A device for training astronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 m.If the force felt by the trainee on her back is 7.80 times her own weight, how fast is she rotating? Express your answer in m/s. Express your answer in rev/s.
To determine the speed at which the trainee is rotating, we can start by calculating the net force acting on her.
The force felt by the trainee on her back is given as 7.80 times her own weight. Since weight can be calculated as the mass multiplied by the acceleration due to gravity, we have:
Force = 7.80 × Weight
Next, we know that the net force acting on an object moving in a circle is given by:
Net Force = (Mass × Speed²) / Radius
Given that the radius of the circle is 12.0 m, we can set up the following equation:
7.80 × Weight = (Mass × Speed²) / 12.0
Since we are interested in finding the speed, we need to rearrange the equation to solve for speed:
Speed² = (12.0 × 7.80 × Weight) / Mass
Taking the square root of both sides of the equation gives us:
Speed = √((12.0 × 7.80 × Weight) / Mass)
To express the speed in m/s, we need to know the value of the weight of the trainee. Let's assume the weight is 100 kg for this calculation:
Speed = √((12.0 × 7.80 × 100 kg × 9.8 m/s²) / 100 kg)
Simplifying the equation gives us:
Speed = √(12.0 × 7.80 × 9.8 m/s²) = √(906.24) = 30.10 m/s
Therefore, the trainee is rotating at a speed of 30.10 m/s.
To express the answer in rev/s (revolutions per second), we need to convert the speed into the number of revolutions the trainee completes in one second.
The distance traveled in one revolution (circumference) is given by 2π times the radius of the circle:
Circumference = 2π × 12.0 m = 75.4 m
The number of revolutions can be calculated as the speed divided by the circumference:
Number of revolutions = Speed / Circumference
Number of revolutions = 30.10 m/s / 75.4 m = 0.399 rev/s
Therefore, the trainee is rotating at a speed of approximately 0.399 rev/s.
To solve this problem, we can use the concept of centripetal force, which is the force that keeps an object moving in a circular path. In this case, the centripetal force is provided by the normal force from the rotating device acting on the trainee.
The normal force is equal to the trainee's weight (mg), where m is the mass of the trainee and g is the acceleration due to gravity. Let's denote the trainee's weight as W.
Given that the force felt by the trainee on her back is 7.80 times her own weight, we can write:
Force felt by trainee = 7.80 * W
The centripetal force required to keep the trainee moving in a circle is equal to the force felt by the trainee on her back. Therefore, we have:
Centripetal force = Force felt by trainee = 7.80 * W
Now, the centripetal force is also equal to the mass of the trainee times the centripetal acceleration (ac), where ac is given by ac = v^2 / r, where v is the speed of rotation and r is the radius of the circular path. Therefore, we can write:
m * ac = 7.80 * W
Substituting the expression for ac, we get:
m * (v^2 / r) = 7.80 * W
Now, let's solve for v, the speed of rotation:
v^2 = (7.80 * W * r) / m
To find v, we need to know the values of W, r, and m. However, since the mass of the trainee is not provided in the problem statement, we cannot determine the exact value of v. We can only solve for v in terms of W, r, and m.
Once you have the values for W, r, and m, you can substitute them into the equation above and calculate v^2. Finally, take the square root of v^2 to find the speed of rotation, v, in m/s.
To convert the speed from m/s to rev/s, divide the speed in m/s by the circumference of the circular path. The circumference of a circle is given by 2πr, where r is the radius of the circular path. So, the speed in rev/s is:
Speed in rev/s = v / (2πr)
Again, substitute the calculated value of v and the given radius to find the speed in rev/s.