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Astronauts use a centrifuge to simulate the acceleration of a rocket launch. The centrifuge takes 40.0 s to speed up from rest to its top speed of 1 rotation every 1.50 s. The astronaut is strapped into a seat 6.20 m from the axis.What is the astronaut's tangential acceleration during the first 40.0 s?How many g's of acceleration does the astronaut experience when the device is rotating at top speed? Each 9.80 m/s^2 of acceleration is 1 g.

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Answer

Time period T = 1.50 s

time t = 40 s

r = 6.2 m

a)

Angular speed ω = 2π/T

                              = [tex]\dfrac{2\pi }{1.5}[/tex]  

                              = 4.189 rad/s

Angular acceleration α = [tex]\dfrac{\omega}{t}[/tex]

                                      = [tex]\dfrac{4.189}{40}[/tex]

                                      = 0.105 rad/s²

Tangential acceleration a = r α = 6.2 x 0.105 = 0.651 m/s²

b)The maximum speed.

       v = 2πr/T

          = [tex]\dfrac{2\pi \times 6.2}{1.5}[/tex]

          = 25.97 m/s

So centripetal acceleration.

        a = [tex]\dfrac{v^2}{r}[/tex]

          = [tex]\dfrac{25.97^2}{6.2}[/tex]

          =  108.781 m/s^2

          = 11.1 g    

in combination with the gravitation acceleration.

[tex]a_{total} = \sqrt{(11.1g)^2+g^2}[/tex]

[tex]a_{total}= 11.145 g[/tex]

Lanuel

a. The astronaut's tangential acceleration during the first 40.0 seconds is [tex]0.651 \;m/s^2[/tex]

b. The number of g's of acceleration the astronaut would experience when the device is rotating at top speed is 11.10g.

Given the following data:

  • Time, t = 40 seconds
  • Radius, r = 6.20 meter
  • Period, T = 1.50 seconds
  • Acceleration, a = 9.80 [tex]m/s^2[/tex] = 1 g.
  • Initial angular velocity = 0 rad/s (since the centrifuge starts from rest).

a. To find the astronaut's tangential acceleration during the first 40.0 seconds:

First of all, we would determine the final angular speed of the astronaut.

[tex]w_f = \frac{2\pi}{T}\\\\w_f = \frac{2\times 3.142}{1.50}\\\\w_f = \frac{6.284}{1.50}[/tex]

Final angular speed = 4.189 rad/s

Next, we would determine the astronaut's angular acceleration:

[tex]w_f = w_o + at[/tex]

Where:

  • [tex]w_f[/tex] is the final angular velocity.
  • [tex]w_o[/tex] is the initial angular velocity.
  • [tex]\alpha[/tex] is the angular acceleration.
  • t is the time.

Substituting the parameters into the formula, we have;

[tex]4.189 = 0 + \alpha 40\\\\4.189 = \alpha 40\\\\\alpha =\frac{4.189}{40} \\\\\alpha = 0.105 rad/s^2[/tex]

Now, we can calculate the astronaut's tangential acceleration:

[tex]\alpha_t = \alpha r\\\\\alpha_t = 0.105\times 6.20\\\\\alpha_t = 0.651 m/s^2[/tex]

b. To find how many g's of acceleration the astronaut would experience when the device is rotating at top speed:

At top speed, the centripetal acceleration of the astronaut is given by:

[tex]\alpha_c = w^2r\\\\\alpha_c = 4.189^2\times 6.20\\\\\alpha_c = 17.55\times 6.20\\\\\alpha_c = 108.81 \;m/s^2[/tex]

In terms of g:

[tex]\alpha_c = \frac{108.81}{9.80} \\\\\alpha_c = 11.10g[/tex]

Read more: https://brainly.com/question/14703849

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