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

1)
`\alpha \approx 8.727* 10^(-3)\,(rad)/(s^(2))`, 2)
`a_(r) = 14.222\,(m)/(s^(2))`, 3)
`a_(t) = 0.072\,(m)/(s^(2))`

Step-by-step explanation:

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

`\alpha = \left((1\,(rev)/(min) )/(12\,min)\right)\cdot \left((2\pi\,rad)/(1\,rev) \right)\cdot \left((1\,min)/(60\,s) \right)`

`\alpha \approx 8.727* 10^(-3)\,(rad)/(s^(2))`

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

`\omega = \left(8.727* 10^(-3)\,(rad)/(s^(2)) \right)\cdot (150\,s)`

`\omega = 1.309\,(rad)/(s)`

The radial component of the linear acceleration is:

`a_(r) = \left(1.309\,(rad)/(s)\right)^(2)\cdot (8.3\,m)`

`a_(r) = 14.222\,(m)/(s^(2))`

3) The tangential component of the linear acceleration is

`a_(t) = \left(8.727* 10^(-3)\,(rad)/(s)\right)\cdot (8.3\,m)`

`a_(t) = 0.072\,(m)/(s^(2))`