# MECHANICS 1

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## M1 maths AQA : Particle moving in a straight line, Kinematics, Dynamics, Statics, Moments, Vectors

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MECHANICS 1
1 Particle moving in a straight line
1.1 Kinematics (motion)
1.1.1 Graphs
1.1.1.1 Speed - time
1.1.1.1.1 Distance Travelled = area under graph
1.1.1.1.1.1 Trapezium rule: Area = average parallel sides x height
1.1.1.1.1.1.1 A = 1/2(a+b)h
1.1.1.1.1.2 Objects meet when they have the same distance travelled
1.1.1.1.2
1.1.1.2 Distance - time
1.1.1.3 Acceleration - time
1.1.2 Constant acceleraton
1.1.2.1 Acceleration = gradient of line
1.1.2.2 In free fall: Acceleration = g (9.8 ms-2)
1.1.2.2.1 Ignore air resistance
1.1.2.3 SUVAT
1.1.2.3.1 s=displacement u=initial velocity v = final velocity a=acceleration t=time
1.1.2.3.2 v=u+at s=(u+v)t/2 v^2=u^2+2as s=ut+1/2at^2 s=vt-1/2at^2
1.1.2.3.3 State which way is positive
1.2 Dynamics (motion of bodies under action of forces)
1.2.1 Momentum, p
1.2.1.1 = mass x velocity
1.2.1.2 Collisions
1.2.1.2.1 Conservation of momentum
1.2.1.2.1.1 Momentum before = momentum after
1.2.1.2.1.1.1 m1u1 + m2u2 = m1v1 + m1v1 + m2v2
1.2.1.2.2
1.2.1.3 Impulse, I
1.2.1.3.1 = force x time
1.2.1.3.1.1 Impulse = area under force - time graph
1.2.1.3.2 = final momentum - initial momentum
1.2.1.3.2.1 =mv-mu
1.2.1.3.3 Newtons 3rd Law: Objects exert equal and opposite forces (and so impulse) on eachother
1.2.2 Newtons 2nd Law: F = mass x acceleration
2 Statics (bodies at rest with forces in equilibrium)
2.1.1 Weight, W
2.1.1.1 = mass x g
2.1.1.2 Due to gravity acting on an abject vertically downwards
2.1.2 Tension, T
2.1.2.1 Being pulled along by a string
2.1.2.1.1 Strings
2.1.2.1.1.1 One string: Tension equal
2.1.2.1.1.2 Two separate strings: Tension different
2.1.2.1.1.3 In-extensible
2.1.2.1.1.3.1 Does not change length so accelerations and velocity of two particles attached are equal
2.1.2.1.1.4 Becomes slack
2.1.2.1.1.4.1 No tension so change in acceleration, must resolve again
2.1.3 Thrust, T
2.1.3.1 Being pushed along by a rod
2.1.3.1.1 Rods
2.1.3.1.1.1 Uniform
2.1.3.1.1.1.1 Weight acts at the centre of the rod
2.1.3.1.1.2 Light
2.1.3.1.1.2.1 Adds no weight to the system
2.1.3.1.1.3 Straight and does not bend, all forces remain perpendicular
2.1.4 Normal reaction, R
2.1.4.1 Perpendicular to the surface in contact with the object
2.1.4.2
2.1.5 Friction, F
2.1.5.1 F = uR

Annotations:

• u is Mue
2.1.5.1.1 In limiting equilibrium (on the point of movement), otherwise equal to or less than
2.1.5.1.2
2.1.5.2 Opposes the motion between two rough surfaces
2.1.5.3 Smooth
2.1.5.3.1 No friction
2.1.5.4 u = coefficient of friction
2.1.5.4.1 0 < u <1
2.2 Balanced, no overall motion, equal and opposite in any direction
2.3 Resolving Forces
2.3.1 Resolve in the direction of the acceleration
2.3.1.1 Then resolve perpendicular to this
2.3.1.1.1 R(^): R - mg
2.3.1.2 R(>): ma - uR
2.3.2 If static
2.3.2.1 Resolve horizontal and vertical or up plane and perpendicular to plane
2.3.3 At an angle
2.3.3.1 Resolve to find the component of the force that acts in the direction of motion
2.3.3.1.1 Component of F = Fcos0

Annotations:

• 0 is Theta
2.3.4 Resultant Force
2.3.4.1 Resolve force in perpendicular directions and then apply pythagoras
3 Moments
3.1 Moment about a point = Force x distance
3.2 The sum of moments
3.2.1 State which way is positive (clockwise or anticlockwise)
3.2.2.1 M(P): clockwise moments - anticlockwise moments
3.2.3
3.3 In equilibrium
3.3.1 In equilibrium the sum of moments about any point is zero
3.3.1.1 M(P) clockwise moments = anticlockwise moments
3.3.2 Resultant force in any direction is zero
3.4 If tilting about a point, other point support force = 0
4 Vectors (quantity with both magnitude and direction)
4.1 i, j notation
4.1.1 i is a unit vector in the x-direction j is a unit vector in the y-direction
4.1.2 Add terms i and j seperately
4.1.3
4.1.4 i and j either bold or underlined
4.2.1.1 AC = AB + BC
4.2.2
4.3 Speed
4.3.1 Calculated using pythagoras for i and j
4.3.2 Magnitude of velocity vector
4.4 Length of line = magnitude
4.4.1 Arrow to show direction
4.5 Equal
4.5.1 Same magnitude and direction
4.6 Parrallel
4.6.1 Same direction
4.7 Vectors involving time
4.7.1 r = r0 + vt
4.7.1.1 r = Position vector at time t
4.7.1.2 r0 = Original position vector at time t=0
4.7.1.3 v = Velocity vector
4.7.2 Bearings
4.7.2.1 From north clockwise 3 significant figures before the decimal
4.7.3 Objects meet when they have the same position vector at the same time