Answer:
The total work done by friction is -2 · μ · m · g · D
Explanation:
Hi there!
The work done by a force is calculated as follows:
W = F · d · cos θ
Where:
W = work.
F = force that does the work.
d = displacement.
θ = angle between the displacement and the force.
If the force is horizontal, as in this case, cos θ = 1
The friction force is calculated as follows:
Ffr = μ · m · g
Where:
μ = friction coefficient.
m = mass of the object.
g = acceleration due to gravity.
Then, in this case, the work done by friction when pushing the block from A to B will be:
W AB = -Ffr · D
W AB = - μ · m · g · D
Notice that the friction force is negative because it is opposite to the pushing force P.
When the block is pushed from B to A, the work done by friction will be:
W BA = Ffr · (-D)
W BA = -μ · m · g · D
Now, the displacement is negative and the friction force is positive (in the opposite direction to -P).
The total work done by friction will be:
W AB + W BA = - μ · m · g · D - μ · m · g · D = -2 μ · m · g · D
I think the key here is to be exquisitely careful at all times, and
any time we make any move, keep our units with it.
We're given two angular speeds, and we need to solve for a time.
Outer (slower) planet:
Angular speed = ω rad/sec
Time per unit angle = (1/ω) sec/rad
Angle per revolution = 2π rad
Time per revolution = (1/ω sec/rad) · (2π rad) = 2π/ω seconds .
Inner (faster) planet:
Angular speed = 2ω rad/sec
Time per unit angle = (1/2ω) sec/rad
Angle per revolution = 2π rad
Time per revolution = (1/2ω sec/rad) · (2π rad) = 2π/2ω sec = π/ω seconds.
So far so good. We have the outer planet taking 2π/ω seconds for one
complete revolution, and the inner planet doing it in only π/ω seconds ...
half the time for double the angular speed. Perfect !
At this point, I know what I'm thinking, but it's hard to explain.
I'm pretty sure that the planets are in line on the same side whenever the
total elapsed time is something like a common multiple of their periods.
What I mean is:
They're in line, SOMEwhere on the circles, when
(a fraction of one orbit) = (the same fraction of the other orbit)
AND
the total elapsed time is a common multiple of their periods.
Wait ! Ignore all of that. I'm doing a good job of confusing myself, and
probably you too. It may be simpler than that. (I hope so.) Throw away
those last few paragraphs.
The planets are in line again as soon as the faster one has 'lapped'
the slower one ... gone around one more time.
So, however many of the longer period have passed, ONE MORE
of the shorter period have passed. We're just looking for the Least
Common Multiple of the two periods.
K (2π/ω seconds) = (K+1) (π/ω seconds)
2Kπ/ω = Kπ/ω + π/ω
Subtract Kπ/ω : Kπ/ω = π/ω
Multiply by ω/π : K = 1
(Now I have a feeling that I have just finished re-inventing the wheel.)
And there we have it:
In the time it takes the slower planet to revolve once,
the faster planet revolves twice, and catches up with it.
It will be 2π/ω seconds before the planets line up again.
When they do, they are again in the same position as shown
in the drawing.
To describe it another way . . .
When Kanye has completed its first revolution ...
Bieber has made it halfway around.
Bieber is crawling the rest of the way to the starting point while ...
Kanye is doing another complete revolution.
Kanye laps Bieber just as they both reach the starting point ...
Bieber for the first time, Kanye for the second time.
You're welcome. The generous bounty of 5 points is very gracious,
and is appreciated. The warm cloudy water and green breadcrust
are also delicious.
Answer:
Explained below:
Explanation:
Water quality is determined in terms of the physical, chemical and biological content of water. The quality of water of lakes and rivers variates with the seasons and also geographic areas, still in the absence of pollution. There are so many factors which affect water quality are as follows :
Pesticides
Temperature
Runoff
Sedimentation
Erosion
pH
Dissolved oxygen
Pesticides
Litter and rubbish
Decayed organic materials
Toxic and hazardous substances
Oils, grease, and other chemicals
The gap between the two flasks is partially evacuated of air creating a near vacuum which significantly reduces heat transfer by conduction or convection
Answer:
as its mass and velocity will less so its momentum will be less than that of baseball
