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Physics / Sears and Zemansky's University Physics with Modern Physics 13 / Chapter 9 / Problem 99P

Sears and Zemansky's University Physics with Modern Physics | 13th Edition | ISBN: 9780321696861 | Authors: Hugh D. Young; Roger A. Freedman; A. Lewis Ford

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CALC A sphere with radius R = 0.200 m has density ? that

Chapter 9, Problem 99P

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QUESTION:

Problem 99P

CALC A sphere with radius R = 0.200 m has density ρ that decreases with distance r from the center of the sphere according to ρ = 3.00 X 103 kg/m3 – (9.00 X 103 kg/m4)r. (a) Calculate the total mass of the sphere. (b) Calculate the moment of inertia of the sphere for an axis along a diameter.

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QUESTION:

Problem 99P

CALC A sphere with radius R = 0.200 m has density ρ that decreases with distance r from the center of the sphere according to ρ = 3.00 X 103 kg/m3 – (9.00 X 103 kg/m4)r. (a) Calculate the total mass of the sphere. (b) Calculate the moment of inertia of the sphere for an axis along a diameter.

ANSWER:

Solution 99P

Step 1:

Data given

Radius of sphere $R=0.200\ m$ $\$

Density $\rho{}=3.00\times{}1{0}^{3}\ kg\ /{m}^{3}-(9\times{}1{0}^{3}\ kg\ /{m}^{4})r$

We need to find the mass of the sphere

This is given by

$dm=\rho{}\times{}dv$

But we know $v=(4π/3){r}^{3}$

Thus we get $dv=4π{r}^{3}dr$

$m=\int\limits_{0}^{0}(0.200\ m)\ 3.00\times{}1{0}^{3}\ kg\ /{m}^{3}-(9\times{}1{0}^{3}\ kg\ /{m}^{4})r\times{}4π{r}^{4}dr$

Substituting values we get

$m=55.3\ kg\$

Thus we have mass of sphere as $55.3\ kg$

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