Physical Quantities and Units
Stick on table from MS word
Scalar and Vector Quantities
Scalar Quantity: A quantity which has magnitude but no direction
Vector Quantity: A quantity which has magnitude and direction
Examples of scalar quantities
Density, temperature, pressure, p.d., frequency, wavelength, power, speed
Examples of vector quantities
Displacement, velocity, acceleration, force, momentum, electric current, magnetic and electric fields
Written using arrows.
The length of the arrow represents the vector's magnitude, whilst its direction shows the direction of the vector.
Remember Pythgoras: A2 + B2 = C2
Multiplying or dividing a vector by another vector can give a vector or scalar quantity.
Multiplying a vector by a scalar ALWAYS gives a vector
SOH CAH TOA = mnemonic for vectors at right angles to one another
The resolution of vectors involves the generation of 2 equivalent vectors from a single vector, i.e. finding the horizontal and the vertical of a force at an angle.
The horizontal in this case is Fsinθ
The vertical is Fcosθ
Together these are known as the two components of the force
The horizontal force is the one which controls the acceleration of an object - the vertical is not required
Definitions in Kinematics
Displacement: Distance moved in a stated direction (vector)
Average Speed: Distance travelled per unit time (scalar)
Velocity: Rate of change of displacement (vector)
Acceleration: Rate of change of velocity (vector)
Instantaneous Speed: The speed at a given instant of time - it is the gradient of the graph of displacement against time at that instant
Graphs of Motion
Instantaneous Velocity: The velocity of an object at a given moment in time
The gradient of a displacement-time graph represents velocity
An upward curve on a displacement-time graph represents acceleration
An upward curve gradually getting less steep on a displacement-time graph represents deceleration
The gradient of a velocity-time graph represents acceleration
The area underneath a velocity-time graph represents displacement
Constant Acceleration Equations
Learn how to derive these equations as well
An object undergoing free fall on Earth has an accleeration g = 9.8118 m s-2
Galileo is said to have measured the accleration of free fall by dropping balls from the top of the Leaning Tower of Pisa. Galileo discovered that the acceleration of free fall is the same for all objects, whatever their mass. Aristotle assumed, without experimenting, that heavier objects would fall faster than lighter objects.
It is different if an object is thrown. All of the horizontal measurements are the same, meaning that horizontal velocity remains constant. Vertical measurements gradually get larger, showing a downward acceleration.
Measurement of g
An electromagnet supports a steel ball. When the current through the electromagnet is switched off, the ball starts to fall and simultaneously an electronic clock is triggered. The ball alls onto a trap door. When it breaks open the trap door, the clock is stopped. the distance that the ball drops is measured and the time, t, for the fall is taken from the clock. The experiement should be repeated several times. Work out the average to arrive at a final value for t. More readings can then be taken at different heights.
Use s = ut + ½ at² where u is zero since the ball starts from rest
This gives: s = ½ gt² and hence... g = 2s/t²
A graph of s against t² will have a gradient of g/2
1. Electromagnet must ONLY JUST support ball; if the current is too high then there will be a delay in releasing the ball after the clock has been triggered
2. If distance is too large/ball too small, air resistance will affect its speed
3. Uncertainties when measuring the the distance of the ball. Should be bottom of the ball to the top of the trap door
Force and the Newton
Net force = mass x acceleration (F = ma)
Remember that forces causes acceleration, not the other way around.
F = ma is not valid at very high speeds because according to Einstein's Theory of Special Relativity, as an object approaches the speed of light, its mass increases.
One Newton: The force that causes a mass of one kilogram to have an accleration of one metre per second every second
Types of force
Gravitational force, Magnetic force, Electrical force.
Motion with Non-Constant Acceleration
Weight: The gravitational force on a body OR mass x acceleration of free fall (W = mg)
Drag: The resistive force which acts on a body when it moves through a fluid
Drag depends on several factors, including velocity, roughness of the surface, cross-sectional area and shape
Terminal Velocity: The velocity at which an object's drag equals its accelerating force. Therefore there is no resultant force and zero acceleration
When an object falls from a great height through the air, the drag on the object increases as it accelerates. Eventually the drag (upwards force) becomes equal to the weight of the object (downwards force) and so the resultant force on teh object is zero. It then travels at a constant velocity. This velocity is called the object's 'terminal velocity'.
Centre of Gravity
Centre of gravity: The point at which the entire weight of an object can be seen to act
To find the centre of gravity of an object
- Object must be of uniform thickness
- Support the object freely on a wire passed through a small hole
- From the wire hang a plumb-line to show the vertical. Mark the line of the string, as the centre of gravity must lie along it
- Repeat the procedure from a different small hole. The centre of gravity will be where the two marked lines intersect
- You can often check this by balancing the object on its centre of gravity on the point of a pencil
Couple: A pair of equal and parallel but opposite forces, which tends to produce rotation only
Torque: One of the forces in a couple x perpendicular distance betweent the forces
Torques produce rotation rather than linear motion and are measured in Newton metres (Nm)
Moment: The turning effect of a single force i.e. force x perpendicular distance from a stated point
The principle of moments states that: For a body in a rotational equilibrium, the sum of the clockwise moments equals the sum of the anticlockwise moments
Density and Pressure
Density: Mass per unit volume (mass over volume)
The symbol for density is 'ρ', pronounced 'rho' and it has the unit kg m-3
Pressure: Force per unit area (force over area)
The units for pressure are Pascals,
One Pascal: The force of 1N spread uniformly over an area of 1 m²
Force upward = force downward
Upward force due to water = Pressure x Area of bottom of ship
Car Stopping Distances
Kinetic energy: ½ mv²
Braking Distance: The distance a vehicle travel while decelerating to a stop
Thinking Distance: The distance travelled from seeing the need to stop to applying the brakes
Stopping Distance: Braking Distance + Thinking Distance
Factors affecting braking distance
- If road is wet / snowy
- Efficiency of brakes
- Weight of vehicle
Factors affecting thinking distance
- Influence of drugs or alcohol
- Day or night
Crumple zones, airbags + seat belts function by increasing the impact time in accidents, thereby decreasing acceleration. As F=ma, force exerted on driver/passengers is decreased, hence decreasing risk of a serious injury.
