4.1c Further Mechanics

SHM & Oscillations

Breithaupt pages 34 to 49

January 4th 2012

AQA A2 SpecificationLessons Topics

1 to 3 Simple harmonic motionCharacteristic features of simple harmonic motion.Condition for shm: a = − (2πf )2 xX = A cos 2πf t and v = ± 2πf √(A2 − x 2)Graphical representations linking x, v, a and t .Velocity as gradient of displacement-time graph.Maximum speed = 2πfA.Maximum acceleration = (2πf )2 A.

4 to 6 Simple harmonic systemsStudy of mass-spring system. T = 2π√(m / k)Study of simple pendulum. T = 2π√(l / g)Variation of Ek, Ep and total energy with displacement, and with time.

7 to 9 Forced vibrations and resonanceQualitative treatment of free and forced vibrations.Resonance and the effects of damping on the sharpness of resonance.Phase difference between driver and driven displacements.Examples of these effects in mechanical systems and stationary wave situations

Oscillations (definitions)An oscillation is a repeated motion about a fixed point.

The fixed point, known as the equilibrium position, is where the oscillating object returns to once the oscillation stops.

The time period, T of an oscillation is the time taken for an object to perform one complete oscillation.

Time = 0 periodTime = T / 8Time = T / 4Time = 3T / 8Time = T / 2Time = 5T / 8Time = 3T / 4Time = 7T / 8Time = 1 period

equilibrium position

The amplitude, A of an oscillation is equal to the maximum value of the displacement, x

Frequency, f in hertz is equal to the number of complete oscillations per second.

also: f = 1 / T

Angular frequency, ω in radians per second is given by:

ω = 2π f or

ω = 2π / Tequilibrium position

x = 0x = + Ax = 0x = - Ax = 0

Simple Harmonic MotionMany oscillating systems undergo a pattern of oscillation that is, or approximately the same as, that known as Simple Harmonic Motion (SHM).

Examples (including some that are approximate):A mass hanging from the end of a springMolecular oscillationsA pendulum or swingA ruler oscillating on the end of a benchThe oscillation of a guitar or violin stringTides Breathing

The pattern of SHM motionThe pattern of SHM is the same as the side view of an object moving at a constant speed around a circular path

The amplitude of the oscillation is equal to the radius of the circle.

The object moves quickest as it passes through the central equilibrium position.

The time period of the oscillation is equal to the time taken for the object to complete the circular path.

+ A

- A

Conditions required for SHMWhen an object is performing SHM:

1. Its acceleration is proportional to its displacement from the equilibrium position.

2. Its acceleration is directed towards the equilibrium position.

Mathematically the above can be written: a = - k xwhere k is a constant and the minus sign indicates that the acceleration, a and displacement, x are in

opposite directions

Acceleration variation of SHM

The constant k is equal to (2πf )2 or ω2 or (2π/T)2

Therefore:a = - (2πf )2 x (given on data sheet)

ora = - ω2 x

ora = - (2π/T)2 x

x

a

+ A- A

- ω2 x

+ ω2 x

gradient = - ω2 or - (2πf )2

QuestionA body oscillating with SHM has a period of 1.5s and amplitude of 5cm. Calculate its frequency and maximum acceleration.

(a) f = 1 / T= 1 / 1.5sfrequency = 0.67 Hz

(b) a = - (2πf )2 xmaximum acceleration is when x = A (the amplitude)a = - (2πf )2 A= - (2π x 0.6667Hz)2 x 0.015mmaximum acceleration = 0.88 ms-2

Displacement equations of SHMThe displacement, x of the oscillating object varies with time, t according to the equations:

x = A cos (2π f t) given on data sheet

orx = A cos (ω t)

orx = A cos ((2π / T) t)

Note: At time, t = 0, x = +A

QuestionA body oscillating with SHM has a frequency of 50Hz and amplitude of 4.0mm. Calculate its displacement and acceleration 2.0ms after it reaches its maximum displacement.

