Question

In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun's energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.3 m .
1) Determine the angular acceleration of a point on the skin of the ship 2.5 min after it started this acceleration.

2) Determine the radial component of the linear acceleration of a point on the skin of the ship 2.5 min after it started this acceleration.

3) Determine the tangential component of the linear acceleration of a point on the skin of the ship 2.5 min after it started this acceleration.

Answer 1

Answer:

1) $\alpha \approx 8.727\times 10^{-3}\,\frac{rad}{s^{2}}$, 2) $a_{r} = 14.222\,\frac{m}{s^{2}}$, 3) $a_{t} = 0.072\,\frac{m}{s^{2}}$

Explanation:

1) Let consider that Apollo spacecraft accelerates at constant rate. The angular acceleration of the spacecraft is:

$\alpha = \left(\frac{1\,\frac{rev}{min} }{12\,min}\right)\cdot \left(\frac{2\pi\,rad}{1\,rev} \right)\cdot \left(\frac{1\,min}{60\,s} \right)$

$\alpha \approx 8.727\times 10^{-3}\,\frac{rad}{s^{2}}$

2) The angular speed of the Apollo spacecraft at t = 2.5 min is:

$\omega = \left(8.727\times 10^{-3}\,\frac{rad}{s^{2}} \right)\cdot (150\,s)$

$\omega = 1.309\,\frac{rad}{s}$

The radial component of the linear acceleration is:

$a_{r} = \left(1.309\,\frac{rad}{s}\right)^{2}\cdot (8.3\,m)$

$a_{r} = 14.222\,\frac{m}{s^{2}}$

3) The tangential component of the linear acceleration is

$a_{t} = \left(8.727\times 10^{-3}\,\frac{rad}{s}\right)\cdot (8.3\,m)$

$a_{t} = 0.072\,\frac{m}{s^{2}}$