# Topic 2 - Mechanics

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## Scalar Quantities

• Just an amount of something
• No direction
• E.g.:
• Mass
• Time
• Energy
• Temperature
• Length
• Speed
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## Vector Quantities

• Has both magnitude and direction
• Drawn as arrows with their size next to them. In the exam, you may see quantities written with arrows above them to show they are vectors
• E.g.:
• Displacement
• Force
• Velocity
• Acceleration
• Momentum
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## Resultant Vectors

• Adding two or more vectors is called finding the resultant of them
• You should start by drawing a diagram: draw the arros 'tip to tail.' If you're doing a vector subtraction, draw the vector you're subtracting with the same magnitude but pointing in the opposite direction
• If the vectors are at right angles to each other, you can use pythagoras and trigonometry to find the resultant
• If the vectors aren't at right angles, you may need to draw a scale diagram

In an exam:

• First, draw the vectors tip-to-tail, then draw a line from the tail of the first vector to the tip of the last vector to give the resultant. If you have drawn to scale, measure the resultant force with a ruler. Now find the bearing of the resultant force (3 digits)
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## Resolving Vectors

• You start from the resultant vector and split it into horizontal and vertical components.
• Use SOHCAHTOA to find the magnitude of the horizontal and vertical (e.g. Vh=v cos θ)
• Think of the vertical and horizontal components seperately once completed
• You can also use scale diagrams, which is easier to do on squared paper
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## Motion with Uniform Acceleration

• Speed - How fast something is moving, regardless of direction
• Displacement (s) - How far an object's travelled from its starting point in a given direction
• Velocity (v) - The rate of change of an object's displacement (its speed in a given direction)
• Acceleration (a) - The rate of change of an object's velocity
• Uniform means 'Constant'
• Uses SUVAT equations:
• a = (v - u)/t                  (v=u+at)
• s = ((u - v)t)/2
• s = ut + 1/2 at^2
• v^2 = u^2 +2as
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## Free Fall

• Free fall is when there's only gravity and nothing else
• Gravitational field strength = 9.81 Nkg^-1
• The force on an object due to gravity is called Weight
• Acceleration and force are related by F = ma
• An object in free fall, a = f/M = g
• You can replace a with g in the equations of motion (a = 9.81)
• G acts downwards, so it is usually negative.
• t is alway positive
• u and v can be positive or negative
• s can be positive or negative
• You can calculate g by performing an experiment and plotting a graph of your results
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## Free Fall Experiments

• 1) Set up the equipment shown in the diagram • 2) Measure the height h from the bottom of the ball bearing to the trapdoor
• 3) Flick the switch to simultaneously start the timer and disconnect the electromagnet, releasing the ball bearing to the trapdoor
• 4) The ball bearing falls, knocking the trapdoor down and breaking the circuit, which stops the timer. Record the time t shown on the timer
• 5) Repeat this experiment three times and avergae the time taken to fall from this height. Repeat from different heights
• 6) You can use these results to find g using a graph
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## Free Fall Experiment - The Graph

• Plot a graph of height (s) against the time, squared
• You know that with constant acceleration, s = ut + 1/2 at^2
• Rearranging this gives 1/2 a = s/t^2, or 1/2 g = s/t^2 (remembe the acceleration is all due to gravity)
• So the gradient of the line of best fit, change in displacement / change in time squared, is equal to 1/2 g
• As you know g, you can calculate the percentage difference between your value of g and the true value, and use this to evaluate the accuracy of your results
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## Projectile Motion

• You have to think of horizontal and vertical motion seperately, using SUVAT twice
• If something's projected at an angle, you'll start with horizontal and vertical velocity:
• Resolve the initial velocity into horizontal and veritcal components
• Often you'll use the vertical component to work out how long its in the air and/or how high it goes, and the horizontal component to work out how far it goes while it's in the air
• You can investigate projectile motion using a video camera:
• Find the frame rate and record a ball's position in each frame
• You can also do this with strobe photography
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## Displacement-Time Graphs

• Acceleration means a curved displacement-time graph
• Bigger accelerations are steeper, deceleration has a decreasing gradient
• The gradient of a displacement-time graph tells you the velocity
• With acceleration, you need to draw a tangent on the graph at the exact point and find its gradient.
• To find the average velocity, just divide the final change in displacement by the final change in time - it doesn't matter if the graph is curved or not
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## Velocity-Time Graphs

Velocity-Time Graphs:

• Acceleration is the gradient of this graph
• Uniform acceleration is always a straight line
• The steeper the gradient, the greater the acceleration
• y = mx + c                     =                           v = u + at         where:
• u = c (y intercept)
• Acceleration isn't always uniform (it is the same as on a D-T graph)
• Displacement = area under a velocity-time graph

Acceleration-Time Graphs:

• Height of graph gives the acceleration at that time, The area under the graph gives the change in velocity
• Negative acceleration is a deceleration
• If a = 0 the object is moving with constant velocity
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## Forces

