Scalar and Vectors
Scalars only have size, Vectors have size and direction.
- Scalars - mass, temp., time, length, speed, energy.
- Vectors - displacement, force, velocity, acceleration, momentum.
You can add vectors using pythagoras (a2 + b2 = c2) and trigonometry (SOHCAHTOA). You can also use pythagoras and trigonometry for adding resultant forces or velocities.
When adding vectors make sure they are tip to tail, and if the vectors aren't at right angles you should do a scale drawing.
Equations of Motion
Uniform acceleration is constant acceleration. Acceleration could mean a change in speed or direction of both.
The four equations of motion -
- V = u + at
V is final velocity, u is intial velocity, a is accleration, t is time
- s = ut + 1/2 at^2
s is displacement
- v^2 = u^2 +2as
- s = (u+v)/2 x t
Newton's Laws of Motion
Newton's 1st Law - The velocity of an object will not change unless a resultant force acts on it. Eg. Apple on a table wont go anywhere because the forces on it are balanced.
Reaction (R) = weight (kg)
Newton's 2nd Law - Acceleration is proportional to the force.
resultant force (N) = mass (kg) x acceleration (ms-2) [F = m x a]
Remember - Resultant force is the vector sum of all the forces. The acceleration is independant of the mass.
All objects fall at the same rate (if you ignore air resistance).
Newton's 3rd Law - Each force has an equal, opposite reaction force. Eg. If an object A exerts a force on object B, then object B exerts an equal but opposite force on object A. The forces are always the same type but the forces are not both applied to the same object.
Weight, Mass and the Centre of Gravity
W = mg
Weight = mass x gravity
Mass is scalar and force (weight) is a vector.
Density is mass per unit volume. Density doesnt vary with size or shape. Is it calculated by p = m/v
Density = mass/ volume (density is measured in gcm-3 or kgm-3)
Centre of gravity - assume all the mass is in one place. An object will be nice and stable if it has a low centre of gravity and a wide base area.
Free body force diagrams show all the forces on a single body. If a body is in equilibrium the forces acting on it will be balanced. Some of the forces to look out for - mg (weight), drag, push, friction, resistance (to earth).
Force calculations - Resolving a force. Resolving a force means spliting it into components - vertical and horizontal. Replace the Force (F) with vertical force (Fv) and horizontal force (Fh) and this is resolving the Force (F). You use trigonometry to find Fv and Fh.
Fh = F cos (the angle)
Fv = F sin (the angle)
Use vector addition to get the resultant force. First add the vectors together and use pythagoras to get the length, then divide your A and B from pythagoras section, take that answer and put it totan-1. You now have the angle, most answers go along the lines of Force 999N at an angle 99 degrees from north.
Work is done whenever energy is transferred.
Work (J) = Force (N) x Distance (m) [W=Fs]
Work is the energy thats been changed from one form to another - not always the total energy. The equation assumes that the direction of force is the same as the direction of movement.
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.
The direction of force isnt always in the same direction as the movement. If not you use horizontal and vertical components.
W = Fs cos (the angle)
Power equals work done per second.
Power (W) = Work Done (J) / Time (s) [P=Wt]
The watt is defined as a rate of energy transfer equal to 1 joule per second.
Power (W) = Force (N) x Velocity (ms-1) [P=Fv]
This equation can be helpful when you are given speed in a question.Sometimes in a power question the force and the motion are in different directions, like with work, we then use horizontal and vertical components.
P = Fv cos (the angle)
Mechanics in the Real World
Many factors affect how quickly a car stops.
Thinking distance + Braking distance = stopping distance
Thinking distance = Speed x Reaction time
Car safety features are usually designed to slow you down more gradually and increase the collision time, some are designed to reduce the force on you. Example features - Seatbelts, airbags, crumple zones (at the front and back of the car) and safety cages (around the car to prevent crushing).
Forces act on sports people. People who bungee and rock climb have forces acting on them all the time, these can be shown using free body diagrams (see forces). Gravity is the only force acting on projectiles.
Conservation of Energy
Principle of Conservation of Energy - 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.
Ek = 1/2 mv2 - Kinetic Energy
(m = mass, v = velocity)
Eg = mgh - Gravitational Potential Energy
(m = mass, g = gravity, h = height)
E = 1/2 Ke2 - Elastic Potential
(e = extension, K= stiffness)
Hookes law states that extension is proportional to force.
F = Ke (K = stiffness constant, e = extension)
Hookes law can apply to springs. The extension or compression of a spring is proportional to the force applied. Tensile - stretch spring, Compressive - squash spring.
Hookes law stops working when the load is great enough. On a graph, if there is a straight line relationship between load and extension then Hookes law is obeyed.
