Physics 1010: The Physics of Everyday Life. TODAY Clocks How do Aeroplanes Turn?

Transcription

1 Physics 1010: The Physics of Everyday Life TODAY Clocks How do Aeroplanes Turn? 1

2 Administrative Mattters Clickers , , , , still unclaimed. If you own one of these clickers please Yu Extra credit: Draft due by midnight tomorrow (only names, 1-sentence description of method); Final report due by midnight Tuesday (HW is due by midnight Monday). 2

3 Extra Credit How Much Grain did Egypt Have to Grow to Build the Great Pyramid of Cheops? My team needs to know: A how many people it took Because: multiply by how many calories each needs B whether or not they used ramps Because: to figure out if there was friction C how many people did it take to support the workers Because: they need to be fed too D how many calories a pound of grain has Because: need to figure out how much grain you need 3

4 For Stationary Water, Pressure Changes with Height p + ρgh = constant Kind of like conservation of energy for non-moving water: Pressure + gravitational potential energy = constant Pressure is energy density (energy per unit volume) Important: the difference in pressures at two points within a static water column depends only on the differences in height of the two points. For example, water pressure increases by about 10,000 Pa (actually 9800 Pa) for every meter increase in depth (on the surface of the Earth), regardless of its container. Remember work done on body to move up a hight h. Also important: this is true for water and most liquids; it is not true for air and other gases. Why? The density ρ is not constant for gases. 4

5 1 P? P? 2 Clicker question : a. Pressure in case 1 is larger than in case 2. b. Pressure in case 2 is larger than in case 1. c. Pressure in case 1 is same as in case 2. d. Pressures can t be determined because the hoses are contorted into complicated shapes. Answer: c. In both cases, the end of the hose is the same distance below the water level. The path and shape of the hose make no difference (in the absence of viscosity, or fluid friction). GRAVITY IS A CONSERVATIVE FIELD; REMEMBER THE RAMP PROBLEMS 5

6 Clicker Question Water streams out from three spigots in an open bucket. From which hole will the water emerge traveling fastest? A B Answer: C The water pressure increases with depth; a greater water pressure exerts a greater force on the water emerging from the spigot, causing greater acceleration, giving the water a greater exit velocity. C D. A, B, and C will spout water at the same speed 6

7 100,000 Pa Note to Last Clicker Question The acceleration of water as it emerges from a spigot is governed not merely by the pressure inside the bucket, but by the pressure difference across the spigot. h 100,000 Pa + ρgh 100,000 Pa p 1 =100,000 Pa + ρgh atmospheric pressure plus water pressure Δp = ρgh resulting flow p 2 = 100,000 Pa atmospheric pressure 7

8 Speed of Water Emerging Under Pressure Water leaves spout with kinetic energy; where does the energy come from? A you robed the bank B gravitational potential energy of the water C the sun D 8

9 Speed of Water Emerging Under Pressure Water leaves spout with kinetic energy; where does the energy come from? Area A Δh m h v 100,000 Pa Δp = ρgh Because of Δp, water exits with speed v v Grav. Potential Energy lost: mgh Kinetic Energy Gained: (1/2)mv 2 Energy Conservation: mgh = (1/2)mv 2 v = 2gh Note: Volume of water leaving tank: A Δh Mass of water leaving tank: m = ρa Δh Does this look familiar? Water exits with same same velocity that an object would gain falling (from rest) a distance equal to the height of the water column. 9

10 Review Pressure = Force / Area Pressure of gases increases with density, temperature (average kinetic energy) Pressure increases with depth (both for water and for air) Boyancy force = weight of displaced fluid Bernoulli s Principle is conservation of energy 10

11 Today Clocks 11

12 What is a clock? Ruler for measuring distance: even ticks in space Clock for measuring time: even ticks in time How do we know what an even tick in time is? Repetitive motion. Astronomy and daily cycle Sun: day & hours (sundial) Moon: months Stars and seasons: years For subdividing the day, need faster repetitive motion 12

13 A spring + mass makes an oscillator, so makes a clock 1) Need some way of counting 2) Need to have oscillation time (period) be the same Mass Mass Mass period time Mass lowest at time t s Mass lowest again at t s Say: period = 1.5s 13

14 Harmonic motion: constant period (Not rigorous; good Kinetic energy (1/2)mv 2 approximation if x small) max Kick mass so has given velocity Mass while passing through equilibrium, then stops and x turns around at x max Conservation of energy sayspotential energy (1/2)kx 2 max mass goes twice as far with twice the initial velocity Mass Twice the distance, with twice the average velocity, so the (1/2)kx 2 max = (1/2)mv 2 max period remains the same! x max 2 = (m/k) v max 2 x max = sqrt(m/k) v max 14

