Rotational DynamicsAngular 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)

mo 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

yv

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

yv

For a 98 TVR ceberacar the maximumEngine 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 maximumrpm is much lower, about60 rpm

60rpm= 1 Hz = f= 2 rads/sec=

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

Lookhere

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

yv

In time t, m moves vtWhich changes the angle (in radians) by vt/rSo = angle/time = vt/rt = v/r

= v/r

= v/r

In fact whilst is the angular speed, thereis also a vector form, called theAngular velocity. As above.

Its direction is along the axis of rotation,such that the object is rotating clockwiselooking along the vector

Basic Concepts - Torque

o To make an object rotate about an axiso Must apply a torque,o A force perpendicular to the radius

A torque in rotationis like a force in linear motion

A torque is a perpendicular forcetimes a radius A = FAdA

B = FBdB

dA

dB

Only the perpendicularcomponent mattersso 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 witha long lever

A short lever would notwork, wheel would fall off UNLESS theforce 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 = maRotation: = Iddt = 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 INTOthe page [the direction ascrew 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, butin 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 vr

Centre ofrotation

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

vrCentre ofrotation

L = mrv

•The train rotates around theturntable axis. •The cab is close to the axis,

r is small, v is smallcontributes little to L.

•The chimney is far from axishigh r, higher vwill 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 = (mr2) The quantity in brackets (mr2) is called the moment of inertiaIt is given the symbol I.

I = mr2 for a single mass

[ I = r2dm for an extended body ]

∫v

Angular Momentum

o The moment of inertia, (mr2) 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 = mv2/2 so E = I2/2

vrCentre ofrotation

L = mrv = (mr2)

o Some examples of moment of inertia are:

A disk rotating around its centre I = mr2/2(if it rotates about the y axis it is I =

mr2/4)

A sphere rotating around any axis through the centre, I = 2mr2/5

A uniform road of length L rotating around its centre I = mL2/12

A simple point mass around an axis is the same as a hollow cylinder I = mr2

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 = li

Where: li is the angular momentum of the ith 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 thisone

Conservation

o The total angular momentum is conservedo Entirely analogous to linear momentum o Spins in opposite directions have opposite signs

As pirouetting skaterpulls in arms, increases

As the circling air drops lower, r decreases so goes up – a tornado