In lecture demonstrations, we do
measurements of the drag force on different objects. The objects are placed in a uniform airstream created by a fan. Calculate the Reynolds number and the drag coefficient.
The calculus of velocity-dependent frictional forces
When a body slides across a surface, the frictional force on it is approximately constant and given by
Unfortunately, the frictional force on a body moving through a liquid or a gas does not behave so simply. This drag force is generally a complicated function of the body’s velocity. However, for a body moving in a straight line at moderate speeds through a liquid such as water, the frictional force can often be approximated by
where
b is a constant whose value depends on the dimensions and shape of the body and the properties of the liquid, and
v is the velocity of the body. Two situations for which the frictional force can be represented this equation are a motorboat moving through water and a small object falling slowly through a liquid.
Let’s consider the object falling through a liquid. The free-body diagram of this object with the positive direction downward is shown in
[link] . Newton’s second law in the vertical direction gives the differential equation
where we have written the acceleration as
As
v increases, the frictional force –
bv increases until it matches
mg . At this point, there is no acceleration and the velocity remains constant at the terminal velocity
From the previous equation,
so
We can find the object’s velocity by integrating the differential equation for
v . First, we rearrange terms in this equation to obtain
Assuming that
integration of this equation yields
or
where
are dummy variables of integration. With the limits given, we find
Since
and
we obtain
and
Notice that as
which is the terminal velocity.
The position at any time may be found by integrating the equation for
v . With
Assuming
which integrates to
Effect of the resistive force on a motorboat
A motorboat is moving across a lake at a speed
when its motor suddenly freezes up and stops. The boat then slows down under the frictional force
(a) What are the velocity and position of the boat as functions of time? (b) If the boat slows down from 4.0 to 1.0 m/s in 10 s, how far does it travel before stopping?
Solution
With the motor stopped, the only horizontal force on the boat is
so from Newton’s second law,
which we can write as
Integrating this equation between the time zero when the velocity is
and the time
t when the velocity is
, we have
Thus,
which, since
we can write this as
Now from the definition of velocity,
so we have
With the initial position zero, we have
and
As time increases,
and the position of the boat approaches a limiting value
Although this tells us that the boat takes an infinite amount of time to reach
the boat effectively stops after a reasonable time. For example, at
we have
whereas we also have
Therefore, the boat’s velocity and position have essentially reached their final values.
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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