answer all parts Consider a sphere of radius R where the volume density rho is n

Question

answer all parts

Consider a sphere of radius R where the volume density rho is not uniform but varies with z as rho = 2R + z a. What must X_com and y_com be? Be sure to justify your answer. For your differential of mass, consider a disc of thickness dz and radius r perpendicular to the z-axis located a distance z from a distance z from the origin. See the figure below. b. What is radius r of this disc? c. What is the mass dm of this disc? d. Set up an integral for the total mass of the sphere in terms of R. At this point, you do not need to do the integral just set it up, Be sure to include the proper limits of integration with your integral. e. Perform the integration. f. Set up an integral for z_com in terms of R. At this point, you do not need to do the integral just set it up. Be sure to include the proper limits of integration with your integral. g. Perform the integration.

Explanation / Answer

`consider differential volume at height z from the origin qas seen in the figure
a. xcm and y cm will vbe on the origin as the mass distribution is the same about these planes
b. radius of this disc, r = sqroot(R^2 - z^2)
c. mass, dm = rho*pi*r^2*dz = (2R + z)*pi*(R^2 - z^2)*dz
d. total mass, M = integrate dm from z = -R to z = R = (integrate)(2R + z)*pi*(R^2 - z^2)*dz = integrate*pi*(2R^3 - 2Rz^2 + zR^2 - z^3)dz = pi(2R^3z - 2Rz^3/3 - z^2R^2/2 - z^4/4)
e. Applying limits
pi(2R^4 - 2R^4/3 - R^4/2 - R^4/4) - pi(-2R^4 + 2R^4/3 - R^4/2 - R^4/4) = 8piR^4/3
f. Zcm = integrate*piz*(2R^3 - 2Rz^2 + zR^2 - z^3)dz/8piR^4/3 from z = R to z = -R
g. z cm = integrate*piz*(2R^3 - 2Rz^2 + zR^2 - z^3)dz/8piR^4/3 from z = R to z = -R = pi(2R^3z^2/2 - 2Rz^4/4 + z^3R^2/3 - z^5/5)/8piR^4/3 from z = R to z = -R
Z cm = 12R/5

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