Welcome to: Physics I. I m Dr Alex Pettitt, and I ll be your guide!

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1 Welcome to: Physics I I m Dr Alex Pettitt, and I ll be your guide!

2 Interference superposition principal: most waves can be added y(x, t) =y 1 (x vt)+y 2 (x + vt) wave 1 + wave 2 = resulting wave y 1 + y 2 =2Acos sin(kx!t 2 2 ) wave 1 + wave 2 cancel Destructive interference y 1 + y 2 = 0 A sin(kx!t)+asin(kx!t + ) waves coincide Constructive interference y 1 + y 2 =2Asin(kx!t)

3 Interference superposition principal: most waves can be added

4 Reflection When a wave hits something it cannot travel through, it must reflect... otherwise, where would the energy go?

5 Reflection fixed end Wave height must be 0 at end. Incident (in-coming) and reflected (out-going) wave must interfere destructively. Reflected wave is inverted.

6 Reflection loose end: ring free to move on pole Wave pulse pushes ring up. Wave height is at maximum at end. Reflected wave not inverted.

7 Standing waves String with 2 fixed ends in-coming wave in-coming wave & reflected wave If exact number of half-wavelengths between fixed ends: superposition Standing wave Only for special types of waves: when their wavelength fits exactly inside the box.

8 Standing waves Mathematically: y(x, t) =y 1 + y 2 = A[cos(kx!t) cos(kx +!t)] in-coming reflected since: cos cos = 2sin apple 1 2 ( + ) sin apple 1 2 ( ) Therefore: y(x, t) =2Asin kx sin!t = kx!t = kx +!t amplitude simple harmonic depends on motion position

9 Standing waves Mathematically: y(x, t) =y 1 + y 2 = A[cos(kx!t) cos(kx +!t)] in-coming reflected since: cos cos = 2sin apple 1 2 ( + ) sin apple 1 2 ( ) Therefore: y(x, t) =2A sin kx sin!t Amplitude 0 at x = 0, x = L: sin m =0 kl = m L = m 2 m = 1, 2, 3,... k =2 /

10 Standing waves L L = m 2 m = 1 m = 2 m = 3 m = 4 m = 5

11 Standing waves The wavelengths of standing waves are called harmonics or modes L = m 2 m is the mode number m = 1 m = 1 is the fundamental mode, the longest possible standing wave m > 1 are overtones nodes do not move anti-nodes oscillate between maximum and minimum

12 Standing waves Musical instruments Stringed instruments (e.g. violin, piano, guitar) have standing waves like those just described (2 fixed ends). Wind instruments (e.g. organ, bassoon, flute) make standing waves in air columns, which have open (not fixed) ends. Open ends are fixed by pressure 2 open ends They are anti-nodes (maximum amplitude)

13 Standing waves Musical instruments Stringed instruments (e.g. violin, piano, guitar) have standing waves like those just described (2 fixed ends). Wind instruments (e.g. organ, bassoon, flute) make standing waves in air columns, which have open (not fixed) ends. odd integer number of quarter-wavelengths 1 open end L = m 4 m = 1, 3, 5...

14 Essential Physics I: E Fluid dynamics Lecture 11:

15 Last lecture: review Wave is a moving oscillation. It carries energy but not matter. s F v = = f = T µ k = 2! = 2 T String If oscillation is SHM: I = P A y(x, t) =A cos(kx ±!t) = P 4 r 2 = 10 log v = I s F I 0 µ Standing waves: 2 fixed 1 fixed, 1 open L = m 2 L = m 4 m = 1, 2, 3,... m = 1, 3, 5...

16 Last lecture: review Travelling wave: y(x, t) =4.0 cos(15x 30t) What is the wave speed? (a) (b) (c) 4 m/s 2 m/s 0.5 m/s (d) (e) 120 m/s 60 m/s

17 Last lecture: review Travelling wave: y(x, t) =4.0cos(15x 30t) What is the wave speed? k! k = 2! = 2 T (a) 4 m/s v = =! (b) 2 m/s T k = =2m/s (c) 0.5 m/s (d) (e) 120 m/s 60 m/s

18 Last lecture: review A boat bobs up and down on a water wave. It moves a vertical distance of 2 m in 1s. A wave crest moves a horizontal distance of 10 m in 2 s. What is the magnitude of the wave speed? (a) (b) (c) 2.0 m/s 5.0 m/s 7.5 m/s (d) 10.0 m/s

19 Last lecture: review A boat bobs up and down on a water wave. It moves a vertical distance of 2 m in 1s. A wave crest moves a horizontal distance of 10 m in 2 s. What is the magnitude of the wave speed? (a) 2.0 m/s Matter (boat) speed Wave speed (b) 5.0 m/s (c) 7.5 m/s v = x t = 10 m 2s =5.0m/s (d) 10.0 m/s

20 Last lecture: review Of the sound sources shown, that which is vibrating with its first harmonic is the... Whistle (a) whistle (b) organ pipe (c) string (d) rod Organ pipe Vibrating spring Vibrating string Vibrating rod (e) spring

21 Last lecture: review Of the sound sources shown, that which is vibrating with its first harmonic is the... Whistle (a) whistle (b) organ pipe (c) string (d) rod Organ pipe Vibrating spring Vibrating string Vibrating rod (e) spring

22 Fluids

23 What is a fluid? A fluid is something that takes the shape of its container. liquids are fluids gases are fluids solids cats are are not fluids. fluids

24 What is a fluid? liquid molecules are close together. Difficult to push closer. Liquids are incompressible. Density is constant: = m V gas molecules are far apart. Easy to push closer. Gases are compressible. Density changes.

