Radius of 2nd Sphere: Find Mass 5x Greater than 1st

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To find the radius of the second sphere with a mass five times greater than the first, it's essential to understand that mass is proportional to volume, not directly to radius. The volume of a sphere is calculated using the formula V = 4/3πr³, indicating that the volume—and thus mass—depends on the cube of the radius. Therefore, if the mass of the second sphere is five times greater, the relationship can be expressed as V2/V1 = 5, leading to the equation (r2/r1)³ = 5. Solving this gives r2 as approximately 7.69 cm, not 22.5 cm, as the initial assumption incorrectly suggested a linear relationship. Understanding the cubic relationship between radius and volume is crucial for solving such problems correctly.
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Homework Statement


Two spheres are cut from a certain uniform rock. One has a radius 4.50cm. The mass of the
other is 5 times greater. Find its radius


Homework Equations


r1 = 4.50cm
r2 = ? when mass is 5x greater


The Attempt at a Solution


Looking at this problem I first thought of somehow integrating the volume of a sphere
which is 4/3\pir3. But then i though i was just thinking too hard. it made more sense for the radius to be 5x the radius as i figure the mass was proportional to the radius.

so: r2 = 5(4.50cm) = 22.5cm

however i got this marked wrong as the answer was 7.69 cm. what wasn't I looking at clearly?
 
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afg_91320 said:

Homework Statement


Two spheres are cut from a certain uniform rock. One has a radius 4.50cm. The mass of the
other is 5 times greater. Find its radius


Homework Equations


r1 = 4.50cm
r2 = ? when mass is 5x greater


The Attempt at a Solution


Looking at this problem I first thought of somehow integrating the volume of a sphere
which is 4/3\pir3. But then i though i was just thinking too hard. it made more sense for the radius to be 5x the radius as i figure the mass was proportional to the radius.

so: r2 = 5(4.50cm) = 22.5cm

however i got this marked wrong as the answer was 7.69 cm. what wasn't I looking at clearly?

You started on the right track. Mass is proportional to volume, so the ratio 5 applies to the two radii how?
 
berkeman said:
You started on the right track. Mass is proportional to volume, so the ratio 5 applies to the two radii how?

well that would be proportional to the mass right? so if r2 is 5x in mass then r1 is 1/5(mass)? set up my equation to isolate r to get the radius...
 
afg_91320 said:
well that would be proportional to the mass right? so if r2 is 5x in mass then r1 is 1/5(mass)? set up my equation to isolate r to get the radius...

Mass is proportional to volume. Write the fraction V1/V2 out fully, and that will show you a ratio involving some form of the radii. That's where the 5x comes into play...
 
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