Simple Harmonic Motion and Gravitation

Simple Harmonic Motion

Description of a bob - Starting from the equilibrium position, it moves in one direction, back to equilibrium and moves in the other direction, and back to equilibrium again.

At centre - displacement = 0 , velocity = max , acceleration = 0

At the maximum displacement - displacement = amplitude , velocity = 0 , acceleration = max

SHM - the acceleration is proportional to the displacement from a fixed point and is directed towards that fixed point.

ω is the constant of proportionality.

The negative sign is directed towards the fixed point.

Damping is the process whereby energy is taken from the vibrating system to minimise the effect of resonance.

Under and heavy damping quickly reduces the amplitude to zero.

Critical damping reduces the amplitude in the shortest possible time.

Over damping takes longer for the amplitude to reach zero than critical damping.

Displacement is the distance in a given direction.

Amplitude is the maximum displacement from equilibrium of an oscillating object.

Period is the time for 1 oscillation.

Frequency of an oscillating object is the number of cycles of oscillations per second.

Phase difference is measured in radians, for 2 objects oscillating with the same time period (T), where phase 

difference is 2 π t / T

Resonance is the amplitude of vibration of an oscillating system which is subjected to a periodic force which is 

largest when the periodic force has the same frequency as the resonant frequency of the system.

Periodic force is a force that varies regularly in magnitude with a definite time period.

Resonant frequency is the frequency of an oscillating system in resonance.

Gravitation

Gravitational fields are only attractive.

Gravitational field strength is the point in a body’s field as the force is exerted on an object placed at that point 

per kilogram of the objects mass.

GPE is the energy of an object due to its position in the gravitational field.

GPE is zero at infinity as the object is so far away that the effect of gravity is negligible.

GPE is negative on the surface of the Earth as an object needs energy to overcome the gravitational field.

GPE is the work done to move a small object from infinity to that point.

Gravitational potential at a point in a gravitational field is the work done per unit mass to move a small object 

from infinity to that point.

Radial Field is where field lines are straight and converge as if from a single point over a large distance.

Uniform field is a region where the field strength is the same in magnitude and direction at every point in the field over a small distance.

The potential gradient at a point in a gravitational field is the change in potential per metre at that point.

Equipotentials are lines of equal potential.

Equipotential surfaces are perpendicular to field lines.

Kepler’s law states that T² is proportional to r³.

Geostationary satellites stay over the equator with a circular orbit which takes 24 hours to fully orbit the Earth in the same direction.

Geosynchronous satellites repeatedly orbit regular points over the Earth over time.

Newton’s law of gravitation states the gravitational force between two point masses at distance apart is given by F = G m M / r².

©2011 Grant Dwyer

Momentum and Circular Motion

Momentum

Newton’s first law states that an object stays at rest unless acted on by an external force.

Newton’s second law states that force is proportional to the rate of change of momentum, unless acted on by an external force.

Newton’s third law states that an action has an equal and opposite reaction.

The Principle of Conservation of Momentum - In a collision, the momentum before the collision is equal to the momentum after the collision, unless acted on by an external force.

The Principle of Conservation of Energy – The total energy after the change is always equal to the total energy before the change, unless acted by an external force.

Area under force-time graph is impulse/momentum.

An elastic collision is where kinetic energy is conserved, but most collisions are inelastic.

Linear momentum is the momentum in a straight line (p = mv).

Circular Motion

In circular motion, linear velocity is at tangents around a circle at any point.

Acceleration and force are directed towards the centre of the circle.

Angular velocity is the velocity around the circle.

An object changing velocity around the circle means that it is accelerating with a constant speed. This is because the direction is constantly changing.

Centripetal force and acceleration are towards the centre of the circle.

Centrifugal force is away from the centre of the circle.

1 radian is equal to 57.3 degrees.

©2011 Grant Dwyer

Nuclear Reactor

The neutrons that are produced in the reaction can then go on and be part of other reactions, creating a chain reaction.

The neutrons that are used are low energy (thermal neutrons) as they can only be captured in this way.

The amount of fissionable material necessary for the chain reaction to sustain itself is called the critical mass.

