A space station consists of two donut-shaped living chambers, A and B, that have the radii shown in the figure. As the station rotates, an astronaut in chamber A is moved 2.06 x 102 m along a circular arc. How far along a circular arc is an astronaut in chamber B moved during the same time?
I have no clue how to solve this.
To solve this problem, you can use the concept of ratios and proportions. Here's a step-by-step explanation:
Step 1: Find the ratio of the radii of chambers A and B.
The ratio of the radii can be found by dividing the radius of chamber B by the radius of chamber A. Let's call the radius of chamber A "rA" and the radius of chamber B "rB". Using the given figure, identify the values of rA and rB.
Step 2: Use the ratio of the radii to find the ratio of the circular arcs.
Since both chambers are rotating at the same time, the ratio of the radii will be the same as the ratio of the circular arcs covered by the astronauts in each chamber. Let's call the distance covered by the astronaut in chamber A "dA" and the distance covered by the astronaut in chamber B "dB". Set up the proportion:
dA / dB = rA / rB
Step 3: Solve the proportion for dB.
Rearrange the proportion from Step 2 to solve for dB:
dB = dA * (rB / rA)
Step 4: Substitute the given values and calculate dB.
Plug in the given values for dA, rA, and rB into the equation from Step 3 and calculate dB.
That's it! By following these steps, you'll be able to calculate the distance along a circular arc that an astronaut in chamber B is moved during the same time as an astronaut in chamber A.