When the graph starts to curve, this is where point E, the Elastic limit is. Increase the load past the elastic limit and the material will be permanently strecthed.
A strecth can be plastic or elastic -
- Elastic - Material returns to its original shape when the forces are removed. For a metal, elastic deformation happens as long as Hookes law is obeyed.
- Plastic - Material is permanently stretched. A metal stretched past its elastic limit shows plastic deformation.
The Young Modulus is stress/strain. Below the limit of proportionality for a material, stress divided by strain is a constant, the constant is called the Young Modulus (E). It is used by engineers to make sure materials can withstand forces.
E = stress/strain = F/A / e/l
To find the Young Modulus you need a very long wire. Use a rule, marker and a pulley with some weights. The longer and thinner the wire, the more it extends for the same force. Take the correct measurements down and you can find the Young Modulus.
Use a stress-strain graph to find E.
Gradient = stress / strain = E (Young Modulus)
The area under a stress-strain graph gives the stored energy. When Hookes law is obeyed the stress-strain graph will have a straight line so you can calculate the energy per unit volume -
energy = 1/2 x strain x stress
Stress and Strain
A stress causes a strain.
Stress = F/A (F = force, A = cross-sectional area)
Strain = e / l (e = extension, l = original length)
A material subjected to a pair of opposite forms might deform. A stress big enough to break the material is called the breaking stress. The maximum stress the material can withstand is called the ulitimate tensile stress. On a graph the breaking stress (B) would be the end of the line and the ultimate tensiles stress (UTS) would be just before it around where the curve goes slightly down.
Elastic strain energy is the energy stored in a stretched material. Elastic strain energy is given by the area under a stress-strain graph.
You can calculate the energy stored in a stretched wire provided it obeys Hookes law.
E = 1/2 Ke2
Behaviour of solids
- Brittle - materials break suddenly without plastically deforming. Eg. chocolate bar, ceramics
- Ductile - materials can be drawn into wires without losing their strength. Eg. copper for wires.
- Malleable - materials change shape but may lose their strength. Eg. Gold, gold rings can be changed shape very easily but a gold bar wouldnt.
- Hard - materials are very resistant to cutting, indentation and abrasions. Eg. cutting tool like a chisel, diamond.
- Stiff - materials have a high resistance to bending and stretching. Stiffness is measured by the Young Modulus - the higher the value, the stiffer the material. Eg. Helmets.
- Tough - materials are really difficult to break. Really tough materials can absorb a lot of energy so are very difficult to break. Eg. Polymers.
Stress-strain graphs for ductile materials curve. A straight line shows it obeys Hookes law, the limit of proportionality (P) is the point where the graph starts to curve, the elastic limit (E) is where the material wouldnt return to its original length and the yield point (Y) is where the material starts to stretch without any extra load.
Velocity - Time Graphs
The gradient of a Velocity-Time graph gives you the acceleration. The distance travelled = area under the graph. Non uniform acceleration (not constant, changing) is a curve on a Velocity-Time graph.
Displacement - Time Graphs
Acceleration means a curved Displacement-Time Graph. Different acceleration have different gradients, bigger accelerations have steeper gradients. Decreasing acceleration has the curve going the other way. The gradient of a Displacement-Time graph gives you the velocity and its the same with curved lines, you just have to draw a tangent.
velocity = change in displacement / time taken
Streamlines and Flow
Streamlines are stable flowlines. A flowline is the path that a particular fluid element takes.
Streamlines are parallel in laminar flow, this usually occurs when a fluid is flowing slowly.
Flowlines are unstable in turbulent flow, this usually occurs when a fluid is flowing quickly. In turbulent flow, the fluid often moves around in miniature whirlpools - called eddy currents.
Both types of flow are used in manufacturing. Laminar flow, smooth flow is used in pipes. Turbulent flow, mixed flow is used in mixing chemicals.
Viscous drag is the force of friction produced by a flowing fluid. Friction opposes motion, so the force acts to slow the flow. The size of the force depends on the viscosity of the fluid - the higher the viscosity, the larger the force. Viscous drag is much larger when the flow is turbulent.
Rate of flow depends on viscosity. The higher the viscosity of a fluid, the slower its rate of flow. Viscosity depends on temperature. The viscosity of most fluids decreasess as the temperature increases, fluids generally flow faster it they're hotter.
Rate of flow = Volume moved / time taken
Viscous drag acts on objects moving through fluids. You can calculate the force due to viscous drag on a sperical object moving through a fluid using stokes law -
F = 6 (pi) n r v
(n = viscosity (Nsm-2), r = radius (m), v = velocity (ms-1)
(pi) is the mathematical symbol and is equal to 3.14159.
Fluids exert upthrust on immersed objects.
Upthrust = weight of fluid displaced.