Question

A carefully designed experiment can measure the gravitational force between masses of \(1 \mathrm{~kg}\). Given that the density of iron is \(7860 \mathrm{~kg} / \mathrm{m}^{3}\) what is the gravitational force between two 1.00 -kg iron spheres that are touching?

Short answer

Answer: The gravitational force between the two 1 kg iron spheres touching each other can be calculated using the following formula: \(F = (6.674 *10^{-11})\dfrac{(1)(1)}{\left(2*\sqrt[3]{\dfrac{3(1/7860)}{4π}}\right)^2}\) By plugging in the known values and solving, we find the gravitational force between the two spheres.

Step-by-step solution

Determine the volume of the spheres using mass and density

Given the density and mass of the iron spheres, we can find their volumes using the formula: Volume = Mass / Density For 1 kg iron, we have: Density = 7860 kg/m^3 Mass = 1 kg Volume = (1 kg) / (7860 kg/m^3)

Find the radius of the spheres using their volume

To find the radius of the sphere, we need to substitute the volume we found in the previous step into the volume formula of the sphere. The formula for the volume of a sphere is: \(V = \dfrac{4}{3}πr^3\) Now, let's solve for r: \(Vr = \dfrac{4}{3}πr^3\) \(r^3 = \dfrac{3V}{4π}\) \(r = \sqrt[3]{\dfrac{3V}{4π}}\) Now plug in the volume from step 1: \(r = \sqrt[3]{\dfrac{3(1/7860)}{4π}}\)

Calculate the gravitational force between the two spheres

Now that we know the radius of the spheres, we can use the formula for the gravitational force between two masses. The gravitational force formula is: \(F = G\dfrac{m_1m_2}{d^2}\) Where, F is the force between two masses, G is the gravitational constant (6.674 × 10^-11 m^3⋅kg^-1⋅s^-2), \(m_1\) and \(m_2\) are the masses of the objects, and d is the distance between the centers of the masses. Since the iron spheres are touching each other, the distance between their centers is twice their radius: \(d = 2r\) Now, let's plug the values into the formula: \(F = (6.674 *10^{-11})\dfrac{(1)(1)}{\left(2*\sqrt[3]{\dfrac{3(1/7860)}{4π}}\right)^2}\) After calculating the result, you will find the gravitational force between the two 1 kg iron spheres touching each other.