The Effect of Scaling on Gravitational Force

Classical Mechanics Level 3

Two solid metal balls are placed in contact with each other. The first ball is made of a material twice as dense as the second ball; the first ball has a radius of 1, the second has a radius of 3. The gravitational force between them is $F_1$ . The balls are somehow enlarged so that they gain mass proportional to their respective densities (density is conserved) and ball one's radius is twice as large as it was and ball two's radius is five times as large as it was. What is the new force relative to $F_1$ ?

12 \ F_1

32.1 \ F_1

55.36 \ F_1

63.45\ F_1

0.06 \ F_1

2 solutions

Peter Macgregor
Apr 24, 2015

Careful reading of the question shows that the distance between the centres of the spheres increases from (1+3) to (2+15), in other words by a factor of $\frac{17}{4}$ .

Since the mass of a sphere is proportional to the cube of its radius, the masses of the spheres increase by factors of 8 and 125, so their product increases by a factor of 1000.

Looking at the formula for F we can see that it will be changed by a factor

$\left(\frac{4}{17}\right)^2\times 1000\approx\boxed{55.36}$

Jared Jones
Apr 8, 2015

F1= G(2 x V1)( 1 x V2)/(4^2)

G is the universal gravitation constant

V1 is 4/3 (pi)

V2 is 27 x (4/3 (pi) )

4 is the sum of the radii 1 and 3

V x D = M

since this is a ratio problem, assume for simplicity that the density of ball 2 is 1kg/ m^3 and the density of ball 1 is 2kg/ m^3

multiply the volume by the density

F1= G( 2 x 4/3pi)(27x4/3pi)/(16)

V(new)=V(old) x (linear scale factor)^3

The first ball has a scale factor of two, so it has 8 times as much volume. The second ball has a scale factor of 5, so it has 125 times as much volume.

The radii are scaled too. The sum of the new radii (2 and 15) is 17

F2= G(8 x (2 x 4/3pi))(125 x 27 x 4/3pi)/(17^2)

therefore,

F2/F1 is the ratio between the new and old force which is (8 x 125/289)/(1/16), or approximately 55.3633218 times as large as the first force.

Quite a good solution.

Actually, this problem is easy, but creates a Strong resonance in your neurons!

Muhammad Arifur Rahman - 6 years, 2 months ago

Thanks! I got the idea from an AP Physics question, but modified it because the original had the balls' size, density and scale factor the same. It didn't require quite as much thought before.

Jared Jones - 6 years, 2 months ago

Another interesting thing is that density ratio is tricky. Its not required here, because they cancel themselves in calculations. That's brilliant!

Muhammad Arifur Rahman - 6 years, 1 month ago

The Effect of Scaling on Gravitational Force

2 solutions

0 pending reports