# Section 7 - Further Mechanics - complete

- Created by: scarlett
- Created on: 20-09-20 13:09

## Radians

- the angle in radians is defined as the arc-length divided by the radius of the circle

- for a complete circle, the arc length is just the circumference of the circle

- dividing this by the radius gives us 2 x pi

- this means there are 2pi radians in a complete circle

- to convert from degrees to radians, multiply by pi/180

- to convert from radians to degrees, multiply by 180/pi

- 1 radian = 57^{o}

- 45^{o} = pi/4 rad

- 90^{o} = pi/2 rad

- 180^{o} = pi rad

## The Angular Speed

- angular speed is defined as the angle turned per unit time

- its unit is rads^{-1} (radians per second)

- you can relate linear speed, v, (sometimes called the tangenital velocity) and angular speed with:

angular speed = linear speed / radius of the circle being turned (m)

Example

In a cyclotron, a beam of particles spirals outwards from a central point, the angular speed of the particles remains constant. The beam of particles in the cyclotron rotates through 360^{o }in 35 microseconds.

A) explain why the linear speed of the particles increases as they spiral outwards, even though they angular speed is constant.

- linear speed depends on r, the radius of the circle being turned as well as as the angular speed

- as r increases, so does v even though angular speed remains constant

B) calculate the linear speed of a particle at a point 1.5m from the centre of rotation

angular speed = angle/time

= 2pi / 35x10^{-6} = 179519.5802

linear speed = 179519.5802 x 1.5 = 2.7 x 10^{5 }ms^{-1}

## Frequency and Period of Circular Motion

- the frequency, f, is the number of complete revolutions per second (revs^{-1} or hertz (Hz))

- the period, T, is the time taken for a complete revolution (in seconds)

- frequency and period are linked by the equation: f = 1/T

- for a complete circle, an object turns through 2pi radians in a time T, so frequency and period are related to the angular speed by:

angular speed = 2pi/T = 2pi x f

## Objects Travelling in Circles

- even if a car is going at a constant speed around a roundabout, its velocity is changing since its direction is changing

- since acceleration is defined as the rate of change of velocity, the car is accelerating even though it isn't going any faster

- this acceleration is called the centripetal acceleration and is always directed towards the centre of the circle

Equations

1) a = v^{2}/r

2) a = angular speed^{2} x r

where:

a = centripetal acceleration, ms^{-1}

v = linear speed, ms^{-1}

angular speed, rads^{-1}

r = radius, m

## Centripetal Force

- from Newton's laws, if there's a centripetal acceleration there must be a centripetal force acting towards the centre of the circle

- since F = ma, the centripetal force must be: F = mv^{2}/r = m x angular speed^{2} x r

- the centripetal force is what keeps the object moving in a circle

- if you remove the force the object would fly off at a tangent

## Simple Harmonic Motion

- an object moving with simple harmonic motion (SHM) oscillates to and fro, either side of a midpoint

- the distance of the object from the midpoint is called its displacement

- there is always a restoring force pulling or pushing the object back towards the midpoint

- the size of the restoring force is directly propottional to the displacement i.e. if the displacement doubles, the restoring force doubles too

- as the restoring force causes acceleration towards the midpoint, we can also say the acceleration is directly proportional to displacement

Condition for SHM

an oscillation in which the acceleration of an object is directly proportional to its displacement from the midpoint, and is directed towards the midpoint

a is proportional to -x

(there's a negative sign as the acceleration is opposing the displacement)

## Restoring Force

- the type of potential energy (E_{p}) depends on what it is that's providing the restoring force

- this will be gravitational E_{p} for pendulums and elastic E_{p} for masses on springs moving horizontally

- as the object moves towards the midpoint, the restoring force does work on the object and so transfers some E_{p} to E_{k}

- when the object is moving away from the midpoint, the E_{k} is transferred back to E_{p} again

