Question

These questions concern a space station, consisting of a long thin uniform rod of mass 4.3 x 10^6 kg and length 769 meters, with two identical uniform hollow spheres, each of mass 1.7 x 10^6 kg and radius 218 meters, attached at the ends of the rod, as shown below. Please note that none of the diagrams shown is drawn to scale.

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 0.11 rad/s, which is just what's needed to produce 1-g of artificial gravity at the endpoints. To achieve this, we use rocket motors to apply a constant force F = 2.2 x 10^6 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 0.11 rad/s? Answer in 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 1,339 kg ad launched at a speed of 1,7300 m/s with respect to the space station. The launch points are each located at the same distance 288 meters from the center of the rod, on opposites sides of the rod. Each time a pair of probes is launched, some angular momentum is imparted to the station, increasing its spin rate. How many such pairs of probes must be launched so that the station's angular velocity will reach the required value of 0.11 rads/s?

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 0.11 rad/s. If the length of the rod is reduced to 311 meters, what will be the new angular velocity of the space station?

D. Let's start again with the station not rotating, and back to its original size, with the rod again at 769 meters long. This time, we would like to get the station rotating at a rate of 0.11 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 2.2x 10^6 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 0.11 rad/s? Answer in minutes please.

Explanations are extremely helpful and thank you so much

axis launch point launch point axis axis axis 4 axiS aus 90 degree perspective shift so you're looking staight along the new rotation axis

Answer 1

ven thin uniform h, l- ,769 m Too identical helloto sphares, each op Radius , γ-218m. the time u兀ed to the motors Are in to ode to brina tha station Prom rest up atn velocit Constant foYce f = 2.2.x O 6 N Now we calculate Situtig values in above euation 5 2. Now substibto vaos in above eauati ton

ne -2 212) -5105 x 1 4.4x106米602.5) t= 卜66155x10" 2.651xr09もー1.66 155x101 62. 6763 62.6763 2 ven of each prohe ,m 1339 Speed, V 12300 m sec The launche d poinls ane located ot sam distan ce.88 ts Station 's ufon velocity冫tagニ0-11 γ0d / Sec the system The Momant of inextia of B Substituting values we 30)(69) 12

let the paǐซ launched be n times by using conserwatton of mementum Substibute. the volues in above equation : 6,7863 n구 times launched poins obes ae uwed to launche Su te length o od is seduced tb 3 meters. Nouo oe colculate hat ill be tha new station Velecity of the Mament of inertia of rod ML 늘 (I.AX 0%, 8尸+ (1-AX106)(#9

aga 9.496 x1010 0 of sotation will be each foyce about dlua OY 7 69 ニ845-q x10 diorect ion, mok Toqua due to both foy ces in anticle ckwise direcb ion force acton od wilt be こ 2 (8 45.qx10 M L

ne 5 e knew that las acceloration where α ukene ; ce, it d anquan velocita t- time velecity after time t 12. T o.il X (4-. 3x106) (769) 12.(169 3x106)