25 What is a fluid?

26 Fluid properties we could use the laws of mechanics to calculate fluid motion......and apply Newton s laws to each molecule in the fluid. But a drop of water contains 1,000,000,000,000,000,000,000 molecules. So it would take the fastest computer many times the age of the Universe to calculate the motion!

27 Fluid properties Consider fluid as continuous, rather than made from discrete (separate) particles. Macroscopic (large-scale) properties: Density: = m V lead 10 4 kg/m 3 water 1000kg/m 3 air 1kg/m 3 space kg/m 3 Pressure: P = F A [N/m 2 ] = [Pa] pascal Pressure is a scalar (non-vector). It applies in all directions. F A

28 Fluid properties Pressure: P = F A [N/m2 ] If the force is applied across a large area, the pressure is small: If the pressure is distributed over many nails, it is not enough to pop the balloon. If just one nail is used, the pressure is high and the balloon pops.

29 Hydrostatic equilibrium If the fluid is at rest ( v =0), Fnet =0 = hydrostatic equilibrium Since: P = F, A F net = F 1 + F 2 = A(P 1 P 2 ) = A P if Fnet =0, P = constant Without external forces, hydrostatic equilibrium needs constant pressure A pressure difference gives a force. increasing P

30 Hydrostatic equilibrium With gravity, the pressure force balances the gravitational force. Since the pressure force comes from pressure difference ( P ), it increases with depth. Consider forces on a column of fluid: F P = F P 0 + F g PA = P 0 A + mg P 0 A since: m = V = A h h PA P 0 A = A hg More generally: P = P 0 + g for liquid (constant ) h P mg P h = g dp dh = g Hydrostatic equilibrium

31 Pascal s Law Since: P = P 0 + g h An increase in pressure here Gives the same increase in pressure everywhere in the fluid Pascal s law: A pressure increase anywhere is felt everywhere in the fluid. Application: hydraulic lift small force F 1, gives pressure P = F 1 A 1 This pressure is felt at the right-hand end to give: F 2 = A 2 P The area is larger, so the force is bigger.

32 Please see handout Q1: i - iii

33 Archimedes Principal Float or sink? Pressure (P) on a volume of fluid. F P Pressure and gravity balance. F P = F g = mg F g If we replace that volume with a solid object, the remaining fluid is the same. Therefore, P (and F P ) is the same. The pressure force from fluid on the object is the buoyancy force. It is equal to the weight of fluid displaced (removed).

34 Archimedes Principal F P F P = F g,f = m (fluid) g F g F P If the object is heavier than the fluid, its gravitational force will be bigger than the buoyancy (P) force. F g F P <F g,o = m (object) g Object will sink. F P If the object is lighter than the fluid, its gravitational force will be smaller than the buoyancy (P) force. F g F P >F g,o = m (object) g Object will float.

35 Archimedes Principal Example A cork has a density of 200kg/m 3. Find the fraction of the volume of the cork that is submerged when the cork floats in water. Buoyant force = weight of water displaced = m W g =( W V 0 )g Gravitational force: c gv To float: Gravitational force = Buoyant force c gv = W gv 0 V 0 V =? V 0 V = c W = 200kg/m3 1000kg/m 3 = 1 5

36 Archimedes Principal Traditional story: Archimedes was asked to check a crown was pure gold Volume of water displaced by crown = Volume of crown Density of crown

37 Archimedes Principal

38 Conservation of Mass Mass of fluid entering in time t = Mass of fluid exiting in time t m = 1 V 1 m = 2 V 2 = 1 A 1 x 1 = 2 A 2 x 2 = 1 A 1 v 1 t = 2 A 2 v 2 t =

39 Conservation of Mass Mass of fluid entering in time t = Mass of fluid exiting in time t m = 1 V 1 m = 2 V 2 = 1 A 1 x 1 = 2 A 2 x 2 = 1 A 1 v 1 t = 2 A 2 v 2 t Av = constant along flow for liquid (constant ) Av = constant along flow mass flow rate volume flow rate (not ok for gases )

40 Please see handout Q1: iv - vi

41 Conservation of Energy Fluid moves along pipe K = 1 2 m(v2 2 v 2 1) v 1 = work done y Fluid to the left exerts pressure force at (1) as fluid moves x 1 1 v 2 2 W 1 = F 1 x 1 = P 1 A 1 x 1 Fluid to the right exerts opposite pressure force at (2) as fluid moves x 2 W 2 = F 2 x 2 = P 2 A 2 x 2 Work done against gravity: W g = V g(y 2 y 1 )

42 Conservation of Energy K = W 1 + W 2 + W g 1 2 m(v2 2 v1)=p 2 1 A 1 x 1 P 2 A 2 x 2 V V V g(y 2 y 1 ) For incompressible fluids: A 1 x 1 = A 2 x 2 = V P 1 P 2 g(y 2 y 1 )= 1 2 (v2 2 v 2 1) P 1 + gy v2 1 = P 2 + gy v2 2 P + gy v2 = constant Bernoulli s equation E(internal) + E(grav.) + E(kin.) = conserved

43 Lift and curve How do aeroplanes fly? Aeroplane wing: The distance travelled above the curved wing is larger than below it. From conservation of mass, the air flow must take the same time to go over and under the wing. Therefore, air above the wing moves faster. From Bernoulli s equation: P + gy v2 = constant A higher velocity gives a lower pressure above the wing. This pressure difference gives an upwards force.

44 Please see handout Q1: vii-viii

45 Lecture 11 : Summary Fluid properties: pressure, density, flow velocity Archimedes Principal: buoyancy force from pressure is equal to the weight of the displaced fluid by an object. Objects less dense that fluid will float Object more dense will sink Continuity Equation: conservation of matter Av = constant along flow Bernoulli's Equation: conservation of energy P + gy v2 = constant (relate flow speed and pressure)

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