- #1

Karol

- 1,380

- 22

## Homework Statement

A spaceship has a shape of a hollow tube with outer radius R

_{1}, internal radius R

_{2}and mass "M". it rotates at angular velocity ω

_{0}. it has 4 short, mass less legs.

In it there is a bar of radius "r" and mass "m". it can be moved instantly to any distance "a" from the center.

The ship starts to fall from height H=10R

_{2}with the legs up and the inner bar at the center. the computer immediately moves the bar to a certain distance "a" upwards, so when the ship touches the ground the legs are at the bottom. it made half a rotation.

Find a formula for "a" so that the ship will land with legs down.

## Homework Equations

Moment of inertia of a cylinder round the center: ##I_c=\frac{1}{2}MR^2##

Steiner's theorem: ##I_a=I_c+Ma^2##

Free fall: ##h=\frac{1}{2}gt^2##

Conservation of angular momentum: ##I_c\omega_0=I_1\omega_1##

## The Attempt at a Solution

Moment of inertia with the bar at the center:

$$I=\frac{1}{2}\left(R_1^2-R_2^2\right)$$

Moment of inertia with the bar at distance a:

$$I=\frac{1}{2}\left(R_1^2-R_2^2\right)+\frac{1}{2}mr^2+m\left(a+\frac{r}{2}\right)^2$$

$$I=\frac{1}{2}\left[ M\left( R_1^2-R_2^2 \right)+m\left( a^2+ra+\frac{5}{4}r^2 \right)\right]$$

The center of mass when the bar is off the center is at distance C:

$$m\left(a+\frac{r}{2}\right)=C(M+m)\rightarrow C=\frac{\left(a+\frac{r}{2}\right)}{M+m}$$

Free fall between the centers of mass at the top and bottom positions:

$$h=\frac{1}{2}gt^2\rightarrow t^2=\frac{2}{g}\left[10R_2-R_1+\frac{\left(a+\frac{r}{2}\right)}{M+m}\right]$$

The ship makes half a turn, i.e. π[rad]. The new angular velocity:

$$\omega_1=\frac{\pi}{t}=\frac{\pi}{\sqrt{\frac{2}{g}\left[10R_2-R_1+\frac{\left(a+\frac{r}{2}\right)}{M+m}\right]}}$$

Conservation of angular momentum:

$$I_c\omega_0=I_1\omega_1\rightarrow \frac{1}{2}\left(R_1^2-R_2^2\right)\cdot \omega_0=\frac{\frac{1}{2}\left[M\left(R_1^2-R_2^2\right)+m\left(a^2+ra+\frac{5}{4}r^2\right)\right]\pi}{\sqrt{\frac{2}{g}\left[10R_2-R_1+\frac{\left(a+\frac{r}{2}\right)}{M+m}\right]}}$$

It's hard to isolate a, i think i am wrong