Download Further Mechanics Revision Notes for AQA A Level Physics and more Study notes Physics in PDF only on Docsity!
Circular Motion
Circular Motion
An object in uniform circular motion has a constant linear speed However, it is continuously changing direction. Since velocity is the speed in a given direction, it, therefore, has a constantly changing velocity o The object therefore must be accelerating o This is because acceleration is defined as the rate of change of velocity This acceleration is called the centripetal acceleration and is perpendicular to the direction of the linear speed o Centripetal means it acts towards the centre of the circular path The centripetal acceleration is caused by a centripetal force of constant magnitude that also acts perpendicular to the direction of motion (towards the centre) o This is a result of Newton's Second Law Therefore, the centripetal acceleration and force act in the same direction
Radians
In circular motion, it is more convenient to measure angular displacement in units of radians rather than units of degrees
The angular displacement θ of a body in circular motion is
defined as: The change in angle, in radians, of a body as it rotates around a circle The angular displacement is the ratio of: Δθ = distance travelled around the circle / radius of circle
Note : both distances must be measured in the same units, e.g. metres A radian (rad) is defined as: The angle subtended at the centre of a circle by an arc equal in length to the radius of the circle Angular displacement can be calculated using the equation: Δθ = s / r Where:
o Δ θ = angular displacement, or angle of rotation (radians)
o s = length of the arc, or the distance travelled around the
circle (m)
o r = radius of the circle (m)
Radians are commonly written in terms of π The angle in radians for a complete circle (360 o ) is equal to: o Circumference of circle / radius = 2πr = 2π Angular Speed Any object travelling in a uniform circular motion at the same speed travels with a constantly changing velocity This is because it is constantly changing direction , and is therefore accelerating The angular speed (⍵) of a body in circular motion is defined as: The rate of change in angular displacement with respect to time Angular speed is a scalar quantity and is measured in rad s- It can be calculated using:
This equation shows how the centripetal acceleration relates to the linear speed and the angular speed Where: a = centripetal acceleration (m s−2) v = linear speed (m s−1) ⍵ = angular speed (rad s−1) r = radius of the orbit (m)
Centripetal Force
An object moving in a circle is not in equilibrium, it has a resultant force acting upon it o This is known as the centripetal force and is what keeps the object moving in a circle
The centripetal force ( F) is defined as:
The resultant force towards the centre of the circle required to keep a body in uniform circular motion. It is always directed towards the centre of the body's rotation. Centripetal force can be calculated using: o F = mv^2 / r = mrω^2 = mvω Where:
o F = centripetal force (N)
o v = linear velocity (m s-1)
o ⍵ = angular speed (rad s-1)
o r = radius of the orbit (m)
Simple Harmonic Motion
Conditions of Simple Harmonic Motion
Simple harmonic motion (SHM) is a specific type of oscillation where:
o There is repetitive movement back and forth through an equilibrium, or central, position, so the maximum horizontal or vertical displacement on one side of this position is equal to the maximum horizontal displacement on the other o The time interval of each complete vibration is the same (periodic) o The force responsible for the motion ( restoring force ) is always directed horizontally or vertically towards the equilibrium position and is directly proportional to the distance from it An oscillation is defined to be SHM when: o The acceleration is proportional to the horizontal or vertical displacement o The acceleration is in the opposite direction to the displacement (directed towards the equilibrium position) The acceleration of an object oscillating in simple harmonic motion is given by the equation: o a = − ⍵^2 x Where:
o a = acceleration (m s-2)
o ⍵ = angular frequency (rad s-1)
o x = displacement (m)
The equation demonstrates: o Acceleration reaches its maximum value when the
displacement is at a maximum ( x = x 0 at
its amplitude) o The minus sign shows that when the object is displaced to the right , the direction of the acceleration is to
the left and vice versa ( a and x are always in opposite
directions to each other)
Calculating Maximum Speed & Acceleration
The maximum speed of an oscillator, vmax, is given by the equation:
o vmax = ωA
Where:
o vmax = maximum speed (m s-1)
o ω = angular frequency (rad s-1)
o A = amplitude (m)
This comes from the SHM speed equation o v = +- √(A^2 – x^2 ) Where:
o v is maximum at the equilibrium position x = 0
o So, vmax = +-ω√A2 = ωA When an oscillator begins its motion at the equilibrium position then the velocity-time graph is a cosine graph v = v0cos(ωt)
The maximum speed of an oscillator is the amplitude , v 0 of
the velocity-time graph For a mass oscillating on a vertical spring:
o vmax occurs when the spring is in its equilibrium position
o v = 0 at the amplitude position
Maximum Acceleration
The maximum acceleration , amax of an oscillator will occur when
the gradient of the velocity-time graph is steepest
o When v = 0 m s−1^ at x = A
Acceleration is zero at the equilibrium position ( x = 0)
The maximum acceleration is given by the equation:
amax = ω^2 A
Where:
o amax = maximum acceleration (m s^2 )
o ω = angular frequency (rad s-1)
o A = amplitude (maximum displacement, x) (m)
This comes from the defining equation of SHM :
a = − ω^2 x
