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MSD 210 SummaryTheme 1 – RotationPlanar Motion  Translation – every point in body remains in the same parallel direction o Rectilinear translation – straight parallel motion o Curvilinear translation – linear motion in curved directions  Rotation – angular motion (relative turning) – also called f...

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MSD 210 Summary Theme 1 – Rotation Planar Motion  Translation – every point in body remains in the same parallel direction o Rectilinear translation – straight parallel motion o Curvilinear translation – linear motion in curved directions  Rotation – angular motion (relative turning) – also called fixed axis rotation  General plane motion – consists of rotation and translation Rotation  Constant motion can be calculated using equations of motion o ω =ω 0+αt o o 

Rotation about a fixed axis o v =rω o

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2 ω2 =ω0 +2 α (θ−θ 0) 1 2 θ=θ 0 +ω 0 t + α t 2

an =r w2=

v2 =vω r

o at =rα Working in 3 dimensions (and for all other questions that aren’t just straight forward) o v =ω ×r o an =ω × (ω × r) o at =α ×r

Example: A motor A accelerates uniformly from 0 to 3600 rev/min in 8 seconds. It turns a drum B using a pulley. The motor has a radius of 75mm and the drum of 200mm. Find: a) The number of revolutions of B during 8s b) The angular velocity of the drum at 4s c) The number of revolutions of B during the first 4 seconds

Relative velocity  When choosing two points on the same rigid body, the relative velocity can only be rotational  Thus: o v a =v b + v a /b v a =v b + ω ×r o Example: Calculate the speed of D and the change in Δx when vc = 0.4 m/s constant velocity

Relative acceleration  The same theory as relative velocity can be applied to acceleration

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a a (¿¿ A/ B)t (¿¿ A/ B)n+ ¿ a A =a B +¿ o

o 

a (¿¿ A/ B)n=ω × ω ×ω ¿ a (¿¿ A/ B)t=α ×r ¿

Always do a velocity analysis of the system first

Theme 2 – Kinetics of a system of particles Newton’s Second Law  Calculating centre of mass

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Proof that the sum of the masses and their relative positions to the centre of mass is zero

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Proof that the sum of the forces is the mass times acceleration

Example: Three monkeys, A, B and C are hanging from a rope suspended from D. A accelerates down at 2 m/s 2, C accelerates down at 1.5 m/s2 and B moves at a constant speed. Find the tension of the rope at D

Work/ Energy  Work to kinetic energy theorem

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When considering internal force and the effect on the work done o Rigid body, or body connected by frictionless hinges – all pairs of internal forces do no work  Each point undergoes the same displacement o System containing elastic elements – the points of application don’t undergo identical displacement  The work is thus not zero  But if no friction can be accounted for using the elastic potential energy of the system 

E.g. springs

1 V e= k x2 2

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Accounting for gravitational potential energy/ elastic potential energy

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Simplifying the kinetic energy equation

Example: two spheres (mass m) connected by a cord length 2b are initially at rest on a horizontal plane. A vertical force F is applied to the centre of the cord, find the maximum velocity that can be achieved when the spheres collide and what is the maximum value for F without lifting the balls off the ground

Impulse and Momentum  Linear momentum of a system proof

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Impulse momentum equation

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Angular momentum about the origin

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Angular momentum about the centre of mass

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Angular momentum about a point P

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Finding the moment sum about P

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Moment is not equal to the derivative of the angular momentum about point p

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Four exceptions when it is true o If p does not move - vp = 0 o If p = G – the cross product of the same value is zero o Vp||VG o Vg = 0

Example: A 16kg carriage moves horizontally with its guide at a speed of 1.2 m/s and carries two assemblies of balls and light rods which rotate around a shaft at O. Each ball weighs 1.6kg. Find the a) kinetic energy, b) the linear momentum and c) the magnitude of HO...