved Quantities  If there is a symmetry in nature, then  There will be a conserved quantity associated with it. (the maths to prove this is beyond the scope of the course) o Examples:  Physics today is the same as physics tomorrow, TIME symmetry The conserved quantity is Energy  Physics on this side of the room is the same as on the other side Linear Translational symmetry The conserved quantity is called Momentum  Physics facing west is the same as physics facing east Rotational symmetry The conserved quantity is called Angular Momentum o What is angular momentum?  For a mass m rotating at speed v, and radius r the angular momentum, L is L = mrv (= I  v r Centre of rotation

Angular Momentum This is (in fact) a familiar quantity: A spinning wheel is hard to alter direction A gyroscope is based on conservation of angular momentum The orbits of planets are proscribed by conservation of a.m. It is claimed that gyroscopic effects help balance a bicycle

Angular Momentum o The angular momentum will depend on  the mass; the speed;  AND the radius of rotation  The radius of rotation complicates things  extended masses (not just particles) have more than one radius  each part of such a mass will have a different value for mr  and a different speed v r Centre of rotation L = mrv The train rotates around the turntable axis. The cab is close to the axis,  r is small, v is small  contributes little to L. The chimney is far from axis  high r, higher v  will contribute more.

To simplify things we use the angular rotation speed,  Where  = radians per second or 2  (rotations per second)  = 2  (v/2pr) = v/r in radians per second [see earlier slides] So we can write v =  r and L = (mr 2 )  The quantity in brackets (mr 2 ) is called the moment of inertia It is given the symbol I. I = mr 2 for a single mass [ I = r 2 dm for an extended body ] ∫ v

Angular Momentum o The moment of inertia, (mr 2 ) is  given the symbol I  can be calculated for any rigid (solid) body  depends on where the body is rotating around  The usual formula for angular momentum is L = I    In rotational dynamics there is a mapping from linear mechanics replace m by I replace v by  replace P by L  then many of the laws of linear mechanics can be used  For example: momentum P = mv so L = I  kinetic energy, E = mv 2 /2 so E = I  2 /2 v r Centre of rotation L = mrv = (mr 2 ) 

o Some examples of moment of inertia are:  A disk rotating around its centre I = mr 2 /2 (if it rotates about the y axis it is I = mr 2 /4)  A sphere rotating around any axis through the centre, I = 2mr 2 /5  A uniform road of length L rotating around its centre I = mL 2 /12  A simple point mass around an axis is the same as a hollow cylinder I = mr 2 Moments of Inertia

o Angular Momentum is a vector  The magnitude of the vector is L = I   The direction is along the rotation axis looking in the direction of clockwise rotation  L = I   Note that this vector does NOT define a position in space  Clearly  is also a vector quantity with a similar definition Vector Form L

o For several objects considered together ‘a system’ o The total angular momentum is the sum of the individual momenta L =  l i Where:  l i is the angular momentum of the i th object   means ‘sum of’  L is the total angular momentum

o Angular Momentum is a vector  The angular momentum vector does NOT define a position in space  The parallel axis theorem says that any rotating body has the same angular momentum around any axis. [NOT changing the rotation axis!] Parallel Axis theorm L This spinning shell has the same A.M. about all the blue axes shown (without moving the shell, position does not matter) But NOT this one

Conservation o The total angular momentum is conserved o Entirely analogous to linear momentum o Spins in opposite directions have opposite signs As pirouetting skater pulls in arms,  increases As the circling air drops lower, r decreases so  goes up – a tornado