- at the midpoint, the objects E_{p} is 0 and its E_{k} is maximum

- at the maximum displacement on both sides of the midpoint, the object's E_{k} is 0 and its E_{p} is at its maximum

- the sum of the potential and kinetic energy is called the mechanical energy and stays constant (as long as the motion isnt damped)

- the energy transfer for one complete cycle of oscillation is E_{p} to E_{k }to E_{p} to E_{k }to E_{p} and then the process repeats

## Displacement, Velocity and Acceleration Graphs

Displacement, x

- varies as a cosine with a maximum value, A (the amplitude)

Velocity, v

- is the gradient of the displacement-time graph

- it has a maximum value of angular frequency of oscillation x A

- it is a quarter of a cycle in front of the displacement

Acceleration, a

- the gradient of the velocity-time graph

- it has a maximum value of angular frequency of oscillation^{2} x A

- it is in antiphase with the displacement

## Frequency, Period and Amplitude

- from maximum positive displacement (e.g. maximum displacement to the right) to maximum negative displacement (e.g. maximum displacement to the left) and back again is called a cycle of oscillation

- the frequency, f, of the SHM is the number of cycles per second (Hz)

- the period, T, is the time taken for a complete cycle (seconds)

- the angular frequency is 2pi x f

~ the formulas for angular frequency are the same as for angular speed in circular motion

- in SHM, the frequency and period are independent of the amplitude (i.e. constant for a given oscillation)

- so a pendulum clock will keep ticking in regular time intervals even if its swing becomes very small

## SHM Equations

1) for an object to be moving with SHM, the acceleration, a, is directly proportional to the displacement, x

2) the constant proportionality depends on ω (angular speed), and the acceleration is always in the opposite direction from the displacement

the defining equation of SHM is a = -ω^{2}x

maximum acceleration = a_{max} = ω^{2}A

3) the velocity is positive is the object's moving in one direction, and negative if it's moving in the opposite direction

v = (+or-)ω x (square root of) A^{2}-x^{2}

maximum speed = ωA

4) the displacement varies with time according to the equation (x = Acos(ωt))

- to use this equaiton, you need to start timing when the pendulum is at its maximum displacement (i.e. when t= 0, x = A)

## Simple Harmonic Oscillator

- when a mass on a spring is pushed to the left or pulled to the right of the equilibrium position, there's a force exerted on it

- the size of the force (in N) is: F = -kx

- where k is the spring constant (stiffness) of the spring in Nm^{-1} and x is the displacement in m

- you also have an formula for the period of a mass oscillating on a spring:

T = 2pi x (sqaure root of) m/k

- where t T is the period in seconds, m is the mass in kg and k is the sping constant in Nm^{-1}

## Checking the Formula Experimentally

Set Up

- attach a clamp and clamp stand to a workbench

- attach a string with a spring onto the clamp and hang the masses on the other end of the spring

- place a position sensor on the floor, underneath the spring and connect it to a computer

- place a 1m ruler between the bench and the spring so you can meaure the initial amplitude

Method

1) set up the equipment

2) pull the masses down a set amount, this will be your initial amplitude

- let the masses go

3) the masses will now oscillate with simple harmonic motion

4) the position sensor measures the displacement of the mass over time

5) connect the position sensor to a computer and create a displacement-time graph

- read off the period T from the graph

- you could also measure the period of an oscillation using a stop watch

- it's sensible to measure the time taken for 5 oscillations for example, then divide by the number of oscillations to get an average, as it'll reduce the random error in the results

- because the spring in this experiment is hung vertically, the potential energy is both elastic and gravitational

## Investigating Factors Which Affect the Period

- change the mass, m, by loading more masses onto the spring

- change the spring stiffness constant, k, by using different combinations of springs

