All of the questions on this concern a space station, consisting of a long thin
ID: 1621054 • Letter: A
Question
All of the questions on this concern a space station, consisting of a long thin uniform rod of mass B)2.8 x 106 kg and length C) 109 meters, with two identical uniform hollow spheres, each of mass D) 1.1 x 106 kg and radius E) 37 meters, attached at the ends of the rod, as shown below. Note that none of the diagrams shown is drawn to scale!
Please help me with all parts!!! Thank you!!
Express the answer to the nearest tenth in the units specified – that is, rounded to the first place after the decimal point.
No scientific notation will be necessary in your answers – just a number, with exactly one digit after the decimal point.
a) Suppose that the station starts out at rest (not rotating). What we want is to get it spinning about an axis passing through its center of mass, at an angular velocity of F) 0.28 rad/s, which is just what’s needed to produce 1-g of artificial gravity at the end points. To achieve this, we use rocket motors to apply a constant force F = G) 4.8x 106 N to each sphere as shown, directed toward the centers of the spheres. How long must the motors fire in order to bring the station from rest up to an angular velocity of F) 0.28 rad/s? Answer: _____ minutes
B) Suppose once again that the space station begins at rest, not rotating. This time, instead of using rocket engines attached to the spherical end modules, we will have small probes periodically launched from two points on the rod-shaped part of the station as shown. The probes will launch in pairs in opposite directions, each individual probe of identical mass H) 4436 kg and launched at a speed of I) 37400 m/s with respect to the space station. The launch points are each located at the same distance J) 41 meters from the center of the rod, on opposite sides of the rod. Each time a pair of probes is launched, some angular momentum is imparted to the station, increasing its spin rate. Question: how many such pairs of probes must be launched, so that the station’s angular velocity will reach the required value of F) 0.28 rad/s? Answer: _____ launched pairs
(c) Now, another feature of this station is that the rod-shaped section can change its length (kind of like an old-fashioned telescope), without changing its overall mass and remaining uniform in its density. Suppose that, however it was accomplished, the station is now rotating at a constant angular velocity of F) 0.28 rad/s. If the length of the rod is reduced to K) 55 meters, what will be the new angular velocity of the space station? Answer: ____rad/s
(d) Let’s start again with the station not rotating, and back to its original size, with the rod again at C) 109 meters long. This time, we would like to get the station rotating at a rate of F) 0.28 rad/s, but now about an axis that passes straight down the length of the rod. We will accomplish this by placing a pair of rocket engines as shown, each again with a thrust of G) 4.8 x 106 N, on one of the spherical end modules. How long must these engines fire in order to get the station’s angular velocity up to F) 0.28 rad/s?
Answer: _______ minutes
Explanation / Answer
As you asked multiple questions, so l am giving answers of first three But for part b image is not given.
(a)
Moment of inertia of system,
l = lrod + 2*lsphere
from theorem of parallel axis ,
lshere = I + ma^2 = (2/3)*mr^2 + m(r + l/2)^2
l = ML^2 / 12 + 2*[(2/3)*mr^2 + m(r + l/2)^2]
l = 2.8*10^6*109^2 / 12 + 2*[(2/3)*1.1*10^6*37^2 + 1.1*10^6*(37 + 54.5)^2]
l = 23199.049*10^6 kg.m^2
we know ,
F*r = torque
T = 4.8*10^6 * 2*(54.5 + 37) = 878.4*10^6 N.m
T = l * alpha
878.4*10^6 = 23199.049*10^6 * alpha
alpha = 0.0378
alpha = dw / dt
dt = dw / alpha
t = 0.28 / 0.0378
t = 7.4 s
(c)
l = 2.8*10^6*55^2 / 12 + 2*[(2/3)*1.1*10^6*37^2 + 1.1*10^6*(37 + 27.5)^2]
l = 11866.246 * 10^6 kg.m^2
from conservation of momentum,
l1*w1 = l2*w2
23199.049 * 0.28 = 11866.246*w2
w2 = 0.547 rad/s