Solutions to Some Random Questions

Question 1:

The height of the center of mass is a mass-weighted sum of the coordinates of the individual centers of mass.

yCM=y1M1+y2M2+y3M3M1+M2+M3y_{CM} = \frac{y_1 M_1 + y_2 M_2 + y_3 M_3}{M_1 + M_2 + M_3}

Values of these quantities:

y1=7RM1=43πR3ρ1y2=4RM2=323πR3ρ2y3=RM3=8R3ρ3y_1 = 7R \hspace{1cm} M_1 = \frac{4}{3} \pi R^3 \rho_1 \\ y_2 = 4R \hspace{1cm} M_2 = \frac{32}{3} \pi R^3 \rho_2 \\ y_3 = R \hspace{1cm} M_3 = 8 R^3 \rho_3

Plugging in:

yCM=7R×43πR3ρ1+4R×323πR3ρ2+R×8R3ρ343πR3ρ1+323πR3ρ2+8R3ρ3=4R283πR4ρ1+1283πR4ρ2+8R4ρ3=163πR4ρ1+1283πR4ρ2+32R4ρ3283πR4ρ1+8R4ρ3=163πR4ρ1+32R4ρ34πR4ρ1=24R4ρ3ρ1ρ3=6πy_{CM} = \frac{7R \times \frac{4}{3} \pi R^3 \rho_1 + 4R \times \frac{32}{3} \pi R^3 \rho_2 + R \times 8 R^3 \rho_3}{\frac{4}{3} \pi R^3 \rho_1 + \frac{32}{3} \pi R^3 \rho_2 + 8 R^3 \rho_3 } = 4R \\ \frac{28}{3} \pi R^4 \rho_1 + \frac{128}{3} \pi R^4 \rho_2 + 8 R^4 \rho_3 = \frac{16}{3} \pi R^4 \rho_1 + \frac{128}{3} \pi R^4 \rho_2 + 32 R^4 \rho_3 \\ \frac{28}{3} \pi R^4 \rho_1 + 8 R^4 \rho_3 = \frac{16}{3} \pi R^4 \rho_1 + 32 R^4 \rho_3 \\ 4 \pi R^4 \rho_1 = 24 R^4 \rho_3 \\ \frac{\rho_1}{\rho_3} = \boxed{\frac{6}{\pi}}

Question 2:

The height of the center of mass is a length-weighted sum of the coordinates of the individual centers of mass of the straight and curved parts. We can use length instead of mass because of the uniform mass density per unit length.

yCM=y1L1+y2L2L1+L2y_{CM} = \frac{y_1 L_1 + y_2 L_2 }{L_1 + L_2 }

Straight Part:

L1=πRy1=0L_1 = \pi R \hspace{1cm} y_1 = 0

Curved Part:

L2=πRy2=2RπL_2 = \pi R \hspace{1cm} y_2 = \frac{2R}{\pi}

Plugging in:

yCM=0πR+2RππRπR+πR=Rπy_{CM} = \frac{0 \pi R + \frac{2R}{\pi} \pi R }{\pi R + \pi R} = \boxed{\frac{R}{\pi}}

Note by Steven Chase
2 years, 1 month ago

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Can you send the pdf of the paper and the name of the book in the background @Steven Chase

ritik agrawal - 2 years, 1 month ago

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I can't, because that image is all I have :)

Steven Chase - 2 years, 1 month ago

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Woah so cengage is famous even in USA?Btw which cengage book are you using?Is it the same as the one used by Indians(IIT advanced ones)?

Ayush Anand - 1 year, 8 months ago

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These questions were sent to me by somebody else.

Steven Chase - 1 year, 8 months ago

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oh alright.

Ayush Anand - 1 year, 8 months ago

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@Ayush Anand lololololol

Kushal Thaman - 9 months, 2 weeks ago

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