# Rotating Rockets Homework: Angular Vel. After 33s

• Tina20
In summary, the rockets generate 4144000 N of torque (70m x 59200 N) and the structure's angular velocity after 33.0 s is 0.3052 rad/s.
Tina20

## Homework Statement

Far out in space, a mass1=146500.0 kg rocket and a mass2= 209500.0 kg rocket are docked at opposite ends of a motionless 70.0 m long connecting tunnel. The tunnel is rigid and has a mass of 10500.0 kg.
The rockets start their engines simultaneously, each generating 59200.0 N of thrust in opposite directions. What is the structure's angular velocity after 33.0 s?

## Homework Equations

center of mass = (m1x1 + m2x2 + m3x3)/m1 + m2 + m3
where m1 = mass of rocket 1, m2 = mass of tunnel, m3 = mass of rocket 3
x(number) = distance to the origin (in this case, rocket one used as origin (set at 0)

Moment of Inertia = m1r1^2 + m2r2^2 (m=mass, r= distance to center of mass)
Moment of inertia of thin, long rod about center = 1/12 ML^2 (M=mass, L=length of rod)

Torque = lengthxForce

angular acceleration = Torque/inertia

angular velocity = angular acceleration x time

## The Attempt at a Solution

So, I found the center of mass, and I know it's correct because the previous question asked for it and was found to be correct. It was 41m.

Now, I think I may be calculating my moment of inertia wrong?
I = m1r1^2 + m3r3^2 + 1/12 ML^2
I = 447 938 063.8 kg*m^2

Am is supposed to include the moment of inertia of the tunnel (rod) in this scenario? Obviously I use the mass and their relative distances to the center of mass for the rockets to find their moment of inertia...but do I just add on the moment of inertia of the rod?

My answer was incorrect, so I am trying to figure out what is wrong about my process. I am assuming my moment of inertia is incorrect.

The torque is 70m x 59200N = 4144000 N*m

(1/12)*M*L^2 is the moment of inertia of the rod about its center. So to find its moment of inertia about the center of mass, use parallel axis theorem of moment of inertia.

Ok, I did that, but the angular velocity still comes out to be 0.305 rad/sec which is wrong. I still don;t know what else I could be wrong. Any ideas?

Tnet = lF = (70m)(59200) = 4144000 N*m

angular acceleration = Tnet/Itotal

w(omega) = angular accelerationxtime
= 0.3052 rad/sAny help will be very much appreciated :)

Last edited:
Tina20 said:
Ok, I did that, but the angular velocity still comes out to be 0.305 rad/sec which is wrong. I still don;t know what else I could be wrong. Any ideas?

Tnet = lF = (70m)(59200) = 4144000 N*m

angular acceleration = Tnet/Itotal

w(omega) = angular accelerationxtime

Any help will be very much appreciated :)

Net angular acceleration is gives by

$$\alpha = F(\frac{1}{m_1r_1} + \frac{1}{m_2r_2} + \frac{1}{m_3r_3})$$

Ok,

so F = lF which is torque, or is it the force of thrust?

I tried both Forces and substituted it into the equation you gave me. I am still not getting a correct answer. I assume r in the equation below is the distance to the center of mass correct? or is it the distance to the centre of the tunnel?

F is the force of the thrust. F= ma. So a = F/m. angular acceleration α is equal to F/(m*r). All the distances are measured from the center of the mass.

Show your calculations and expected result.

Ok, so F = 59200N

angular a = F (1/m1r1 + 1/m2r2 + 1/m3r3)
angular a = 59200N (1/m1r1 + 1/m2r2 + 1/m3r3) where m1=rocket 1, m2=mass of tunnel, m3=mass of rocket 2. r1 = rocket 1 to centre of mass, r2= centre of tunnel to centre of mass, r3 = rocket 2 to centre of mass.

I solved for angular a and multiplied it by the time and got the right answer! Thank you so much. I don't understand why I was getting it wrong last night...probably too late and I was making a silly mistake.

## 1. What is angular velocity?

Angular velocity is a measure of the rate of change of angular displacement over time. It is often represented by the symbol ω and measured in radians per second (rad/s).

## 2. How is angular velocity different from linear velocity?

Angular velocity describes the rotational motion of an object, while linear velocity describes the straight-line motion of an object. Angular velocity is measured in radians per second, while linear velocity is measured in meters per second.

## 3. How is angular velocity calculated?

Angular velocity is calculated by dividing the change in angular displacement by the change in time. The formula is ω = Δθ/Δt, where ω is angular velocity, Δθ is change in angular displacement, and Δt is change in time.

## 4. How does angular velocity affect a rotating rocket?

Angular velocity affects a rotating rocket by determining the speed at which it rotates. The higher the angular velocity, the faster the rocket will rotate. This can impact the stability and control of the rocket during flight.

## 5. How is angular velocity related to centripetal acceleration?

Angular velocity and centripetal acceleration are directly related. The centripetal acceleration of an object in circular motion is equal to the square of its angular velocity multiplied by the radius of the circle. This relationship is expressed by the formula a = ω²r, where a is centripetal acceleration, ω is angular velocity, and r is the radius of the circle.

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