Maximum displacement = amplitude, A = 4.0mm(a) x = A cos (2π f t) = 4.0mm x cos (2π 50 x 2.0ms) = 4.0mm x cos (2π 50 x 0.0020s) = 0.6283displacement = 0.628 mm

(b) a = - (2πf )2 x= - (2π x 50)2 x 0.6283mm= - (100π)2 x 0.0006283macceleration = - 62 ms-2

Velocity equations of SHM

The velocity, v of an object oscillating with SHM varies with displacement, x according to the equations:

v = ± 2πf √(A2 − x 2) given on data sheet

orv = ± ω√(A2 − x 2)

orv = ± (2π / T) √(A2 − x 2)

Question

f = 1 / T= 1 / 4.0ms= 1 / 0.004s= 250Hz

(a) v = ± 2πf √(A2 − x 2) maximum speed occurs when x = 0vmax = ± 2πf √(A2) = ± 2πf A= ± 2π x 250 x 30μm= ± 2π x 250 x 0.000 030mmaximum speed = 0.0471 ms-1

(b) v = ± 2πf √(A2 − x 2) = ± 2π x 250 x √((0.000 030m)2 − (0.000 015m)2) = ± 500π x √((0.000 030m)2 − (0.000 015m)2) = ± 500π x √((9 x 10 -10) − (2.25 x 10 -10))= ± 500π x √(6.75 x 10 -10)= ± 500π x 2.598 x 10 -5

speed = 0.0408 ms-1

A body oscillating with SHM has a period of 4.0ms and amplitude of 30μm. Calculate (a) its maximum speed and (b) its speed when its displacement is 15μm.

Variation of x, v and a with time

The acceleration, ax depends on the resultant force, Fspring on the mass.Note that the acceleration ax is always in the opposite direction to the displacement, X.

SHM time graphs

T/4 T/2 3T/4 T 5T/4 3T/2

x = A cos (2π f t)

a = - (2π f )2 A cos (2π f t)

v = - 2π f A sin (2π f t)

+ A

- A

xva

time

vmax = ± 2π f A

amax = ± (2π f )2 A

Note: The velocity curve is the gradient of the displacement curve and the acceleration curve is the gradient of the velocity curve.

a

v

x

SHM summary table

Displacement Velocity Acceleration

+ A 0 - (2π f )2 A

0 ± 2π f A 0

- A 0 + (2π f )2 A

A ? B ?

C ?

E ?

D?

E ?

SHM graph question

T/4 T/2 3T/4 T 5T/4 3T/2

a

time

a

The graph below shows how the acceleration, a of an object undergoing SHM varies with time. Using the same time axis show how the displacement, x and velocity, v vary in time.

x

x

acceleration is proportional to minus displacement

v

v

velocity is the gradient of displacement

The spring-mass systemIf a mass, m is hung from a spring of spring constant, k and set into oscillation the time period, T of the oscillations is given by:

T = 2π√(m / k)

AS Reminder: Spring Constant, k:This is the force in newtons required to cause a change of length of one metre.

k = F / ΔLunit of k = Nm-1

Mass on spring - Fendt

QuestionA spring extends by 6.0 cm when a mass of 4.0 kg is hung from it near the Earth’s surface (g = 9.8ms-2). If the mass is set into vertical oscillation state or calculate the period (a) near the Earth’s surface and (b) on the surface of the Moon where g = 1.7 ms-2.

