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Chapter 5: Problem 56

A space station is shaped like a ring and rotates to simulate gravity. If the radius of the space station is \(120 \mathrm{m},\) at what frequency must it rotate so that it simulates Earth's gravity? [Hint: The apparent weight of the astronauts must be the same as their weight on Earth.]

Short Answer

Expert verified
Answer: The space station should rotate at a frequency of approximately 0.0125 Hz to simulate Earth's gravity.

Step by step solution

01

Determine the centripetal acceleration formula

The formula for centripetal acceleration (a_c) is given by: \(a_c = \frac{v^2}{r}\), where v is the linear speed of rotation and r is the radius of the circular path.
02

Relate centripetal acceleration with gravitational acceleration

We need the centripetal acceleration to be equal to Earth's gravitational acceleration (g) to simulate Earth's gravity. Therefore, we set the centripetal acceleration equal to g: \(a_c = g \Rightarrow \frac{v^2}{r} = g\)
03

Relate linear speed (v) to frequency (f)

The linear speed (v) can be expressed in terms of frequency (f) using the formula \(v = 2\pi rf\). Substitute this into our previous equation: \(\frac{(2\pi rf)^2}{r} = g\)
04

Solve the equation for frequency (f)

We need to find the frequency, so we will solve the above equation for f: \(\frac{(2\pi rf)^2}{r} = g \Rightarrow (2\pi rf)^2 = rg \Rightarrow f^2 = \frac{g}{(2\pi r)^2}\) Now take the square root: \(f = \sqrt{\frac{g}{(2\pi r)^2}}\)
05

Plug in the values and calculate the frequency

We are given the radius of the space station, \(r = 120 \,\text{m}\). Earth's gravitational acceleration, \(g = 9.81 \,\text{m/s}^2\). Substitute these values into the frequency equation: \(f = \sqrt{\frac{9.81}{(2\pi \cdot 120)^2}}\) Now, calculate the frequency: \(f \approx \sqrt{\frac{9.81}{(753.98)^2}} \approx 0.0125 \,\text{Hz}\) The space station must rotate at a frequency of approximately \(0.0125\) Hz to simulate Earth's gravity.

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Most popular questions from this chapter

Objects that are at rest relative to the Earth's surface are in circular motion due to Earth's rotation. (a) What is the radial acceleration of an object at the equator? (b) Is the object's apparent weight greater or less than its weight? Explain. (c) By what percentage does the apparent weight differ from the weight at the equator? (d) Is there any place on Earth where a bathroom scale reading is equal to your true weight? Explain.
You place a penny on a turntable at a distance of \(10.0 \mathrm{cm}\) from the center. The coefficient of static friction between the penny and the turntable is \(0.350 .\) The turntable's angular acceleration is $2.00 \mathrm{rad} / \mathrm{s}^{2} .$ How long after you turn on the turntable will the penny begin to slide off of the turntable?
A bicycle is moving at \(9.0 \mathrm{m} / \mathrm{s} .\) What is the angular speed of its tires if their radius is \(35 \mathrm{cm} ?\)
A high-speed dental drill is rotating at $3.14 \times 10^{4} \mathrm{rad} / \mathrm{s}$ Through how many degrees does the drill rotate in 1.00 s?
A carnival swing is fixed on the end of an 8.0 -m-long beam. If the swing and beam sweep through an angle of \(120^{\circ},\) what is the distance through which the riders move?
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