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to get a flat uniform cylindrical satellite spinning at the correct rate, scientists fire 4 tangential rockets. suppose that the satellite has a mass of 3600 kg and a radius of 4m, and that the rockets each add a mass of 250 kg. what is the steady force required of each rocket if the satellite is to reach 32 rpm in 5 min starting from rest.

Respuesta :

Answer:

The steady force required of each rocket is 31.36 N

Explanation:

Given that,

Mass of satellite = 3600 Kg

Radius = 4 m

Mass of rocket = 250 kg

Time = 5 min

Angular velocity = 32 rpm

We need to calculate the moment of inertia

Using formula of moment of inertia

[tex]I=\dfrac{1}{2}Mr^2+4mr^2[/tex]

Where, M = mass of satellite

r = radius

m = mass of rocket

Put the value into the formula

[tex]I=\dfrac{1}{2}\times3600\times4^2+4\times250\times4^2[/tex]

[tex]I=44800\ kg-m^2[/tex]

We need to calculate the angular acceleration

Using formula of angular acceleration

[tex]\alpha=\dfrac{\omega}{t}[/tex]

Where, [tex]\omega[/tex] = angular velocity

t = time

Put the value into the formula

[tex]\alpha=\dfrac{2\pi\times32}{3600\times5}[/tex]

[tex]\alpha=0.0112\ rad/s^2[/tex]

We need to calculate the steady force required of each rocket

Using formula of torque

[tex]\tau=I\alpha[/tex]

[tex]4rF=I\alpha[/tex]

[tex]F=\dfrac{I\alpha}{4r}[/tex]

Put the value into the formula

[tex]F=\dfrac{44800\times0.0112}{4\times4}[/tex]

[tex]F=31.36\ N[/tex]

Hence, The steady force required of each rocket is 31.36 N

batolisis

The steady force required of each rocket is = 31.36 N

Given data:

mass of satellite ( M ) = 3600 kg

radius of satellite ( r )  = 4 m

mass added by each rocket ( m ) = 250 kg

Angular speed of satellite = 32 rpm

Time = 5 mins  

First step : Calculate the value of the moment of inertia

I = [tex]\frac{1}{2} Mr^2 + 4mr^2[/tex] ----- ( 1 )

where : M = 3600 kg,  m = 250 kg,  r = 4m.

Insert values into equation ( 1 )

I = 44800 kg-m^2

Next step : Determine the angular acceleration

∝ = ω / t

  = ( 2π * 32 ) /  3600 * 5

  = 0.0112 rad/s²

Final step : Determine the steady force required by each rocket

Τ = I * ∝    ----- ( 1 )

where T = 4rF

Equation ( 1 ) becomes

4rF = I * ∝

∴ F = I*∝ / 4r

       = ( 44800 * 0.0112 ) / 4*4

      = 31.36 N .

Hence we can conclude that The steady force required of each rocket is = 31.36 N

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