- change the amplitude, A, by pulling the masses down by different distances

T and m

- straight line through the origin

- T is proportional to the square root of m

- T^{2} is proportional to m

T and 1/k

- straight line through the origin

- T is proportional to the square root of 1/k

- T^{2} is proportional to 1/k

T and A

- T doesn't depend on A

## Simple Pendulum & SHO

1) attach a pendulum to an angle sensor connected to a computer

2) displace the pendulum from its rest position by a small angle (<10^{o}) and let it go

- the pendulum will oscillate with SHM

3) the angle sensor measures how the bob's displacement from the rest position varies with time

4) use the computer to plot a displacement-time graph and read off the period, T, from it

- make sure you calculate the average period over several oscillations to reduce the percentage uncertainty in your measurement

5) change the mass of the pendulum bob, m, the amplitude of displacement, A, and the lengh of the rod, l, independently to see how they affect the period T

T and l

- straight line through the origin

- T^{2} proportional to l

- T proportional to square root of l

T and m

- T does not depend on m

T and A

- T does not depend on A

- the formula for the period of a pendulum is: T = 2pi x (square root of) l/g)

## Free Vibrations and Forced Vibrations

Free Vibrations

- if you stretch and release a mass on a spring, it oscillates at its resonant frequency

- if no energy's transferred to or from the surroundings, it will keep oscillating with the same amplitude forever

- in practice this never happens, but a spring vibrating in air is called a free vibration

Forced Vibrations

- a system can be forced to vibrate by a periodic external force

- the frequency of this force is called the driving frequency

- if the driving frequency is much less than the resonant frequency then the two are in phase - the oscillator just follows the motion of the driver

- but, if the driving frequency is much greater than the resonant frequency, the oscillator wont be able to keep up and the driver will be completely out of phase wil the oscillator

- at resonance, the phase difference between the driver and the oscillator is 90o

## Resonance

- when the driving frequency approaches the resonant frequency, the system gains more and more energy from the driving force and so vibrates with a rapidly increasing amplitude

- when this happens the system is resonating

Examples

a) organ pipe

- the coloumn of air resonates, setting up a stationary wave in the pipe

b) swing

- a swing resonates if its driven by someone pushing it at its resonant frequency

c) glass smashing

- a glass resonates when driven by a sound wave at the right frequency

d) radio

- a radio is tuned so the electric circuit resonated at the same frequency as radio broadcasts

## Damping

- in practice, any oscillating system loses energy to its surroundings

- this is usually down to frictional forces like air resistance

- these are called damping forces

- systems are often deliberately amped to stop them oscillating or to minimise the effect of resonance

example

- shock absorbers in a car suspension provide a damping force by squashing oil through a hole when compressed

## Effects of Damping

- the degree of damping can vary from light damping (where the damping force is small) to overdamping

- damping reduces the amplitude of the oscillation over time

- the heavier the damping, the quicker the amplitude is reduced to 0

- critical samping reduces the amplitude in the shortest possible time

- car suspension systems and moving coil meters are critically damped so that they don't oscillate but return to equilibrium as quickly as possibe

- systems with even heavier damping are overdamped

- they take longer to return to equilibrium than a critically damped system

- plastic deformation of ductile materials reduces the amplitude of oscillations in the same way as damping

- as the material changes shape, it absorbs energy, so the oscillation will be smaller

## Damping Affecting Resonance

- lightly damped systems have a very sharp resonance peak

- their amplitude only increases dramatically when the driving frequency is very close to the resonant frequency

- heavily damped systems have a flatter response

- their amplitude doesn't increase very much near the resonant frequency and they aren't as sensitive to the driving frequency

- structures are damped to avoid being damaged by resonance

- taipei 101 is a very tall skyscraper which uses a giant pendulum to damp oscillations caused by strong winds

- damping can also be used to improve performance

- for example, loudspeakers in a room create sound waves in the air

- these reflect off of the walls of the room, and at certain frequencies stationary sound waves are created between the walls of the room

- this causes resonance and can affect the quality of the sound

- some frequencies are louder than they should be

- places like recordng studios use soundproofing on their walls which absorb the sound energy and convert it into heat energy

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