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 1.50 x 102 m along a circular arc. How far along a circular arc is an astronaut in chamber B moved during the same time?
To find out how far along a circular arc an astronaut in chamber B is moved during the same time, we must first find the ratio of the radii of chambers A and B.
From the figure, we see that the radius of chamber A (rA) is 5 times smaller than the radius of chamber B (rB):
rA : rB = 1 : 5
Now, we can say that:
arc_lengthA / arc_lengthB = rA / rB
Given that the astronaut in chamber A is moved 1.50 x 10^2 m along a circular arc, we can plug the values into the equation and solve for arc_lengthB:
(1.50 x 10^2) / arc_lengthB = 1 / 5
To solve for arc_lengthB, we multiply both sides by 5, and we get:
arc_lengthB = 5 * (1.50 x 10^2)
arc_lengthB = 7.5 x 10^2 m
So, during the same time, an astronaut in chamber B is moved 7.5 x 10^2 meters along a circular arc.
To determine how far along a circular arc an astronaut in chamber B is moved during the same time, we can use the concept of rotational motion and the ratio of the radii of the two chambers.
The distance traveled along a circular arc can be calculated using the formula:
Distance = Radius × Angle
In this case, we are given the distance traveled in chamber A (1.50 x 10^2 m) and we need to find the distance traveled in chamber B. Let's denote the radius of chamber A as rA and the radius of chamber B as rB.
The ratio of the radii of two circles is equal to the ratio of their corresponding circular arcs. Therefore, we can set up the following equation:
rA / rB = Distance A / Distance B
Substituting the given values, we have:
rA / rB = 1.50 x 10^2 m / Distance B
Now, we need to isolate Distance B to solve for it. Rearranging the equation, we have:
Distance B = (rB / rA) × Distance A
Since we are given the radius of chamber A and need to find the distance in chamber B, we can rewrite the equation as:
Distance B = (rB / rA) × 1.50 x 10^2 m
The figure provided does not show the specific values for the radii of the chambers, so you'll need to refer to the given information or the diagram to determine these values. Once you have the radii, substitute them into the equation to find the distance an astronaut in chamber B is moved during the same time.
To solve this problem, we need to use the concept of angular displacement. The angular displacement is the angle through which an object rotates or moves in a circular path.
Given:
- The radius of chamber A: Ra
- The radius of chamber B: Rb
- The distance moved by the astronaut in chamber A: dA = 1.50 x 10^2 m
The formula for angular displacement is:
θ = s / r
Where:
- θ is the angular displacement in radians
- s is the arc length or distance traveled
- r is the radius of the circular path
To find the angular displacement of chamber A, we can use:
θA = dA / Ra
Now, we want to find the arc length or distance moved by the astronaut in chamber B (dB) during the same time. We can use the angular displacement calculated for chamber A to find the distance moved by the astronaut in chamber B.
dB = θB * Rb
Now, let's calculate the angular displacement of chamber A and then find the distance moved by the astronaut in chamber B.
Step 1: Calculate the angular displacement of chamber A
θA = dA / Ra
θA = (1.50 x 10^2 m) / Ra
Step 2: Calculate the distance moved by the astronaut in chamber B
dB = θB * Rb
dB = θA * Rb
dB = [(1.50 x 10^2 m) / Ra] * Rb
Therefore, the distance moved by the astronaut in chamber B during the same time is [(1.50 x 10^2 m) / Ra] * Rb .