1. Equilibrium Systems ‘in balance’ o Static Equilibrium  Balanced Forces  Balanced Moments  Centre of Gravity o Dynamic Equilibrium  Constant Linear Momentum  Constant Angular Momentum

2. 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

3. Moments To move the world, Archimedes needed a lever To move a large mass: o Use a large force OR o A small force + lever

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

5. Moments F D The MOMENT of the force F about the pivot P is D x F Moment = DF P

6. This is a simple lever, it is used to Amplify a force

7. 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

8. d D F To lift 1kg: DdF 50cm30cm1kg  9.81  30 / 50 Newtons = 5.9N 1m20cm1kg  9.81  20 / 100 Newtons = 2.0N 2km1cm1kg  9.81  1 / 2.10 5 Newtons = 4.9x10 -5 N Which is the lever a fruit fly would need! A 1kg weight makes a 9.81N force

9. 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

10. 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 practical Same principle apply to pulleys

11. o In these  The A force moves 1m o L moves 1m/4 = 25cm o Max load = 4f  The B force moves 1m o L moves 1m/5 = 20cm o Max load = 5f The usual form is a ‘block and tackle’ o Several pulley wheels next to each other oThe amplification factor is The number of supporting ropes Excluding the one being pulled

12. 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  = 2dF The pivot ‘does nothing’ BUT this is NOT a pure torque because the pivot point will also be applying a force to the lever. CAREFUL! This is not completely obvious

13. Centre of Gravity The 10 balls act as if they were a single mass Acting 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.

14. Centre of gravity To balance the centre of gravity must be above the support point.

15. r1r1 r2r2 r3r3 R = (m 1 r 1 +m 2 r 3 +m 3 r 3 ) m 1 +m 2 +m 3 R is the position of the centre of gravity R

16. Static Equilibrium Is when The (vector) sum of forces is zero moments is zero momentum is zero angular momentum is zero

17. d 1 d 2 F1F1 F2F2 o d 1 F 1 = d 2 F 2 for equilibrium  Or lever will rotate  Even if the two forces are equal Equilibrium in Moments

18. d 1 d 2 F1F1 F2F2 o d 1 F 1 = d 2 F 2 for equilibrium Equilibrium in Torque (and Force) Consider the same situation FpFp o Also F p = F 1 + F 2 (to prevent translation of lever  F i =0) oAbout Rod centre of gravity  Anticlockwise moment is = F 1 (d 1 + d cog ) = F 2 d 2 + F 1 d cog  Clockwise moment = (F 1 + F 2 )d cog +(d 2 - d cog )F 2 = F 2 d 2 +F 1 d cog o So NO Torque – equilibrium in moments and in Torque are the same thing o (when any forces from the pivot are considered) Centre of gravity for the rod (same as centre of mass) d cog

19. Static Equilibrium DO NOT try this at home

20. Dynamic Equilibrium A spinning gyroscope has angular momentum But it does not change magnitude (no friction case) This is a form of dynamic equilibrium The axis of the gyroscope will precess around the direction of gravity, but the magnitude of the total angular momentum will not change. (this means the axis will rotate around at a constant angular speed) [a calculation is beyond the scope of this course] Momentum and angular momentum do not need to be zero

21. The term ‘dynamic equilibrium’ can also refer to any situation in 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 water it 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.