# Physics AS Mechanics G481

## 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

## Vector Calculations

Written using arrows.

The length of the arrow represents the vector's magnitude, whilst its direction shows the direction of the vector.

Remember Pythgoras: A^{2} + B^{2} = C^{2}

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

## Vector Resolution

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

## Free Fall

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

## Turning Forces

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

*Floating*

Force upward = force downward

Therefore...

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
- Tiredness
- Day or night

## Car Safety

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

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

*Uncertainty*

- 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

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