Question

As their booster rockets separate, Space Shuttle astronauts typically feel accelerations up to 3 g, where g = 9.80 m/s². In their training, astronauts ride in a device where they experience such an acceleration as a centripetal acceleration. Specifically, the astronaut is fastened securely at the end of a mechanical arm, which then turns at constant speed in a horizontal circle. Determine the rotation rate, in revolutions per second, required to give an astronaut a centripetal acceleration of 3.00g while in circular motion with radius 9.45 m.

Answer 1

To give an astronaut a centripetal acceleration of 3.00g in a circular motion with a radius of 9.45 m, a rotation rate of approximately 0.280 revolutions per second is required.

Step-by-step explanation:

To determine the rotation rate required to give an astronaut a centripetal acceleration of 3.00g at a radius of 9.45 m, we first need to convert the acceleration to meters per second squared (m/s²). Three times the acceleration due to gravity (3g) is:
3 × 9.80 m/s² = 29.4 m/s².

Next, using the formula for centripetal acceleration
`a_c`= rω², where r is the radius and ω is the angular velocity in radians per second, we can solve for ω. Following this, we will convert ω to revolutions per second since 1 revolution is 2π radians:
ω = √(29.4 m/s² / 9.45 m) = 1.76 rad/s.

Lastly, to convert radians per second to revolutions per second, we divide by 2π radians:
1.76 rad/s ÷ 2π rad/rev = 0.280 rev/s.

Answer 2

The rotation rate required to give an astronaut a centripetal acceleration of 3g while in a circular motion with a radius of 9.45 m is approximately 0.28 revolutions per second.

Step-by-step explanation:

To calculate the rotation rate required for an astronaut to experience a centripetal acceleration of 3g at a radius of 9.45m, we first need to establish the centripetal acceleration which is given by the formula a = rω2, where 'a' is the centripetal acceleration, 'r' is the radius, and 'ω' is the angular velocity.

Given that the desired acceleration is 3g (which is 3 × 9.80 m/s2), and the radius is 9.45 m, we can rearrange the formula to solve for 'ω'.

The formula for angular velocity based on centripetal acceleration is ω = √(a/r).

So, ω = √(3×9.80m/s2/9.45m) = √(3.132 m/s2) = 1.77 rad/s.

To convert radial velocity to revolutions per second, we use the relationship 1 revolution = 2π radians. Therefore, the rotation rate in revolutions per second is ω/(2π) which equals 1.77 rad/s / 2π = 0.28 rev/s, or approximately 0.28 revolutions per second.