The spring constant, k is given by:k = F / ΔL = (4.0 x 9.8) / 0.060m= 653.3 Nm-1

(a) T = 2π√(m / k)= 2π √(4.0 / 653.3)= 2π √(0.006122)time period = 0.49s

(b) g does not appear in the time period equation of the spring.Therefore the period on the Moon is the same = 0.49s

The simple pendulumA simple pendulum consists of:• a point mass • undergoing small oscillations (less than 10°)• suspended from a fixed support • by a massless, inextendable thread of length, L • within a gravitational field of strength, g

The time period, T is given by:

T = 2π√(L / g)

Simple Pendulum - Fendt

QuestionCalculate: (a) the period of a pendulum of length 20cm on the Earth’s surface (g = 9.81ms-2) and (b) the pendulum length required to give a period of 1.00s on the surface of the Moon where g = 1.67ms-2.

(a) T = 2π√(L / g)= 2π √(0.20m / 9.81ms-2)= 2π √(0.204)time period = 0.90s

(b) T = 2π√(L / g)becomes: T2 = 4π2 (L / g)becomes: L = T2g / 4π2

= ((1.00)2 x 1.67) / 4π2

length = 0.0423m (3.23cm)

Free oscillationA freely oscillating object oscillates with a constant amplitude.

The total of the potential and kinetic energy of the object will remain constant.EP + ET = a constant

This occurs when there are no frictional forces acting on the object such as air resistance.

Energy variation in free oscillation

Position 1 – Maximum displacement

Potential energy, EP = Total energy, ET

Kinetic energy, EK = 0

21

equilibrium position

3

54

Position 2

ET = EP + EK

Position 3 – Equilibrium position

EP = 0

EK = Total energy, ET

Position 4

ET = EP + EK

Position 5 – Maximum displacement

EP = Total energy, ET

EK = 0

position EP EK

1 ET 0

3 0 ET

5 ET 0

QuestionA simple pendulum consists of a mass of 50g attached to the end of a thread of length 60cm. Calculate: (a) the period of the pendulum (g = 9.81ms-2) and (b) the maximum height reached by the mass if the mass’s maximum speed is 1.2 ms-1.

(a) T = 2π√(L / g)= 2π √(0.60m / 9.81ms-2)= 2π √(0.06116)time period = 1.56s

(b) The maximum speed occurs at the equilibrium position, when the mass is at its lowest position so that its potential energy. EP = 0

kinetic energy, EK = ½ m v2

= ½ x (0.050kg) x (1.2ms-1)2

= 0.036 J

The maximum height occurs when EP = EK

so m x g x h = 0.036 Jh = 0.036 / (0.050 x 9.81)maximum height = 0.074m

Energy variation with an oscillating spring

The strain potential energy is given by:

EP = ½ k x2

Therefore the maximum potential energy of an oscillating spring system= ½ k A2

= Total energy of the system, ET

But: ET = EP + EK

½ k A2 = ½ k x2 + EK

And so the kinetic energy is given by:

EK = ½ k A2 - ½ k v2

EK = ½ k (A2 - x2)

Energy versus displacement graphs

The potential energy curve is parabolic, given by:

EP = ½ k x2

The kinetic energy curve is an inverted parabola, given by:

EK = ½ k (A2 - x2)

The total energy ‘curve’ is a horizontal line such that:

ET = EP + EK

- A 0 + A

energy

displacement, x

EP

EK

ET

Energy versus time graphsDisplacement varies with time according to:x = A cos (2π f t)Therefore the potential energy curve is cosine squared, given by:EP = ½ k A2 cos2 (2π f t)

½ k A2 = total energy, ET.and so: EP = ET cos2 (2π f t)Kinetic energy is given by:EK = ET - EP

EK = ET - ET cos2 (2π f t)EK = ET (1 - cos2 (2π f t))EK = ET sin2 (2π f t)EK = ½ k A2 sin2 (2π f t)

ET

0 T/4 T/2 3T/4 T

energy

time, t

½ kA2

EP

EK

DampingDamping occurs when frictional forces cause the amplitude of an oscillation to decrease.

The amplitude falls to zero with the oscillating object finishing in its equilibrium position.

The total of the potential and kinetic energy also decreases.

The energy of the object is said to be dissipated as it is converted to thermal energy in the object and its surroundings.

