MECHANICS 1

MECHANICS 1

1. Particle moving in a straight line

   • Kinematics (motion)
     • Graphs
       • Speed - time
         • Distance Travelled = area under graph
           • Trapezium rule: Area = average parallel sides x height
             • A = 1/2(a+b)h
           • Objects meet when they have the same distance travelled
       • Distance - time
       • Acceleration - time
     • Constant acceleraton
       • Acceleration = gradient of line
       • In free fall: Acceleration = g (9.8 ms-2)
         • Ignore air resistance
       • SUVAT
         • s=displacement u=initial velocity v = final velocity a=acceleration t=time
         • v=u+at s=(u+v)t/2 v^2=u^2+2as s=ut+1/2at^2 s=vt-1/2at^2
         • State which way is positive
   • Dynamics (motion of bodies under action of forces)
     • Momentum, p
       • = mass x velocity
       • Collisions
         • Conservation of momentum
           • Momentum before = momentum after
             • m1u1 + m2u2 = m1v1 + m1v1 + m2v2
       • Impulse, I
         • = force x time
           • Impulse = area under force - time graph
         • = final momentum - initial momentum
           • =mv-mu
         • Newtons 3rd Law: Objects exert equal and opposite forces (and so impulse) on eachother
     • Newtons 2nd Law: F = mass x acceleration

2. Statics (bodies at rest with forces in equilibrium)

   • Additional forces
     • Weight, W
       • = mass x g
       • Due to gravity acting on an abject vertically downwards
     • Tension, T
       • Being pulled along by a string
         • Strings
           • One string: Tension equal
           • Two separate strings: Tension different
           • In-extensible
             • Does not change length so accelerations and velocity of two particles attached are equal
           • Becomes slack
             • No tension so change in acceleration, must resolve again
     • Thrust, T
       • Being pushed along by a rod
         • Rods
           • Uniform
             • Weight acts at the centre of the rod
           • Light
             • Adds no weight to the system
           • Straight and does not bend, all forces remain perpendicular
     • Normal reaction, R
       • Perpendicular to the surface in contact with the object
     • Friction, F
       • F = uR
         • u is Mue
         • In limiting equilibrium (on the point of movement), otherwise equal to or less than
       • Opposes the motion between two rough surfaces
       • Smooth
         • No friction
       • u = coefficient of friction
         • 0 < u <1
   • Balanced, no overall motion, equal and opposite in any direction
   • Resolving Forces
     • Resolve in the direction of the acceleration
       • Then resolve perpendicular to this
         • R(^): R - mg
       • R(>): ma - uR
     • If static
       • Resolve horizontal and vertical or up plane and perpendicular to plane
     • At an angle
       • Resolve to find the component of the force that acts in the direction of motion
         • Component of F = Fcos0
           • 0 is Theta
     • Resultant Force
       • Resolve force in perpendicular directions and then apply pythagoras

3. Moments

   • Moment about a point = Force x distance
   • The sum of moments
     • State which way is positive (clockwise or anticlockwise)
     • Add up the moments about a point
       • M(P): clockwise moments - anticlockwise moments
   • In equilibrium
     • In equilibrium the sum of moments about any point is zero
       • M(P) clockwise moments = anticlockwise moments
     • Resultant force in any direction is zero
   • If tilting about a point, other point support force = 0

4. Vectors (quantity with both magnitude and direction)

   • i, j notation
     • i is a unit vector in the x-direction j is a unit vector in the y-direction
     • Add terms i and j seperately
     • i and j either bold or underlined
   • Adding vectors
     • Triangle law of addition
       • AC = AB + BC
   • Speed
     • Calculated using pythagoras for i and j
     • Magnitude of velocity vector
   • Length of line = magnitude
     • Arrow to show direction
   • Equal
     • Same magnitude and direction
   • Parrallel
     • Same direction
   • Vectors involving time
     • r = r0 + vt
       • r = Position vector at time t
       • r0 = Original position vector at time t=0
       • v = Velocity vector
     • Bearings
       • From north clockwise 3 significant figures before the decimal
     • Objects meet when they have the same position vector at the same time

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