Download An Opportunity

An Opportunity
You can switch to the calculus stream and study
more physics
(if you want!)
– 16.107 and beyond –
if you get a B (B+ for honours physics) in 16.102
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Fluids, the story so far...
Density = Mass/Volume
Density of substance
Specific gravity =
Density of water at 4◦C
Pressure = Force/Area
Increase of pressure with depth:
P2 = P1 + !gh
Pascal’s Principle: pressure changes are communicated uniformly
throughout the fluid
Archimedes’ Principle: the buoyant force is equal to the weight of
displaced fluid
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11.38
A duck is floating on a lake with 25% of its volume beneath
the water. What is the average density of the duck?
Fb= buoyant force
0.25V
mg = weight of duck
The duck floats, so the weight of the duck is equal to the weight of
water with a volume of 25% of the volume of the duck.
That is, mg = !water × 0.25 ×V g
But, m = !duck ×V
Therefore, !duck = 0.25!water = 250 kg/m3
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11.46 A solid object has an apparent weight of 15.2 N when
T
completely immersed in ethyl alcohol. When completely
submerged in water, its apparent weight is 13.7 N. What is the
volume of the object? Density of ethyl alcohol = 806 kg/m3.
Fb
Weigh the object by supporting it by a string and
measuring the tension in the string.
mg
In ethyl alcohol: T1 = mg − Fb1 = 15.2 N
In water:
T2 = mg − Fb2 = 13.7 N
So, mg = 15.2 + Fb1 = 13.7 + Fb2
Fb2 − Fb1 = 1.5 N and, Fb2 − Fb1 = (!H20 − !ethyl )V g
Therefore, 1.5 = (1000 − 806)gV
1.5
V=
= 7.9 × 10−4 m3
194g
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11.44
What is the smallest number of logs (ρ = 725 kg/m3), radius
0.08 m, length = 3 m, that can be used to carry four people,
each of mass 80 kg?
The volume of each log is πr2L = 0.0603 m3, and its mass is m =
(0.0603 m3) × (725 kg/m3) = 43.72 kg.
When fully immersed in water, the buoyant force on a log is:
Fb = (1000 kg/m3) × (0.0603 m3)g = 590.9 N
With n logs, the total buoyant force is 590.9n and the total weight,
including the people, is W = (4×80 + 43.72n)g. Set these forces to be
equal:
590.9n = (320 + 43.72n)g
n = 19.3, so need at least 20 logs.
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Fluids in Motion
Streamline (steady) flow – the velocity of a fluid at some point does
not change with time. Streamlines show the path of fluid particles.
Streamlines do not cross.
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Fluids in Motion
Unsteady flow – the velocity at a point in the fluid changes with
time.
Turbulent flow is an extreme case of unsteady flow for a fast-moving
fluid – the velocity changes erratically from moment to moment, as
at sharp obstacles or bends.
Most liquids are incompressible, or nearly so – the density is almost
independent of pressure.
Gases are highly compressible, but their density is sometimes
constant enough in a flowing gas for them to be considered
incompressible.
Viscous flow – a type of friction impeding the relative motion of
layers of a fluid.
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The Equation of Continuity
What goes in must come out.
!m2
!m1
Mass flowing in = mass flowing out
Mass flowing in during time !t is, !m2 = "2 A2 v2!t
Mass flowing out during time !t is, !m1 = "1 A1 v1!t
!m1 = !m2 and so, "1 A1 v1 = "2 A2 v2 Equation of continuity
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The Equation of Continuity
!m
= " A v = mass flow rate
!t
If the fluid is incompressible, ρ1 = ρ2
and v1 A1 = v2 A2
Q = v A = volume flow rate
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