Types of damping1. Light DampingIn this case the amplitude gradually decreases with time.

The period of each oscillation will remain the same.

The amplitude, A at time, t will be given by: A = A0 exp (- C t)where A0 = the initial amplitude and C = a constant depending on the system (eg air resistance)

2. Critical DampingIn this case the system returns to equilibrium, without overshooting, in the shortest possible time after it has been displaced from equilibrium.

3. Heavy DampingIn this case the system returns to equilibrium more slowly than the critical damping case.

light damping

heavy damping

critical damping

displacement

time

A0

Forced oscillationsAll undamped systems of bodies have a frequency with which they oscillate if they are displaced from their equilibrium position.This frequency is called the natural frequency, f0.

Forced oscillation occurs when a system is made to oscillate by a periodic force. The system will oscillate with the applied frequency, fA of the periodic force.

The amplitude of the driven system will depend on:1. The damping of the system.2. The difference between the applied and natural frequencies.

Resonance

The maximum amplitude occurs when the applied frequency, fA is equal to the natural frequency, f0 of the driven system.

This is called resonance and the natural frequency is sometimes called the resonant frequency of the system.

Resonance curves

applied force frequency, fA

amplitude of driven system, A

driving force

amplitude

f0

more damping

light damping

very light damping

Notes on the resonance curvesIf damping is increased then the amplitude of the driven system is decreased at all driving frequencies.

If damping is decreased then the sharpness of the peak amplitude part of the curve increases.

The amplitude of the driven system tends to be: - Equal to that of the driving system at very low frequencies.- Zero at very high frequencies.- Infinity (or the maximum possible) when fA is equal to f0 as damping is reduced to zero.

Phase differenceThe driven system’s oscillations are always behind those of the driving system.

The phase difference lag of the driven system depends on:1. The damping of the system.2. The difference between the applied and natural frequencies.

At the resonant frequency the phase difference is π/2 (90°)

Phase difference curves

applied force frequency, fA

phase difference of driven system compared with driving system f0

more damping

less damping

- 90°

- 180°

Examples of resonance• Pushing a swing• Musical instruments (eg

stationary waves on strings)

• Tuned circuits in radios and TVs

• Orbital resonances of moons (eg Io and Europa around Jupiter)

• Wind driving overhead wires or bridges (Tacoma Narrows)

The Tacoma Narrows Bridge Collapse

YouTube Videos:http://www.youtube.com/watch?v=3mclp9QmCGs – 4 minutes – with commentaryhttp://www.youtube.com/watch?v=IqK2r5bPFTM&feature=related – 3 minutes – newsreel footagehttp://www.youtube.com/watch?v=j-zczJXSxnw&feature=related – 6 minutes - music background only

An example of resonance caused by wind flow.

Washington State USA, November 7th 1940.

Internet Links (SHM)Motion in 2D - PhET - Learn about velocity and acceleration vectors. Move the ball with the mouse

or let the simulation move the ball in four types of motion (2 types of linear, simple harmonic, circle). See the velocity and acceleration vectors change as the ball moves.

Pendulum Lab - PhET - Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It's easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of g on planet X. Notice the anharmonic behavior at large amplitude

SHM & circular motion compared - NTNU

SHM (spring) and circular motion - netfirms

SHM with a spring & pendulum - Explore Science

Mass on spring - Fendt

Mass on a spring - NTNU

Mass on spring oscillations - NTNU

Spring oscillation with varying mass and k - NTNU

Spring oscillation showing graphs of s, v & a - NTNU

Simple Pendulum - Fendt

Pendulum in an accelerated car - NTNU Pendulum with rotating suspension point - © 1998 Franz-Josef Elmer

Internet Links (Forced Oscillation & Damping)

Spring Oscillation - PhET - A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