15 A mass of 1.5 kg passes through equilibrium of the spring going 2 m/s to the right. The spring has spring constant 75 N/m. How far will the mass go before stopping? a) 50 m b) 37.5 m c) 0.28 m d) m e) 0.75 m Answer: C Conservation of energy (1/2)kx 2 = (1/2)mv 2 x 2 = (m/k)v 2 x = sqrt(m/k)v x = sqrt(1.5/75)2 = 0.28 m 15

16 Want clock to record seconds, how do we choose mass and spring? Period given by T = 2π (m/k) Pick m and k so that 2π (m/k) = 1s 16

17 Two-parter: I hang a mass of 1 kg on a spring and find that it stretches 0.15 m. How much mass should I hang to have a 1s oscillator? A) 3.3 kg B) 2.0 kg C) 6.6 kg D) 5.2 kg First find spring constant Then use period/mass relationship to get mass 17

18 Two-parter: I hang a mass of 1 kg on a spring and find that it stretches 0.15 m. How much mass should I hang to have a 1s oscillator? A) 3.3 kg B) 2.0 kg C) 6.6 kg D) 5.2 kg First find spring constant Then use period/mass relationship to get mass Answer: C F=-kx k=-f/x = 9.8N/0.15m = 65.3 N/m T = 2π (m/k) T 2 = 4π 2 (m/k) M = k T 2 /4π 2 = 65.3 N/m 1 / (4* ) = 6.6 kg 18

19 The pendulum: sideways displacement varies with force Similar to horizontal spring Longer is easier, more gravity is harder k = (g/l)m 19

20 Period of a pendulum Pulling a pendulum to the side is like pulling a spring: the force increases with distance Force also increases with mass and gravity, but decreases with length Spring constant k = (g/l)m Period = 2π (m/k) period = 2" m k = 2" m mg / L = 2" L g 20

21 A pendulum on a clock completes an oscillation in 0.5 s. How long is it? Period = T = 2π (L/g) a) m b) m c) 0.49 m d) 0.98 m e) 1.96 m Answer: A (T/2π) = sqrt(l/g) (T/2π) 2 = L/g L = g(t/2π) 2 L = 9.8(0.5/2π) 2 L = m = 6.2 cm 21

22 But pendulum slows down, and how does one count the oscillations? 22

23 Escapement mechanism allows gear to move one tooth for each pendulum period Just releases one, but traps other side Count ticks of gear? Use dial Put 60 teeth on the gear Get second hand Revolves in 1 minute Second hand 23

24 Give power to the clock by hanging weight over a pulley Hanging weights put force on the gear to cause it to rotate Nudges pendulum to make up for frictional loss Second hand 24

25 Clocks were important for determining longitude: the longitude problem Longitude - how far around the earth. Measured in degrees, with 360 degrees all the way around Earth rotates once in 24 hours, so moving around earth by how much causes the sun to come up 1 hour later? John Harrison invents the chronometer: a very accurate clock 25

26 What can we adjust on a pendulum clock to correct its time? Second hand a) Mass b) Length c) Enclosure d) a & b e) All of the above Increase? Answer: B Period = 2π (L/g) Unaffected by mass Cannot change g Can change only L 26

27 Quartz oscillators are used in watches Piezoelectric crystals: coupling between electronics and crystal like pendulum and weights Tuning: shave off some mass with a laser Usually refer to crystals by frequency: One oscillation in 1/6 of a second; period = 1/6 s 6 oscillations in 1 second; frequency = 6 Hz Frequency = 1/period 27

28 Can you answer? If a harmonic oscillator has a larger oscillation amplitude, will its period be shorter, longer, or the same? How is the period related to The spring constant and the mass for a spring oscillator? The length, mass, and gravitational acceleration for a pendulum? How long must a pendulum be to have a period of 1/4 s? What keeps a pendulum going? 28

29 How do Aeroplanes Turn? In straight flight, lift equals weight lift In order to turn, aeroplane needs centripetal force, F c =mv 2 /R It achieves this by banking lift F c weight F v weight 29

30 How do Aeroplanes Turn? What else does an aeroplane do while turning? A flaps up and down B accelerate C change direction D rotate 30

31 How do Aeroplanes Turn? While turning it also rotates (spins) For an object to start rotating, it needs A torque B C D 31

32 How do Aeroplanes Turn? While turning it also rotates (spins) For an object to start rotating, it needs to have torque applied; T=Iα It achieves this by using the rudder F 32

33 How do Aeroplanes Turn? In order to bank, it also has to rotate It achieves this by using the aerilons (flaps) Extend aerilons, increase lift on wing weight 33

34 How do Aeroplanes Turn? Word of caution: Aeroplanes (especially large jets) have large moments of inertia (T=Iα) They need a lot of torque to start rotating, and once they start rotating they will keep rotating until an oposite torque is applied using the rudder Sharp turns can place severe stress on the tail section of a large jet (needs large torque, so large force from the rudder) 34