The moderator slows down or/and absorbs neutrons.

The moderator needs to be able to slow down some neutrons enough to cause further fission and that absorbs more neutrons the higher the temperature will decrease the chance of meltdown.

Water can be used as a moderator.

The control rods control the chain reaction by limiting the number of neutrons in the reactor. These absorb neutrons so that the rate of fission is controlled. Boron can be used as control rods.

Control rods can be inserted by varying amounts to control the reaction rate but in an emergency the reactor will be shut down automatically by the release of the control rods into the reactor.

Boron can be used as control rods.

Coolant is sent around the reactor to remove heat produced in the fission.

Water can also be used as a coolant.

The heat from the reactor can then be used to make steam for powering electricity-generating turbines.

When a neutron is released, one or more of the following events can occur:

• slowed down by the moderator
• taking about 50 collisions to reach thermal speeds
• absorbed by uranium-235 to cause fission event
• one neutron released goes on to cause a further fission is the critical condition
• a neutron may leave the reactor core without further interaction
• be absorbed by uranium-238
• be absorbed by a control rod
• be scattered by uranium-238
• be scattered by uranium-235

The reactor core is thick steel to withstand high temperature and pressure.

The core is within thick concrete walls which absorb neutrons and gamma radiation.

High level waste– spent fuel rods are stored in cooling ponds for at least a year. They are also stored in sealed containers underground.

Intermediate level waste – stored in sealed drums and then encased in concrete, then stored in specially constructed buildings with walls of reinforced concrete.

Low level waste – sealed in metal drums and stored in trenches.

©2011 Grant Dwyer

Properties of Radiation

Properties	Alpha Radiation (α)	Beta Radiation (β)	Gamma Radiation (γ)
Nature	2 Protons and Neutrons	β+ = Electron β- = Positron	Photon of Energy of the Order of MeV
Range in Air	Fixed Range (Depends on Energy which can be up to 100mm	Range up to about 1m	Follows the Inverse Square Law (See Equations)
Deflection in a Magnetic Field	Easily Deflected	Opposite Direction to α particles and more easily deflected	Not Deflected
Absorption	Stopped by Paper or Thin Metal Foil	Stopped by Approximately 5mm of Aluminium	Stopped by Several Centimetres of Lead
Ionisation	Produces about 104 Ions per mm in Air at Standard Pressure	Produces about 100 Ions per mm in Air at Standard Pressure	Very Weak Ionising Effect
Energy of Each Particle/Photon	Constant for a given Source	Varies up to a Maximum for a given Source	Constant for a given Source
Danger Inside and Outside of the Body	Inside: Dangerous Outside: Less Dangerous than β and γ	Inside: Less Dangerous than α Outside: Dangerous	Inside: Less Dangerous than α Outside: Dangerous

©2011 Grant Dwyer

Kinetic Theory

The assumptions for an ideal gas for kinetic theory are:

• the gas contains a large number of particles
• the particles move rapidly and randomly
• the motion of the particles follows Newton’s laws
• collisions between particles themselves or at the walls of a container are perfectly elastic
• there are no attractive forces between particles
• any forces that act during collisions are instantaneous
• particles have a negligible volume compared with the volume of the container

A particle moves with speed, c, in a square box of length L

The components of c in three dimensions are u, v and w.

You can show that c² = u² + v² + w² - An extension of pythagoras in 3 dimensions.

Particle of mass, m, and velocity, u₁.

Momentum = m u₁

Change in momentum = -2m u₁

F = change in momentum / time = -2m u₁ / t

u₁ = 2L / t

t = 2L / u₁

F = -m u₁² / L

P = F / A = (m u₁² / L) / A = m u₁² / V

Total Pressure = (m / V) (u₁² + …...+ u_N²) = m N u² / V

u² is only in one direction so c is in 3 direction which gives u² + v² + w²

c_rms² = u_rms^{2 +} v_rms^{2 +} w_rms²

u_rms² = ⅓ c_rms²

P = ⅓m N c­_rms² / V

P V = ⅓m N c­_rms²

©2011 Grant Dwyer