Two golf balls each have a 4.30 -cm diameter and are 1.00 m apart. What would be the gravitation

Two golf balls each have a 4.30 -cm diameter and are 1.00 m...

Key Concepts

Newton's Law of Universal Gravitation

This law states that every pair of masses in the universe attracts each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It is expressed mathematically as F = G*(m1*m2)/r^2, where G is the gravitational constant, m1 and m2 are the masses, and r is the distance between them. This fundamental principle is essential for calculating gravitational forces in any system involving mass.

Density and Mass Calculation

In many physics problems, the mass of an object is derived from its density and volume. When dealing with objects composed of a specific material, such as nuclear matter, the density is used in conjunction with the volume (calculated from the object's dimensions) to determine its mass via the relation m = density × volume. This calculation is critical when actual mass values are not provided directly.

Volume of a Sphere

The volume of a sphere is calculated using the formula V = (4/3)?r^3, where r is the radius of the sphere. This geometric concept is vital when computing the mass of spherical objects by combining it with the material’s density. It ensures an accurate representation of mass distribution in problems involving spherical bodies.

Nuclear Matter Density

Nuclear matter refers to a state of matter with an extremely high density, typically found in the cores of atomic nuclei. When objects are assumed to be made of nuclear matter, their mass is vastly greater than that of ordinary materials. This concept plays a crucial role in understanding how material properties can significantly affect gravitational interactions.

Transcript

00:01 Question 8, a little bit of a different one, tells us that we have two golf balls with a specific diameter, which i'm calling d, and they are one meter apart here given by x.

00:13 We want to find what the gravitational force exerted on each ball would be, would be on the other ball, if we can assume that these golf balls are made from nuclear materials.

00:25 So, of course, the equation we want to solve for, or the final equation, is the gravitational force equation.

00:34 So we have g times m, we can assume they're both the same size.

00:38 So this is just m1 times m2, which is just m squared, divided by the distance between the object's centers squared.

00:48 So in terms of my labeling here, d is the diameter of each golf ball, x is the difference, the distance between the two edges of the golf balls, and s is the center -to -center distance, which is, of course, what we want when we're dealing with gravitational force between two objects.

01:03 So s, we can easily find, and g, of course, is a constant.

01:07 So all we need to solve for is the mass of these golf balls.

01:12 So while it may seem easy, there's a little bit of work to go into this.

01:17 So we're going to assume that both these golf balls are made of nuclear material.

01:24 So in order to do this, we need to find the nuclear density of each of these, assuming they're made of protons and neutrons.

01:31 Or i guess in this scenario, i think i'm just going to assume they're made out of neutrons just for the math, make the math a little bit easier.

01:37 And since the mass between a new one, neutron and proton isn't that drastically different than i don't expect much of a concern for that.

01:46 So if we can think, if you need to solve for the density of what these objects are, again, if they are made from nuclear material, and of course that's the mass over the volume.

01:59 We're not given any sort of mass in terms of grams, but again, if we are assuming these are made of neutrons, we can simply just say it's the mass per neutron multiplied by atomic number or how many neutrons are present.

02:14 The volume, the golf ball, so it's very easy to assume that they are just perfect spheres.

02:20 That's 4 over 3 pi are cubed.

02:29 4 pi over 3 is r cubed.

02:34 However, again, we're assuming their nuclear matter, so we can, you can tell, or we've found that their radius of a nucleus is given by the relationship of a, this little constant, times the atomic number, to the power of one -third.

02:48 So i can go ahead and rearrange this.

02:51 I'll put my three on top now.

02:53 Times the mass of a neutron over a, sorry, times a, it's given by 4 pi little a cubed, times capital a, a time is one, one third cube, which is just one.

03:12 So these atomic numbers actually, actually very luckily, cancel out.

03:16 We're left with our nuclear density of these golf holes would be three times the mass of the neutron over 4 pi a cube.

03:27 This we can actually to figure out we can substitute numbers in.

03:31 So three times the mass per neutron, 1 .67 times 10 to the negative 27 kilograms divided by 4 pi, 1 .2 times 10 to the negative 15 meters cubed.

03:59 And we find that the nuclear density of these golf balls, again assuming they're made of nuclear materials, is 2 .31 times 10 to the 17 kilograms meters.

04:16 Per meter cubed.

04:22 So this is the nuclear density of our materials of each golf ball, i should say.

04:27 So if this is the nuclear density, then we could calculate the mass based on just the volume...

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