NASA wants to build a centrifuge to train its astronauts to handle the strong accelerations they will experience when the space shuttle lifts off. A composite material capable of withstanding a normal force of 120000N without rupturing is used to manufacture the base of the cabin on which the astronaut's seat rests. What is the maximum mass that this base can support without rupturing when the apparatus turns at a velocity of 250km/h and the astronaut's cabin is located at a distance of 12.5m from the centre of the rotation?

Question

NASA wants to build a centrifuge to train its astronauts to handle the strong accelerations they will experience when the space shuttle lifts off. A composite material capable of withstanding a normal force of 120000N without rupturing is used to manufacture the base of the cabin on which the astronaut's seat rests. What is the maximum mass that this base can support without rupturing when the apparatus turns at a velocity of 250km/h and the astronaut's cabin is located at a distance of 12.5m from the centre of the rotation?

So first I converted 250km/h into 250000m/3600s = 69m/s

Then since the normal force FN = 120000N, then I guess the centripetal force Fc = 120000N as well, just in the opposite direction. (So would it make sense that I interpret the FN being the force that resists against the Fc and they are equal magnitude?)

Then the equation:
Fc = mv^2/r
120000N = ((m)(69m/s)^2)/12.5m
m = 311.04kg = 310kg

Correct or not? If not, please explain where I went wrong.

Answer 1

mass=1.2E5*12.5/(69)^2

recheck that.

Answer 2

But my final answer is correct?

Answer 3

Your approach is almost correct, but there is a small mistake in the equation. The centripetal force (Fc) and the normal force (FN) are not equal in magnitude. The normal force is the force exerted by the base of the cabin in the upward direction to balance the weight of the astronaut and the seat. The centripetal force is the force acting towards the center of rotation required to keep the astronaut moving in a circular path.

The correct equation to use is:

Fc = m*((v^2)/r)

In this equation:
- Fc is the centripetal force
- m is the mass of the astronaut and the seat
- v is the velocity (in meters per second)
- r is the radius (distance from the center of rotation)

So, using your values:

Fc = 120,000 N
v = 69 m/s
r = 12.5 m

Substituting these values into the equation:

120,000 N = m*((69 m/s)^2 / 12.5 m)

Now, solve for m:

m = (120,000 N * 12.5 m) / (69 m/s)^2

Calculating this, you should get:

m ≈ 441.72 kg

So, the maximum mass that the base can support without rupturing is approximately 441.72 kg.