<span>It's pretty easy problem once you set it up.
Earth------------P--------------Moon
"P" is where the gravitational forces from both bodies are acting equally on a mass m
Let's define a few distances.
Rep = distance from center of earth to P
Rpm = distance from P to center of moon
Rem = distance from center of earth to center of moon
You are correct to use that equation. If the gravitational forces are equal then
GMearth*m/Rep² = Gm*Mmoon/Rpm²
Mearth/Mmoon = Rep² / Rpm²
Since Rep is what you're looking for we can't touch that. We can however rewrite Rpm to be
Rpm = Rem - Rep
Mearth / Mmoon = Rep² / (Rem - Rep)²
Since Mmoon = 1/81 * Mearth
81 = Rep² / (Rem - Rep)²
Everything is done now. The most complicated part now is the algebra,
so bear with me as we solve for Rep. I may skip some obvious or
too-long-to-type steps.
81*(Rem - Rep)² = Rep²
81*Rep² - 162*Rem*Rep + 81*Rem² = Rep²
80*Rep² - 162*Rem*Rep + 81*Rem² = 0
We use the quadratic formula to solve for Rep:
Rep = (81/80)*Rem ± (9/80)*Rem
Rep = (9/8)*Rem and (9/10)*Rem
Obviously, point P cannot be 9/8 of the way to the moon because it'll be
beyond the moon. Therefore, the logical answer would be 9/10 the way
to the moon or B.
Edit: The great thing about this idealized 2-body problem, James, is
that it is disguised as a problem where you need to know a lot of values
but in reality, a lot of them cancel out once you do the math. Funny
thing is, I never saw this problem in physics during Freshman year. I
saw it orbital mechanics in my junior year in Aerospace Engineering. </span>
sylent_reality
· 8 years ago
I attached the missing picture.
We can figure this one out using the law of conservation of energy.
At point A the car would have potential energy and kinetic energy.

Then, while the car is traveling down the track it loses some of its initial energy due to friction:

So, we know that the car is approaching the point B with the following amount of energy:

The law of conservation of energy tells us that this energy must the same as the energy at point B.
The energy at point B is the sum of car's kinetic and potential energy:

As said before this energy must be the same as the energy of a car approaching the loop:

Now we solve the equation for

:
Answer:
part a : <em>The dry unit weight is 0.0616 </em>
<em />
part b : <em>The void ratio is 0.77</em>
part c : <em>Degree of Saturation is 0.43</em>
part d : <em>Additional water (in lb) needed to achieve 100% saturation in the soil sample is 0.72 lb</em>
Explanation:
Part a
Dry Unit Weight
The dry unit weight is given as

Here
is the dry unit weight which is to be calculated- γ is the bulk unit weight given as

- w is the moisture content in percentage, given as 12%
Substituting values

<em>The dry unit weight is 0.0616 </em>
<em />
Part b
Void Ratio
The void ratio is given as

Here
- e is the void ratio which is to be calculated
is the dry unit weight which is calculated in part a
is the water unit weight which is 62.4
or 0.04 
- G is the specific gravity which is given as 2.72
Substituting values

<em>The void ratio is 0.77</em>
Part c
Degree of Saturation
Degree of Saturation is given as

Here
- e is the void ratio which is calculated in part b
- G is the specific gravity which is given as 2.72
- w is the moisture content in percentage, given as 12% or 0.12 in fraction
Substituting values

<em>Degree of Saturation is 0.43</em>
Part d
Additional Water needed
For this firstly the zero air unit weight with 100% Saturation is calculated and the value is further manipulated accordingly. Zero air unit weight is given as

Here
is the zero air unit weight which is to be calculated
is the water unit weight which is 62.4
or 0.04 
- G is the specific gravity which is given as 2.72
- w is the moisture content in percentage, given as 12% or 0.12 in fraction

Now as the volume is known, the the overall weight is given as

As weight of initial bulk is already given as 4 lb so additional water required is 0.72 lb.
Answer:
Part A. The magnitude of the normal force is equal to the magnitude of the weight of the suitcase minus the magnitude of the force of the pull.
Part B. The magnitude of normal force acting on the suitcase is equal to the sum of the weight of the suitcase and the man.
Explanation:
Part A. This is because when the man pulls on the suit upwards, he exerts a force in the upward direction. This takes part of the force of weight of the suitcase and decreases the force the suitcase is exerting on the ground. Thus, the normal force (force exerted by suitcase on the ground) also decreases by the same force as the pull.
Part B. The statements for this part were not given in the question, but the answer reflects what is going to happen in that scenario. Since the man sits on the suitcase, the total weight acting on the ground through the suitcase is that of the suitcase plus the man. Since this force (acting on the ground) is normal force, the statement given in the answer is correct.
Answer:
The average magnitude of magnetic field B= 0.0433/ d Tesla
(You have not provided length of side of loop, so if you divide this value by length you will get value of magnetic field.)
Explanation:
Induced emf
where B= magnetic field
d= breadth of rectangular piece
V= velocity with which the rectangular piece = o.o6m/s
n= no of turns = 10
EMF = 26mV
since d (breadth of the frame) is not given, I will use it as a variable
EMF= n×B×d×V ------------------(1) (EMF induced due to multiple turns)
From eq 1, we get
B= (EMF)/(n d V)
B= (26 X 0.001) / (10 d 0.06)
B= 0.0433/ d Tesla