A device for training astronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 10.2 m. If the force felt by the trainee on her back is 7.61 times her own weight, how fast is she rotating?
(in m/s)
Express your answer in revolutions per second. Do not enter units.
To find the speed at which the trainee is rotating, we can use the concept of centripetal force. The centripetal force is the force exerted towards the center of the circular path, which in this case is experienced by the trainee on her back.
The centripetal force is given by the equation:
F = m * (v^2) / r
Where:
F is the centripetal force
m is the mass of the trainee
v is the velocity (speed) of rotation
r is the radius of the circular path
In this case, the force felt by the trainee on her back is 7.61 times her own weight, which can be expressed as:
F = 7.61 * W
Where:
W is the weight of the trainee
Now, the weight of an object can be calculated using the equation:
W = m * g
Where:
W is the weight of the object
m is the mass of the object
g is the acceleration due to gravity (approximately 9.8 m/s^2)
In this case, the trainee's weight is given as 7.61 times her own weight, so we can substitute the value of W:
7.61 * W = m * g
Next, we can substitute the value of W from the weight equation in the centripetal force equation:
F = 7.61 * m * g
Now, we can substitute this value of F in the centripetal force equation:
7.61 * m * g = m * (v^2) / r
Canceling the mass from both sides:
7.61 * g = v^2 / r
Finally, solving for v (velocity), we can rearrange the equation:
v = √(7.61 * g * r)
Substituting the given values:
g = 9.8 m/s^2 (acceleration due to gravity)
r = 10.2 m (radius of the circular path)
v = √(7.61 * 9.8 * 10.2)
v ≈ 26.1226 m/s (rounded to four decimal places)
Now, to express the answer in revolutions per second, we need to convert the speed from m/s to the number of revolutions per second.
To do this, we need to know the circumference of the circular path, which can be calculated using the formula:
C = 2 * π * r
Substituting the value of r:
C = 2 * π * 10.2
C ≈ 64.0419 m (rounded to four decimal places)
The number of revolutions per second can be calculated by dividing the velocity by the circumference:
Revolutions per second = v / C
Substituting the values:
Revolutions per second ≈ 26.1226 / 64.0419
Revolutions per second ≈ 0.407 (rounded to three decimal places)
Therefore, the trainee is rotating at approximately 0.407 revolutions per second.