Undamped and undriven pendulum - © 1998 Franz-Josef Elmer Driven spring oscillator with one mass - Explore Science Driven spring oscillator with two masses - Explore Science Forced oscillation - Fendt Coupled Penduala - Fendt Pendulum driven by a periodic force - © 1998 Franz-Josef Elmer Horizontally driven pendulum - © 1998 Franz-Josef Elmer Vertically driven pendulum - © 1998 Franz-Josef Elmer Resonance in a string - netfirms RCL Resonance - netfirms

Core Notes from Breithaupt pages 34 to 491. What is an oscillation?2. In the context of oscillations explain the meaning of the following: (a) equilibrium; (b)

displacement; (c) amplitude; (d) time period; (e) frequency; (f) angular frequency and (g) phase difference.

3. Explain the conditions (in terms of acceleration and displacement) required for an object to be undergoing simple harmonic motion.

4. State the defining equation for SHM shown on page 37.5. Sketch the graphs shown on page 36 and explain the mathematical relationships

between them.6. (a) State the equation for the time period for a mass-spring system. (b) Calculate the

mass required to give a mass-spring system of spring constant 25N/m a time period of one second.

7. (a) State the equation for the time period for a simple pendulum. (b) Calculate the length required to give a pendulum a time period of one second near the Earth’s surface.

8. How, if at all would your answers to Q6(b) and Q7(b) be different if you were on the Moon’s surface? (gMoon = 1.7 N/kg)

9. Explain how potential, kinetic and total energy vary during the oscillation of a simple pendulum. (assume no resistive forces).

10. What is damping? How would it change your answers to Q9?11. What is meant by ‘resonance’?12. Sketch and explain the resonance curves on page 47.13. Explain how damping affects the phase difference between the driving force and the

driven system.

Notes from Breithaupt pages 34 & 35Oscillations

1. What is an oscillation?2. In the context of oscillations explain the meaning of the

following: (a) equilibrium; (b) displacement; (c) amplitude; (d) time period; (e) frequency; (f) angular frequency and (g) phase difference.

3. Try the summary questions on page 35

Notes from Breithaupt pages 36 & 37The principles of simple harmonic motion

1. Explain the conditions (in terms of acceleration and displacement) required for an object to be undergoing simple harmonic motion.

2. State the defining equation for SHM shown on page 37.3. Sketch the graphs shown on page 36 and explain the

mathematical relationships between them.

4. Try the summary questions on page 37

Notes from Breithaupt pages 38 & 39More about sine waves

1. Explain the ways in which circular motion and SHM correspond with each other.

2. Try the summary questions on page 39

Notes from Breithaupt pages 40 to 43Applications of simple harmonic motion

1. (a) State the equation for the time period for a mass-spring system. (b) Calculate the mass required to give a mass-spring system of spring constant 25N/m a time period of one second.

2. (a) State the equation for the time period for a simple pendulum. (b) Calculate the length required to give a pendulum a time period of one second near the Earth’s surface.

3. How, if at all would your answers to Q1(b) and Q2(b) be different if you were on the Moon’s surface? (gMoon = 1.7 N/kg)

4. Derive the equations for (a) the spring-mass system and (b) the simple pendulum.

5. Try the summary questions on page 43

Notes from Breithaupt pages 44 to 46Energy and simple harmonic motion

1. Explain how potential, kinetic and total energy vary during the oscillation of a simple pendulum. (assume no resistive forces).

2. What is damping? How would it change your answers to Q1?

3. Repeat Q1, this time for a mass-spring system.4. Describe with the aid of suitable graphs the different

types of damping.5. Try the summary questions on page 46

Notes from Breithaupt pages 47 to 49Forced oscillations and resonance

1. What is meant by ‘resonance’?2. Sketch and explain the resonance curves on page 47.3. Explain how damping affects the phase difference

between the driving force and the driven system.

4. Explain the demonstration ‘Barton’s Pendulum’ on page 48.

5. Give and explain an example of resonance with bridges.6. Try the summary questions on page 49