At its Ames Research Center, NASA uses its large '20-G' centrifuge to test the effects of very large accelerations ('hypergravity') on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5g. (a) How fast must the astronaut's head be moving to experience this maximum acceleration?

Verified step by step guidance

1. Identify the given values: the length of the centrifuge arm (radius, r) is 8.84 m, and the maximum acceleration (a) is 12.5 times the acceleration due to gravity (g = 9.8 m/s^2).
2. Calculate the maximum acceleration in meters per second squared by multiplying 12.5g by the acceleration due to gravity (g).
3. Use the formula for centripetal acceleration, \( a = \frac{v^2}{r} \), where \( v \) is the linear speed of the astronaut's head and \( r \) is the radius of the motion (length of the arm).
4. Rearrange the formula to solve for \( v \): \( v = \sqrt{a \cdot r} \).
5. Substitute the values of \( a \) and \( r \) into the equation to find the speed \( v \) of the astronaut's head.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the acceleration directed towards the center of a circular path that an object follows. It is necessary for maintaining circular motion and is calculated using the formula a = v²/r, where 'a' is the centripetal acceleration, 'v' is the tangential speed, and 'r' is the radius of the circular path. In the context of the centrifuge, this acceleration is what the astronaut experiences as they are spun around.

Gravitational Force and g-Forces

The term 'g-force' refers to the force of gravity on an object, expressed in multiples of the acceleration due to Earth's gravity (approximately 9.81 m/s²). When an astronaut experiences 12.5g, it means they feel an acceleration equivalent to 12.5 times that of Earth's gravity. This concept is crucial for understanding the physical stresses placed on the body during high acceleration scenarios, such as those experienced in a centrifuge.

Tangential Speed

Tangential speed is the linear speed of an object moving along a circular path and is directly related to the radius of the circle and the angular velocity. It can be calculated using the formula v = ωr, where 'v' is the tangential speed, 'ω' is the angular velocity in radians per second, and 'r' is the radius. In the case of the centrifuge, determining the tangential speed at which the astronaut's head must move to achieve the desired acceleration is essential for understanding the dynamics of the system.