Physics 2065A/B Study Guide - Midterm Guide: Ground Speed, Sagitta, Viscoelasticity

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31 Jan 2013
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Helium Football Notes
Intro: kicking a ball is a skill and highly variable; in experiment kicker couldn’t tell if ball did or didn’t
have helium in it
Results: avg helium ball went 3yds further; bc of variability btwn kicks, there is statistically no difference
Helium effect on ball: weight empty, full of air, full of helium- 410g, 3.2 g at 13psi, -0.9% less than empty
ball
Reasoning: “helium is only half as dense as air so sound travels through it twice as fast”
Speed of sound in a gas: Cs= square root of [(ϒ/p) times P] where ϒ is the constant, P is pressure and p is
density; Fhe ~ 3 Fair ratio of densities
Football facts: regulation NFL football 13 psi, 90kPa; circumference middle btwn 52.7 and 54, and
around the ends 70.5-72.4 with the length 14-14.2 (all in cm); weight is 397-425g; shape of prolate
ellipsoid
Weight of gas added to football: first find volume of football, then the amount of gas added to this
volume (SLIDE 12)
Pressure is the force per unit area exerted by a body; when you are underwater the relevant weight is
the column of water above you; atmospheric pressure results from the weight of the air in a column
above you (barometer measures atmospheric pressure); units- force per area= N/m², 1 N/m²= 1 Pascal
(Pa); room pressure on avg is 101 300 N/m²= 101.3 kPa (equivalents include 1kg/cm², 1.013 bar (greek
baros, weight), 1 atmopshere, 14.7psi)
Gauge Pressure pressure on tire pump is relative to atmospheric pressure (gauge pressure); absolute
pressure of football is 13psi + 14.7psi= 27.7 psi
Moles: 1 mole= same # particles as in 12g of C12 atoms which is Avogadro’s number, 6.022x10(23)
particles/mole; mole of gas has a mass equal to its molecular number... so Helium which is “2” has molar
mass of 4g/mol; air is 79%N (“14”, 28g/mol) and 21% O (“16”, 32g/mol) so air is 28.96g/mol
Ideal Gas Law: PV=nRT (P is pressure in Pa, V is volume in m³, n is number of moles of gas, R is gas
constant of 8.314m³Pa/K/mol, T is temperature in Kelvin (0degress=273.1K) SLIDE 13
Mass of gas in football: application of PV=nRT
P=13psi, 90kPa, V=0.00430 m³, T=20 degrees= 293.1K, R=8.314 m³Pa/K/mol
number of moles added to football n=0.14; mass air is 28.96g/mol x n=4.5g, mass helium is 4g/mol x
n=0.36g
Weight: Wair=411g+4.5g=416g; He football weight is ~0.9%less than empty footballs (407g);
Buoyancy: He molecules are lighter than air molecules which causes upward force on balloon buoyancy;
with sufficient He in balloon Fbuoyancy (greater than) Fgravity and balloon rises
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Archimedes’ Principle: the buoyancy force on a submerged object is equal to weight of fluid that is
displaced by the object SLIDE 23
Imparting Energy: He football has more ‘spring’ to it so kicker ‘imparts more energy’ to it
Coefficients of Restitution: =1 for elastic collisions, =0 for completely inelastic collisions (objects stick
together), 0<CR<1 for inelastic collisions
The Kicking Experiment: both air and He football struck with blow from same hammer moving at 5.64
m/s; for each trial the ball left the collision with a speed of 9.52 m/s due to this force; as expected no
changes to the coefficient of restitution were observed since pressure in balls was the same
Theory of Kicking Experiment: assume all the hammer hits is the ball and stops, so all its momentum is
transferred to the ball, then pair=phammer mairvair=mhammervhammer and similarly for He football
mHevHe=mhammervhammer
Since air has greater mass than He, vhe/vair=mhe/mair= 415.5g/411.6g 1.0095; horizontal distance ball
travels, the range, is proportional to the square of velocity; thus, when kicked with the same force the
He ball should travel 2% further than the air ball since it is going 1% faster (0.8yd for a 44yd punt,
negligible increase)
Air Drag and Helium Football: without air drag...the kicked football’s trajectory is fully determined by the
(constant) force of gravity, which pulls all objects to the surface at the same rate; if given the same
initial speed, the air and He footballs will travel the same distance, with the same hang time; if the
applied force is the same (as with a real kicker), the He football moves 1% faster and thus, travels 2%
further due to conservation of momentum
Two important things happen when the air drag force is included:
1. the acceleration is no longer constant but changes as the drag changes, since drag is proportional to
the square of velocity; the drag in the vertical direction is in the same direction as the force of gravity as
the football moves up, but, the vertical drag is in the opposite direction as the force of gravity when the
football descends
2. the drag force is independent of mass but the acceleration due to the drag force depends on mass,
since a=F/m, unlike gravity whose acceleration is independent of mass; thus, the acceleration due to the
drag force is less for a more massive object; the acceleration of the football is a= (Fgrav-Fdrag)/m=g-
(Fdrag/m); as m increases the drag term gets smaller; drag decreased distance of kick by about 15yd,
with peak height decreases by about 7yd
Drag force vs Gravity: vertical drag force is downward as the ball goes up and is initially about half as
large as gravity; the drag force is upward as the ball goes down; by the time the ball is near the ground
its magnitude is about 35% of gravity; negative means drag points downward
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Model Same Initial Force: if assuming ball leaves kickers foot at same speed the air football travels
slightly further than the He football (greater inertia, acceleration due to drag decreases with increasing
mass); if assume kicker kicks each ball with same force He football goes slightly further )He ball has
higher initial speed), but not significantly farther SLIDE 42
Fluid Dynamics Notes
A fluid is... liquid, gas and plasma; continually flows under an applied force; in common usage liquids are
fluid substances that have no shape when not in a container; the distinction btwn solids and fluids is
often blurred viscoelastic fluid, Silly Putty (flows, but over extended time period)
Gas as a Fluid:
- Difference btwn gas and liquid: though molecules in liquids are not organized in lattice
formation like solids, the forces btwn them are strong; molecules in gases have small forces
btwn them
- The major difference in their compressibility: the change in volume of a liquid or solid with
changing pressure is small compared to gas
Fluid Dynamics: study of fluids in motion; assumptions for an ideal fluid include...fluids are
incompressible, temperature is constant (no convection), steady flow (velocity and pressure do not
change with time, velocity and pressure can change from point to point, but smoothly)
Streamlines: an imaginary path a particle would follow if released in a fluid...can streamlines cross?
Flow Types
- Laminar: a fluid element travels on a smooth path predictable by Newton’s laws
- Turbulent: the velocity and pressure fluctuate rapidly about the mean flow (no complete theory
of turbulence exists)
When streamlines pinch off and form vortices, the flow is rational; when streamlines are straight or
gently bend the flow is irrotational
Rotantional Flow: tangential and normal to streamlines uniform circular flow is rotantional; when a
vortex is present the components rotate opposite to one another, so perpendicular sticks fasten
together exhibiting irroational flow by vortex; uniform flow btwn parallel walls is often rotational; flow
can be rotational but not turbulent
Simplifications for Ideal Fluid: 1. the fluid is incompressible (changes in pressure don’t affect volume)
2. Temperature is steady 3. Flow is steady, so velocity and pressure do not change with time
4. Flow is laminar, not turbulent 5. The flow is irrotational 6. No friction in the fluid (viscosity)
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