a device for training astronauts and jet fighter pilots is designed to rotate the trainee in a horizontal circle of radius 10 m. if the force felt by the trainee is 7.75 times her own weight, how fast is she rotating? express in both m/s and rev/s
v^2/r=7.75g
v= sqrt (77.5g)
To determine the speed at which the trainee is rotating, we can start by finding the net force acting on the trainee.
The force felt by the trainee is 7.75 times her own weight. Let's denote her weight as "W." Therefore, the force felt by the trainee is 7.75W.
In circular motion, the net force acting on an object moving in a circle is the centripetal force, given by the equation:
F = (m * v^2) / r
Where:
F is the centripetal force,
m is the mass of the object,
v is the velocity of the object, and
r is the radius of the circular path.
Since the trainee is rotating horizontally, the force felt by the trainee is the centripetal force.
Now, we equate the force felt by the trainee (7.75W) to the centripetal force:
7.75W = (m * v^2) / r
Next, we need to relate the weight (W) to the mass (m) of the trainee. Weight is given by the formula:
W = m * g
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting this relationship into our equation, we have:
7.75 * (m * g) = (m * v^2) / r
Now we can solve for the velocity (v) by rearranging the equation:
v^2 = (7.75 * g * r)
Taking the square root of both sides:
v = √(7.75 * g * r)
Substituting the values:
g ≈ 9.8 m/s^2
r = 10 m
v = √(7.75 * 9.8 * 10) = √(757.15) ≈ 27.5 m/s
The trainee is rotating at a speed of approximately 27.5 m/s.
To convert this into rev/s (revolutions per second), we divide the velocity by the distance traveled in one revolution.
The circumference of the circular path is given by:
C = 2πr
C = 2π * 10 = 20π
Now, we divide the velocity by the circumference to obtain the speed in rev/s:
v_rev/s = v / C = 27.5 / (20π) ≈ 0.437 rev/s
Therefore, the trainee is rotating at a speed of approximately 27.5 m/s or 0.437 rev/s.
To determine the rotational speed of the trainee, we need to consider the centripetal force acting on her and relate it to the gravitational force acting on her.
1. First, let's calculate the gravitational force acting on the trainee. The weight of an object is given by the equation: weight = mass * gravitational acceleration.
Since the force felt by the trainee is 7.75 times her own weight, we can write:
7.75 * weight = mass * gravitational acceleration
2. The centripetal force acting on an object moving in a circle is given by the equation: centripetal force = mass * (velocity)^2 / radius
In this case, the centripetal force is the force felt by the trainee, which is 7.75 times her weight. Substituting the values, we get:
7.75 * weight = mass * (velocity)^2 / radius
3. Rearrange the equation to solve for velocity:
velocity = sqrt((7.75 * weight * radius) / mass)
4. Convert the velocity from m/s to rev/s.
Since one revolution is a full circular path of 2π radians, we can convert the velocity:
velocity (rev/s) = velocity (m/s) / (2π * radius)
Now, we can substitute the given values into the formulas to calculate the rotational speed:
Given:
weight = trainee's weight
radius = 10 m
force felt by the trainee = 7.75 times her weight
1. Calculate the gravitational force:
weight = mass * gravitational acceleration
7.75 * weight = mass * gravitational acceleration
2. Solve for velocity:
velocity = sqrt((7.75 * weight * radius) / mass)
3. Convert the velocity to rev/s:
velocity (rev/s) = velocity (m/s) / (2π * radius)
With the trainee's weight provided, we can use the formulas to calculate the rotational speed.