• Free-body force diagrams show all forces on a single body
• Includes all the forces that act on the body, but not the other way around
• Forces are vectors and should therefore the arrow labels should show size AND direction
• Resolving a force means splitting it into components (see Resolving Vectors if unsure)
• This makes 'awkward' forces easier to deal with
• You can add the components back together to get the resultant force (see Resultant Vectors)
• Use directions that make sense for the situation you're dealing with. If you've got an object on a slope, choose your directions along the slope and at right angles to it. You can turn the paper a bit to help
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## Newton's Laws of Motion

• 1st Law:
• The velocity of an object will not change unless a resultant force acts on it
• A body will stay still (or 'at rest') or move in a straight line at a constant speed, unless there's a resultant force acting on it
• 2nd Law:
• Resultant force (F (N) = m (kg) x a (ms^-2))
• The more force you have acting on an object of certain mass, the more acceleration you get
• This is ignoring air resistance, so ignore air resistance unless stated
• 3rd Law:
• If an object A exerts a force on object B, then object B exerts an equal but opposite force on object A
• They MUST be the same type of force!!!!! (gravitational or electrical)
• If you push against a wall, the wall will push back against you, just as hard
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## Drag and Terminal Velocity NOT DONE

• Two types of friction: solid surfaces or in a fluid
• Air resistance is a type of fluid resistance
• Fluid = liquid or gas
• depends on the viscosity of the fluid
• It increases as speed increases (directly proportional)
• Depends on the shape of the object moving through it
• A projectile is slowed down by air resistance
• You need to know: frictional forces always act in the opposite direction to the motion of the object, they can never speed things up or start moving something, and they convert kinetic energy into heat and sound
• Terminal velocity:
• Terminal velocity is when something is falling, and accelerates until the air resistance equals its weight
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## Momentum

• p = mv (p = momentum in kg m^-1, m = mass in kg, and v = velocity in m^-1)
• Momentum is always conserved
• The total momentum of two objects before they collide equals the total momentum after the collision
• To investigate momentum, an air track and light gates are used
• Newton's 2nd Law:
• F = mv/t
• F = resultant force in N
• m = mass in kg
• v = velocity in ms^-1
• t = time in s
• Also useful to use Newton's 3rd law:
• If object A collides with object B and exerts a force F on B for a time T, Newton's 3rd law states that object B will also exert a force -F on A for a time T
• The change in A's momentum is equal to -FT, and the change in B's momentum is FT, so the overall change in momentum is (-FT) + FT = 0, so momentum is conserved
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## Work and Power

• Work is done whenever energy is transferred, measured in joules
• W = Fs
• Work is the energy that's been changed from one form to another - not necessarily the total energy
• The distance needs to be measured in metres
• The force F will be a fixed value in any calculations, either because its constant or because its the average force
• The equation assumes that the direction of the force is the same as the direction of movement (but it isn't always)
• Definition of a joule: One joule is the work done when a force of 1 newton moves an object through a distance of 1 metre
• Power = work done per second (P = E/t or P = W/t)
• The Watt is defined as a rate of energy transfer equal to 1 joule per second
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## Conservation of Energy and Efficiency

• The principle of conservation of energy says that: 'energy cannot be created or destroyed. Energy can be transferred from one form to another but the total amount of energy in a closed system will not change'
• All energy transfers involve losses - maybe as heat or sound, like in frictional forces or when you use a computer (electrical energy -> light, heat and sound)
• Efficiency is the ratio of useful energy output to total energy output
• Efficiency = useful energy(or power) output / total energy(or power) input
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## Mass, Weight and Centre of Gravity

• Weight = mass x gravitational field strength (W = mg)
• The centre of gravity of an object is the single point that you can consider its whole weight to act together. The object will always balance around this point, although in some cases the centre of gravity will fall outside the object
• You can find the centre of gravity by using symmetry or by experiment (in an irregular object)
• Hang the object freely from a point, draw a vertical line downwards from the point of suspension, then hang from a different point and draw another line of suspension. The centre of gravity is where these two lins cross.
• How high the centre of gravity is tells you how stable the object is
• More stable if it has a low centre of gravity and a wide base area
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## Moments

• Moment of a force (Nm) = force (N) x perpendicular distance from the line of action of the force to the axis of rotation (m)
• Moments must be balanced or the object will turn
• The principle of moments states hat for a body to be in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point
• In a lever, an effort force acts against a load force by means of a rigid object rotating around a pivot
• If the centre of gravity is to one side of the pivot, then there will be a clockwise moment due to the weight of the broomstick acting at a distance x from the pivot. There is no anticlockwise moment, so the broomstick will rotate clockwise. However, if the centre of gravity is directly above the pivot, then there are no clockwise or anticlockwise moments and so the broomstick is in equilibrium.
• Sometimes you may need to resolve forces to calculate moments (in CGP book)
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