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Ostrovityanka [42]
3 years ago
11

49. \ A rectangular plate is rotating with a constant angular speed about an axis that passes perpendicularly through one corner

, as the drawing shows. The centripetal acceleration measured at corner A is n times as great as that measured at corner B. What is the ratio L1/L2 of the lengths of the sides of the rectangle when n
Physics
1 answer:
faust18 [17]3 years ago
4 0

Answer:

    \frac{L_1}{L_2} = \sqrt{(n^2 - 1)}

Explanation:

For this interesting problem, we use the definition of centripetal acceleration  

      a = v² / r  

angular and linear velocity are related  

     v = w r  

we substitute  

    a = w² r

the rectangular body rotates at an angular velocity w  

We locate the points, unfortunately the diagram is not shown. In this case we have the axis of rotation in a corner, called O, in one of the adjacent corners we call it A and the opposite corner A  

the distance OB = L₂  

the distance AB = L₁

the sides of the rectangle  

It is indicated that the acceleration in in A and B are related  

      a_A = n \ a_B  

we substitute the value of the acceleration  

    w² r_A = n r_B  

the distance from the each corner is  

    r_B = L₂  

    r_A = \sqrt{L_1^2 + L_2^2}  

we substitute  

   \sqrt{L_1^2 + L_2^2} = n L₂  

    L₁² + L₂² = n² L₂²  

    L₁² = (n²-1) L₂²  

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If, as is typical, each of them breathes about 500 cm3 of air with each breath, approximately what volume of air (in cubic meter
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Answer:

<em>a) 12614.4 m^3</em>

<em>b) 28.8 m</em>

Explanation:

The complete question is

Four astronauts are in a spherical space station. (a) If, as is typical, each of them breathes about 500 cm^3 of air with each breath, approximately what volume of air (in cubic meters) do these astronauts breathe in a year? (b) What would the diameter (in meters) of the space station have to be to contain all this air?

The average breathing rate is 12 breaths per minute

there are 60 minutes x 24 hours x 365 days in a year = 525600 minutes in a year

if an average human takes 12 breath per minute, then in a year an average human take 12 x 525600 = 6307200 breath

For the four astronauts, the amount of breath = 4 x 6307200 = 25228800 breath in total.

The volume of air per breath = 500 cm^3

1 cm^3 = 10^-6 m^3

500 cm^3 = x m^3

x = 500 x 10^-6 = 5 x 10^-4 m^3

Therefore in a year, the volume of these astronauts breath in a year = 5 x 10^-4 x 25228800 = <em>12614.4 m^3</em>

b) If the space station is spherical, the volume will be given as = \frac{4}{3} \pi r^3

where r is the radius of the space station

This volume of the space station must be able to contain all the volume of breath produced by the astronauts which is = 12614.4 m^3

Equating, we have

12614.4 = \frac{4}{3} \pi r^3

12614.4 = \frac{4}{3}*3.142*r^3

12614.4 = 4.1893r^{3}

r^{3} = 12614.4/4.1893 = 3011.1

r = \sqrt[3]{3011.1} =<em> 14.4 m</em>

<em>diameter of the space station = 14.4 m x 2 =  28.8 m</em>

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<u />

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