(II) A device for training astronauts and jet fighter pilots...

(II) A device for training astronauts and jet fighter pilots is designed to rotate a trainee in a horizontal circle of radius 12.0 $\mathrm{m}$ . If the force felt by the trainee on her back is 7.85 times her own weight, how fast is she rotating? Express your answer in both $\mathrm{m} / \mathrm{s}$ and rev/s.

Key Concepts

Circular Motion

Circular motion refers to the movement of an object along a curved path, typically a circle. This type of motion requires a constant change in the direction of velocity, which is achieved by an inward acceleration known as centripetal acceleration.

Centripetal Acceleration and Force

Centripetal acceleration is the acceleration directed toward the center of a circle that keeps an object moving along its circular path, defined by the relationship a = v²/r, where v is the speed and r is the radius. The corresponding centripetal force, given by F = m·v²/r, represents the net inward force needed to sustain the circular motion.

Expressing Acceleration in Terms of g

In many practical applications, especially in aerospace and aviation, acceleration is often expressed as a multiple of the gravitational acceleration g (approximately 9.8 m/s²). This provides an intuitive understanding of the forces experienced by individuals, comparing them directly to the familiar weight force due to gravity.

Conversion Between Linear and Rotational Quantities

The relationship between the linear speed of an object moving along a circle and its rotational quantities, such as revolutions per second, is established via the circumference of the circle. By dividing the linear speed by the circumference, one can convert the speed to a rotation rate in rev/s, which is a common measure in systems involving rotational motion.

Transcript

00:01 Question number 11 asks us, what is the speed of an astronaut in a rotational device that exerts 7 .85 times her weight on her back? to get an idea of what's happening here, let's draw our astronaut in the seat of the rotational device here.

00:20 Here it is.

00:23 There's the arm of the device right there.

00:27 And in the back of her is where a force is applied.

00:30 And she's traveling forward at some velocity.

00:35 And there's the central shaft right there.

00:39 Now, the length of the arm attached to the central shaft and the astronaut's chair is what we're going to call r.

00:49 And force is applied to the back of her at a magnitude of 7 .85 times her weight.

00:59 And this force is rotational.

01:01 Now, since she's being spun in a horizontal circle, the force she would feel would be equal to her centripetal acceleration.

01:10 So, f sub r is going to be equal to m times v squared over r.

01:18 We also know that f sub r is equal to 7 .85w, 7 .85 times the astronaut's weight.

01:29 Her weight can be thought of as the product of her mass and the gravitational constant.

01:35 So we can rewrite this as 7 .85 mg.

01:42 Now we can set 7 .85 times mg equal to m times v squared over r.

01:51 We don't know what the mass of our astronaut is, but we don't need to, because both the ms cancel out on...

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