The word "Per" means divide
"miles per gallon" is the same as "miles / gallon"
The truck went 1,200 miles
on 55 gallons
1,200 ÷ 55 = 21.81
Answer:
Originally : Level = log I / I0
Currently: Level = 10 log I / I0
Level = 10 log 600 = 10 * 2.78 = 27.8
Note the term 1 bel = 10 decibels
1. GPE - 40 * 2 * 10 = 800j
Answer:
The angular velocity is ![w_f = 1.531 \ rad/ s](https://tex.z-dn.net/?f=w_f%20%3D%20%201.531%20%5C%20rad%2F%20s)
Explanation:
From the question we are told that
The mass of each astronauts is ![m = 50 \ kg](https://tex.z-dn.net/?f=m%20%3D%20%2050%20%5C%20kg)
The initial distance between the two astronauts ![d_i = 7 \ m](https://tex.z-dn.net/?f=d_i%20%20%3D%20%207%20%5C%20%20m)
Generally the radius is mathematically represented as ![r_i = \frac{d_i}{2} = \frac{7}{2} = 3.5 \ m](https://tex.z-dn.net/?f=r_i%20%20%3D%20%20%5Cfrac%7Bd_i%7D%7B2%7D%20%3D%20%5Cfrac%7B7%7D%7B2%7D%20%20%3D%20%203.5%20%5C%20%20m)
The initial angular velocity is ![w_1 = 0.5 \ rad /s](https://tex.z-dn.net/?f=w_1%20%3D%200.5%20%5C%20%20rad%20%2Fs)
The distance between the two astronauts after the rope is pulled is ![d_f = 4 \ m](https://tex.z-dn.net/?f=d_f%20%3D%20%204%20%5C%20%20m)
Generally the radius is mathematically represented as ![r_f = \frac{d_f}{2} = \frac{4}{2} = 2\ m](https://tex.z-dn.net/?f=r_f%20%20%3D%20%20%5Cfrac%7Bd_f%7D%7B2%7D%20%3D%20%5Cfrac%7B4%7D%7B2%7D%20%20%3D%20%202%5C%20%20m)
Generally from the law of angular momentum conservation we have that
![I_{k_1} w_{k_1}+ I_{p_1} w_{p_1} = I_{k_2} w_{k_2}+ I_{p_2} w_{p_2}](https://tex.z-dn.net/?f=I_%7Bk_1%7D%20w_%7Bk_1%7D%2B%20I_%7Bp_1%7D%20w_%7Bp_1%7D%20%3D%20I_%7Bk_2%7D%20w_%7Bk_2%7D%2B%20I_%7Bp_2%7D%20w_%7Bp_2%7D)
Here
is the initial moment of inertia of the first astronauts which is equal to
the initial moment of inertia of the second astronauts So
![I_{k_1} = I_{p_1 } = m * r_i^2](https://tex.z-dn.net/?f=I_%7Bk_1%7D%20%3D%20I_%7Bp_1%20%7D%20%3D%20%20m%20%2A%20%20r_i%5E2)
Also
is the initial angular velocity of the first astronauts which is equal to
the initial angular velocity of the second astronauts So
![w_{k_1} =w_{p_1 } = w_1](https://tex.z-dn.net/?f=w_%7Bk_1%7D%20%3Dw_%7Bp_1%20%7D%20%3D%20w_1)
Here
is the final moment of inertia of the first astronauts which is equal to
the final moment of inertia of the second astronauts So
![I_{k_2} = I_{p_2} = m * r_f^2](https://tex.z-dn.net/?f=I_%7Bk_2%7D%20%3D%20I_%7Bp_2%7D%20%3D%20%20m%20%2A%20%20r_f%5E2)
Also
is the final angular velocity of the first astronauts which is equal to
the final angular velocity of the second astronauts So
![w_{k_2} =w_{p_2 } = w_2](https://tex.z-dn.net/?f=w_%7Bk_2%7D%20%3Dw_%7Bp_2%20%7D%20%3D%20w_2)
So
![mr_i^2 w_1 + mr_i^2 w_1 = mr_f^2 w_2 + mr_f^2 w_2](https://tex.z-dn.net/?f=mr_i%5E2%20w_1%20%2B%20mr_i%5E2%20w_1%20%3D%20mr_f%5E2%20w_2%20%2B%20mr_f%5E2%20w_2)
=> ![2 mr_i^2 w_1 = 2 mr_f^2 w_2](https://tex.z-dn.net/?f=2%20mr_i%5E2%20w_1%20%3D%202%20mr_f%5E2%20w_2)
=> ![w_f = \frac{2 * m * r_i^2 w_1}{2 * m * r_f^2 }](https://tex.z-dn.net/?f=w_f%20%3D%20%20%5Cfrac%7B2%20%2A%20m%20%2A%20r_i%5E2%20w_1%7D%7B2%20%2A%20m%20%2A%20%20r_f%5E2%20%7D)
=> ![w_f = \frac{3.5^2 * 0.5}{ 2^2 }](https://tex.z-dn.net/?f=w_f%20%3D%20%20%5Cfrac%7B3.5%5E2%20%2A%20%200.5%7D%7B%20%202%5E2%20%7D)
=> ![w_f = 1.531 \ rad/ s](https://tex.z-dn.net/?f=w_f%20%3D%20%201.531%20%5C%20rad%2F%20s)