Period of Simple Pendulum Period of Simple Pendulum A simple pendulum consists of a string and a bob at the end o The bob is a weight, generally spherical and considered a point mass o The bob moves from side to side o The string is light and inextensible remaining in tension throughout the oscillations o The string is attached to a fixed point above the equilibrium position The time period of a simple pendulum for small angles of oscillation is given by: o T = 2π √L/g o T = time period (s) o L = length of string (from pivot to centre of mass of the bob) (m) o g = gravitational field strength (Nkg-1) The time period of a pendulum does depend on the gravitational field strength , meaning its period would be different on the Earth and the Moon Energy in SHM Simple harmonic motion also involves an interplay between different types of energy : potential and kinetic o The swinging of a pendulum is an interplay between gravitational potential energy and kinetic energy o The horizontal oscillation of a mass on a spring is an interplay between elastic potential energy and kinetic energy
o E = EP + EK
Where:
o E = total energy in joules (J)
o EP = potential energy in joules (J)
o EK = kinetic energy in joules (J)
Remember the equations for potential and kinetic energy:
o Gravitational potential energy: Ep = mgh
o Elastic potential energy, Ep = ½ kx^2
o Kinetic energy, Ek = mv^2
Energy-Displacement Graph The kinetic and potential energy transfers go through two complete cycles during one period of oscillation o One complete oscillation reaches the maximum displacement twice (on both the positive and negative side of the equilibrium position) The key features of the energy-displacement graph are: o Displacement is a vector, so, the graph has
both positive and negative x values
o The potential energy is always maximum at
the amplitude positions x = A, and 0 at the equilibrium
position x = 0
This is represented by a ‘U’ shaped curve o The kinetic energy is the opposite: it is 0 at the amplitude
positions x = A, and maximum at the equilibrium position x =
This is represented by an ‘n’ shaped curve o The total energy is represented by a horizontal straight line above the curves The key features of the simple pendulum energy-time graph are :
o Both the kinetic and gravitational potential energy transfers are represented by periodic functions (sine or cosine) which vary in opposite directions to one another o When the gravitational potential energy is 0, the kinetic energy is at its maximum and vice versa o The total energy is represented by a horizontal straight line directly above the energy curves at the maximum kinetic and gravitational potential energy value o Energy is always positive so there are no negative values on the y-axis (Any SHM energy graph drawn with negative energy values is incorrect)
Forced Vibrations & Resonance
Damping
Damping is defined as: The reduction in energy and amplitude of oscillations due to resistive forces on the oscillating system Damping continues to have an effect until the oscillator comes to rest at the equilibrium position A key feature of simple harmonic motion is that the frequency of damped oscillations does not change as the amplitude decreases o For example, a child on a swing can oscillate back and forth once every second, but this time remains the same regardless of the amplitude Types of Damping There are three degrees of damping depending on how quickly the amplitude of the oscillations decrease: o Light damping o Critical damping o Heavy damping Light Damping
Heavy Damping When a heavily damped oscillator is displaced from the equilibrium, it will take a long time to return to its equilibrium position without oscillating The system returns to equilibrium more slowly than the critical damping case o For example, door dampers are used on doors to prevent them slamming shut Key features of a displacement-time graph for a heavily damped system: o There are no oscillations. This means the displacement does not pass zero o The graph has a slow decreasing gradient from when the oscillator is first displaced until it reaches the x axis o The oscillator reaches the equilibrium position (x = 0) after a long period of time, after which the graph remains a horizontal line for the remaining time Free & Forced Oscillations Free oscillations occur when there is no transfer of energy to or from the surroundings o This happens when an oscillating system is displaced and then left to oscillate Therefore, a free oscillation is defined as: An oscillation where there are only internal forces (and no external forces) acting and there is no energy input A free vibration always oscillates at its resonant frequency In order to sustain oscillations in a simple harmonic system, a periodic force must be applied to replace the energy lost in damping o This periodic force does work against the resistive force responsible for decreasing the oscillations
o It is sometimes known as an external driving force These are known as forced oscillations (or vibrations), and are defined as: Oscillations acted on by a periodic external force where energy is given in order to sustain oscillations Forced oscillations are made to oscillate at the same frequency as the oscillator creating the external, periodic driving force For example, when a child is on a swing, they will be pushed at one end after each cycle in order to keep swinging and prevent air resistance from damping the oscillations o These extra pushes are the forced oscillations, without them, the child will eventually come to a stop Resonance The frequency of forced oscillations is referred to as the driving
frequency ( f) or the frequency of the applied force
All oscillating systems have a natural frequency ( f 0 ) , this is
defined as this is the frequency of an oscillation when the oscillating system is allowed to oscillate freely Oscillating systems can exhibit a property known as resonance When the driving frequency approaches the natural frequency of an oscillator, the system gains more energy from the driving force o Eventually, when they are equal, the oscillator vibrates with its maximum amplitude, this is resonance Resonance is defined as: When the frequency of the applied force to an oscillating system is equal to its natural frequency, the amplitude of the resulting oscillations increases significantly
A graph of driving frequency f against amplitude A of oscillations is
called a resonance curve. It has the following key features:
o When f < f 0 , the amplitude of oscillations increases
o At the peak where f = f 0 , the amplitude is at its maximum.
This is resonance
