Equilibrium Systems ‘in balance’

o Static Equilibrium Balanced Forces Balanced Moments Centre of Gravity

o Dynamic Equilibrium Constant Linear Momentum Constant Angular Momentum

o Aim of the lecture Concepts in Static Equilibrium

Force balance Moments Moment balance

Force Amplification Use of levers Simple Pulley

Concepts in Dynamic Equilibrium Constant momentum Constant angular momentum

o  Main learning outcomes familiarity with

Moments Levers Centre of Mass for a system Equilibrium

Lecture 9

Moments

To move the world, Archimedes needed a lever

•To move a large mass:o Use a large force ORo A small force + lever

Moments

• The force needed to lift the load (effort) x D = The load x d• The pivot point is called a fulcrum

d

D

Moments

F

D

The MOMENT of the force F about the pivot P is D x F

Moment = DF

P

•This is a simple lever,• it is used to

Amplify a force

The same effect can be achieved in other ways:

•Note that in this case: The force amplification is less than 1 The distance travelled by the load is greater than the force

d D

F

To lift 1kg:

D d F50cm 30cm 1kg 9.81 30 / 50 Newtons = 5.9N

1m 20cm 1kg 9.81 20 / 100 Newtons = 2.0N

2km 1cm 1kg 9.81 1 / 2.105 Newtons = 4.9x10-5N

Which is the lever a fruit fly would need!

A 1kg weight makes a 9.81N force

But not much movement in mass

•And not very practical either•A 2km lever would be very heavy•Especially for a fruit fly

Distance 1kg moves Distance fruit fly moves 1cm 2km

=

0.1mm

20m

o Note that energy is conserved:

Force x distance 1kg moves = Force fruit fly uses x distance fruit fly moves

Work Done on 1kg mass = Work done by fruit fly

Levers not always practicalSame principle apply to pulleys

o In these The A force moves 1m

o L moves 1m/4 = 25cmo Max load = 4f

The B force moves 1mo L moves 1m/5 = 20cmo Max load = 5f

The usual form is a ‘block and tackle’

o Several pulley wheels next to each otheroThe amplification factor is

The number of supporting ropes Excluding the one being pulled

Moments and Torques

oTwo moments acting on one body such that: The total force is zero The total moment is not zero

o Produce a torque [see earlier]

F

F

d d

This is a torque of value= 2dFThe pivot ‘does nothing’

BUT this is NOT a puretorque because the pivotpoint will also be applying aforce to the lever.

CAREFUL!This is not completelyobvious

Centre of Gravity

The 10 balls act as if they were a single massActing at one distance along the lever.The single distance is directly below the centre of gravity

An extended mass, or a collection of masses can often be represented by a single mass equal to the total mass located at one point the point is called the centre of gravity.

Centre of gravity

To balance the centre of gravitymust be above the support point.

r1 r2

r3

R = (m1r1+m2r3+m3r3)

m1+m2+m3

R is the position of the centre of gravity

R

Static Equilibrium

Is when

•The (vector) sum of •forces is zero•moments is zero•momentum is zero•angular momentum is zero

d1 d2

F1

F2

o d1F1 = d2F2 for equilibrium Or lever will rotate Even if the two forces are equal

Equilibrium in Moments

d1 d2

F1

F2

o d1F1 = d2F2 for equilibrium

Equilibrium in Torque (and Force)

Consider the same situation

Fp

o Also Fp = F1+ F2 (to prevent translation of lever Fi=0)oAbout Rod centre of gravity

Anticlockwise moment is = F1(d1+ dcog) = F2d2+ F1dcog

Clockwise moment = (F1+ F2)dcog+(d2 - dcog)F2 = F2d2+F1dcog

o So NO Torque – equilibrium in moments and in Torque are the same thingo (when any forces from the pivot are considered)

Centre of gravityfor the rod (sameas centre of mass)

dcog

Static Equilibrium

DO NOT try this at home

Dynamic Equilibrium

A spinning gyroscope has angular momentumBut it does not change magnitude (no friction case)This is a form of dynamic equilibrium

The axis of the gyroscope will precess aroundthe direction of gravity, but the magnitude of thetotal angular momentum will not change.(this means the axis will rotate around at a constantangular speed)[a calculation is beyond the scope of this course]

Momentum and angular momentumdo not need to be zero

The term ‘dynamic equilibrium’ can also refer to any situationin which the properties relevant to describing the system do not change, even if the parts making up the system do, or are moving.

o Eg Water vapour is in dynamic equilibrium with the liquid waterit is above.

Molecules are constantly exchanged between vapour and liquid But the total number in the liquid (and gas) remains constant

o A rotating Ferris wheel is in dynamic equilibrium.

o A static Ferris wheel is in static equilibrium.

o As it accelerates from static to rotating it is neither.