Crumple zones are parts of a car designed to collapse in a collision, increasing the distance over which the force is acting - passengers continue to move for 0.5m extra
Seat belts stop you from continuing to move forward relatively slowly in comparison to a windscreen - they are wide and soft compared to a small edge of glass/steel
Air bags consist of a nylon bag folded into the steering wheel/dashboard. When the front end of the spring in an accelerometer is stopped by an acceleration of around -10g, occurring only in an accident, the mass on the end of the spring continues to move forward, & hits a switch, igniting NaN3 and KNO3 to form N2 gas to fill the bag
GPS - Satellite A sends out a signal and it arrives after a known time at a GPS. Given the speed of electromagnetic radiation, the distance can be found. Receiver must be on circle X. This is repeated for satellite B, so must be at one of 2 intersections on X and Y. Once satellite C is used, the position of the receiver is found at point Z, intersecting with X and Y
Work and the Joule
Work: Force x Distance moved in the direction of the force
Work is measured in joules
1 Joule: The work done when a force of 1 newton moves its point of application 1 metre in the direction of the force
The Conservation of Energy
Energy: The stored ability to do work
Conservation of energy: Describes the situatino in any closed system, where energy may be converted from one form into another, but cannot be created or destroyed
Work done = energy tranferred
Examples of energy in different forms
- Kinetic energy, when an object has speed
- Gravitational potential energy, where an object is at a high level in the Earth's gravitational field
- Internal energy; the molecules in all objects have random movement and have some potential enegy when they are close to one another. Sometimes called heat energy
Potential and Kinetic Energies
Derivation of g.p.e. equation
If F = ma, then W = mg. The gravitational potential energy lost in falling a distance, h, is:
Force x distance = Wh, hwich gives the general equation, where g is a constant...
g.p.e = mgh
By the law of the conservation of energy, kinetic energy should be equal to gravitational potential energy, provided air resistance is negligible:
mgh = ½ mv²
Where v = speed and h = distance fallen to give:
v = √2gh
Power and the Watt
Power: The rate of doing work
One watt: One joule per second
1kW = 1000W = 1000 J s-1
1 kWh =1000 J s-1 x 3600s = 3,600,000 J
1kWh of energy = approx 15p
Efficiency and Deformation Definitions
Efficiency = (Useful output energy / Total input energy) x 100%
No device can ever have 100% efficiency because some energy is always lost as heat
Elastic Deformation: The object will return to its original shape when the deforming force is removed
Plastic Deformation: The object will not return to its original shape when the deforming force is removed; it becomes permanently distorted
Elastic Limit: The point at which elastic deformation becomes plastic deformation
Tensile Forces: Usually 2 equal and opposite forces acting no a wire in order to stretch it. When both forces have the value T, the tensile force is also T, not 2T
Compressive Forces:Two or more forces that have the effect of reducing the volume of the object on which they are acting or reducing the length of a spring
Deformation of Materials
An experiment to stretch a wire
A long thin copper wire is held in place in a clamp at one end. The other end supports a hanger weight after passing over a pulley. The hanger must be just heavy enough to keep the wire taut.
A marker is attached to the wire and a ruler is fixed in position below the marker. Gradually add mass to the weight (thereby increasing tension). Record the tension/N, reading on the ruler/mm and cumulative extension/mm.
Plot a graph of tension/N (Y) against extension/mm (X). The continual but slow increase in extension nearing the end of the experiment is known as 'creep'.
Graph should show a straight line followed by a curved section. The end of the straight line is known as the limit of proportionaility, and is very close to where the wire ceases to be elastic.
Hooke's Law: the extension of an elastic body is proportional to the force that causes it
In equation form, this is: F = kx, where F = force, x = extension and k is the force constant
Force constant is expressed in newtons per metre e.g. is k = 6 N mm-1, then it takes 6 newtons to cause an extension of 1 mm.
The area underneath a tension-extension graph is equal to the work done, so work done is equal to the area of the triangle, = ½ Fx
Since F = kx, work done therefore = ½kx²
Elastic potential energy = ½kx²
The Young Modulus
Stress: Force per unit cross-sectional area (force / area)
Strain: Extension per unit length (extension / length)
Young Modulus: Stress divided by strain ( (force x length) / (area x extension). This is equal to (force / extension) x (length / area) )
The experiment to measure tension against extension can be modified to find the Young modulus of the material of the wire. Length = length of wire section from clamp to marker. Area = area of cross section of the wire, calculated by measuring its diameter using a micrometer screw gauge.
F/x = gradient of the straight line part of the graph
- Extension can be measured more accurately using a travelling microscope
- Use the manufacturer's tabulated data from their products as this means you don't have to measure the wire's diameter and square it - squaring a value doubles the uncertainty
Categories of Materials
Ultimate Tensile Strength: The maximum tensile force that can be applied to an object before it breaks
Brittle: A material that distorts very little even when subject to a large stress and does not exhibit any plastic deformation e.g. concrete
Ductile: Materials that have a large plastic region (therefore they can be drawn into a wire) e.g. copper
Polymeric: A material made of many smaller molecules bonded together, often making tangled long chains. These materials often exhibit very large strains (e.g. 300%) e.g. rubber
Stick on graphs