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Chapter 9: Problem 36

What is the centripetal acceleration of the Moon? The period of the Moon's orbit about the Earth is 27.3 days, measured with respect to the fixed stars. The radius of the Moon's orbit is \(R_{M}=3.85 \cdot 10^{8} \mathrm{~m}\).

Short Answer

Expert verified
Question: Determine the centripetal acceleration of the Moon given that its orbit has a radius of \(3.85 \cdot 10^8\) meters, and it takes 27.3 days to complete one orbit. Answer: To determine the centripetal acceleration of the Moon, follow these steps: 1. Calculate the circumference of the Moon's orbit using its radius. 2. Find the velocity of the Moon using the circumference and period of the orbit. 3. Compute the centripetal acceleration using the formula \(a_c = \dfrac{v^2}{r}\). After completing these steps, you will find that the centripetal acceleration of the Moon is approximately \(2.72 \times 10^{-3}\, m/s^2\).

Step by step solution

01

Find the Circumference of the Moon's orbit

We can find the circumference of the Moon's orbit using the formula \(C = 2\pi R_M\). Let's calculate the circumference: \(C = 2 \pi (3.85 \cdot 10^{8})\)
02

Find the Velocity of the Moon

To find the velocity of the Moon, we can use the formula \(v = \dfrac{C}{T}\), where \(T\) is the period of the Moon's orbit. We are given that \(T\) is 27.3 days, but we need to convert it to seconds. \(T = 27.3 \cdot 24 \cdot 60 \cdot 60\) seconds Now, we can find the velocity: \(v = \dfrac{2 \pi (3.85 \cdot 10^{8})}{27.3 \cdot 24 \cdot 60 \cdot 60}\)
03

Calculate the Centripetal Acceleration

Now that we have calculated the velocity \(v\), we can find the centripetal acceleration using the formula \(a_c = \dfrac{v^2}{R_M}\). Substitute the values and compute the result: \(a_c = \dfrac{(\dfrac{2 \pi (3.85 \cdot 10^{8})}{27.3 \cdot 24 \cdot 60 \cdot 60})^2}{3.85 \cdot 10^{8}}\) Simplify and compute the result to get the centripetal acceleration of the Moon.

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In a tape recorder, the magnetic tape moves at a constant linear speed of \(5.6 \mathrm{~cm} / \mathrm{s}\). To maintain this constant linear speed, the angular speed of the driving spool (the take-up spool) has to change accordingly. a) What is the angular speed of the take-up spool when it is empty, with radius \(r_{1}=0.80 \mathrm{~cm} ?\) b) What is the angular speed when the spool is full, with radius \(r_{2}=2.20 \mathrm{~cm} ?\) c) If the total length of the tape is \(100.80 \mathrm{~m}\), what is the average angular acceleration of the take-up spool while the tape is being played?

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