1- A stick has a linear mass density that varies with location according to the
ID: 1522336 • Letter: 1
Question
1- A stick has a linear mass density that varies with location according to the formula lamda= Cx. What is the total mass of the stick if it is 3 meters long?
a- 3C
b- 9C
c- C/3
d- C/9
e- 9C/2
2- There Is more biomass in the Earth’s northern hemisphere than its southern hemisphere. That means every Spring, when plants and trees begin to grow, trillions of tons of nutrients get a few meters farther from the Earth’s center, and the actually slows down. Likewise, the rotation of the Earth speeds up a tiny bit in autumn.
3- Astronomers using the Hubble Space Telescope have recently deduced the presence of what they think is a black hole in a distant galaxy. They measured the speed of orbiting gas, at a distance of 5.7*10^17 meters (60 light years) to be 780 kilometers/sec. Deduce the mass of the central object, and compare it to the mass of our sun.
a- 2.6 thousands suns.
b- 2.6 billion suns.
c- Approximately one sun.
d- 2.6 trillion suns.
e- 2.6 quadrillion suns.
4- A killer asteroid is placed at rest at a distance from the earth sufficient to avoid immediate detection by a very mean ET civilization (distance = 10^11 meters). It has a mass of 10^16 kg, and will fall towards the Earth because of gravity (ignore all other effects, including the orbital speed of the Earth, and gravity from the sun and other planets). How fast will this object be moving when it hits the earth?
a- 7,920 m/s
b- 11,200 m/s.
c- 9,800 m/s
d- 4.4 m/s
e- 1,400,000 m/s.
Explanation / Answer
Hi,
1.- The linear mass density can be calculated as:
= m/x :::::: where m is the mass of the object and x is the length of said object
If the value of is Cx, then the mass will be equal to:
m = x = Cx2 ::::::: if x is equal to 3, then :::::::::: m = 9C
The answer is letter b.
2.- Here there is no question to be answer as they don't ask anything, they are just stating facts. We could talk about the veracity of the things they say supporting our discussions in the definition of moment of inertia and angular momentum, but as it is not asked, we will do nothing.
3.- In this case we could use a combination of universal gravitation and circular movement. If we assume that the orbiting gas is moving at constant speed then we have the following:
The force over the gas by the central object can be written in two ways:
F = G mM/r2
F = mv2/r
In the previous equations, m is the mass of the gas, M is the mass of the central object, r is the distance of the gas from the central object, G is the gravity constant and v is the speed of the gas.
G mM/r2 = mv2/r ::::::::: M = v2r / G = (780*103 m/s)2(5.7*1017 m)/ (6.673*10-11 Nm2/kg2) = 5.20*1039 kg
Considering that the Sun's mass is about 1.99*1030 kg, the supposed black hole is:
n = 5.20*1039 kg / 1.99*1030 kg = 2.6*109 , times the sun.
If we consider that a billion is a thousand millions, then the answer will be letter b.
4.- If we ignore all other effects and we only consider gravitational effects, this problem can be solved using conservation of energy. In that event, the gravitational potential energy of the asteriod in orbit will be transform fully into kinetic energy. Therefore, to find the final speed of the asteroid can be calculated as:
(1/2) mv2 = mgh :::::::: v = (2gh)1/2 = (2 * 9.8 m/s2 * 10*11 m)1/2 = 4.4*106 m/s
This answer means that no letter has the right answer (the letter d could be the answer if it had the *106 , but this is only a suppositions).
I hope it helps.
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