Suppose the surface (radius = r) of the space station in the figure is rotating at 38.6 m/s. What must be the value of r for the astronauts to weigh one-half of their earth-weight?
To determine the value of r for the astronauts to weigh one-half of their earth-weight on the rotating surface of a space station, we can use the concept of centripetal force and gravitational force.
First, let's consider the situation when the astronauts are standing on the stationary surface of the Earth. Their weight is given by the formula W = mg, where W is the weight, m is the mass of the astronaut, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, let's focus on the rotating surface of the space station. When it rotates, there is an additional force acting on the astronauts called the centripetal force. This force is directed towards the center of rotation and is responsible for keeping the astronauts moving in a circular path.
The centripetal force can be calculated using the formula F = (m * v^2) / r, where F is the centripetal force, m is the mass of the astronaut, v is the velocity of the rotating surface, and r is the distance from the center of rotation.
In order for the astronauts to weigh one-half of their earth-weight, the gravitational force and the centripetal force must be equal. Therefore, we can equate the two equations:
mg = (m * v^2) / r
To simplify, we can cancel out the mass (m) from both sides of the equation:
g = v^2 / r
Now, we can solve for r:
r = v^2 / g
Substituting the given values v = 38.6 m/s and g = 9.8 m/s^2, we can calculate the value of r:
r = (38.6 m/s)^2 / 9.8 m/s^2
r ≈ 152.58 meters
Therefore, in order for the astronauts to weigh one-half of their earth-weight on the rotating surface of the space station, the radius of the surface (r) should be approximately 152.58 meters.
To solve this problem, we need to use the concept of centripetal force and equate it to the gravitational force.
Step 1: Determine the centripetal force acting on the astronauts.
The centripetal force can be calculated using the formula:
F_c = m*(v^2)/r
Where F_c is the centripetal force, m is the mass of the astronauts, v is the velocity of the rotating surface, and r is the radius of the space station's surface.
Step 2: Determine the gravitational force acting on the astronauts.
The gravitational force acting on the astronauts can be calculated using the formula:
F_g = m*g
Where F_g is the gravitational force, m is the mass of the astronauts, and g is the acceleration due to gravity.
Step 3: Equate the centripetal force to the gravitational force.
Since we want the astronauts to weigh one-half of their earth-weight, the centripetal force should be half of the gravitational force:
F_c = 1/2 * F_g
Substituting the formulas from steps 1 and 2:
m*(v^2)/r = 1/2 * m*g
The mass of the astronauts cancels out:
v^2/r = 1/2 * g
Step 4: Solve for the radius.
Rearrange the equation to solve for r:
r = v^2 / (1/2 * g)
Substitute the given value for the velocity (v = 38.6 m/s), and the acceleration due to gravity on Earth (g = 9.8 m/s^2):
r = (38.6^2) / (1/2 * 9.8)
r = 150.12 m
Therefore, the value of r for the astronauts to weigh one-half of their earth-weight on the rotating space station must be approximately 150.12m.