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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

Two gears \(A\) and \(B\) are in contact. The radius of gear \(A\) is twice that of gear \(B\). (a) When \(A\) 's angular velocity is \(6.00 \mathrm{Hz}\) counterclockwise, what is \(B^{\prime}\) s angular velocity? (b) If \(A\) 's radius to the tip of the teeth is \(10.0 \mathrm{cm},\) what is the linear speed of a point on the tip of a gear tooth? What is the linear speed of a point on the tip of \(B\) 's gear tooth?
Verify that all three expressions for radial acceleration $\left(v \omega, v^{2} / r, \text { and } \omega^{2} r\right)$ have the correct dimensions for an acceleration.
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.

In an amusement park rocket ride, cars are suspended from 4.25 -m cables attached to rotating arms at a distance of \(6.00 \mathrm{m}\) from the axis of rotation. The cables swing out at an angle of \(45.0^{\circ}\) when the ride is operating. What is the angular speed of rotation?
A \(0.700-\mathrm{kg}\) ball is on the end of a rope that is \(1.30 \mathrm{m}\) in length. The ball and rope are attached to a pole and the entire apparatus, including the pole, rotates about the pole's symmetry axis. The rope makes an angle of \(70.0^{\circ}\) with respect to the vertical. What is the tangential speed of the ball?
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