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Chapter 4: Problem 19

A bowling ball (mass \(=7.2 \mathrm{~kg},\) radius \(=0.11 \mathrm{~m}\) ) and a billiard ball (mass \(=0.38 \mathrm{~kg}\), radius \(=0.028 \mathrm{~m}\) ) may each be treated as uniform spheres. What is the magnitude of the maximum gravitational force that each can exert on the other?

Short Answer

Expert verified
The maximum gravitational force between the two spheres is approximately \(9.59 \times 10^{-9}\) Newtons.

Step by step solution

01

Understand the Problem

We need to find the gravitational force between a bowling ball and a billiard ball. This gravitational force can be calculated using Newton's law of universal gravitation.
02

Identify and Write Down Known Quantities

\[m_1 = 7.2 \text{ kg (mass of the bowling ball)}, \ m_2 = 0.38 \text{ kg (mass of the billiard ball)}, \ r_1 = 0.11 \text{ m (radius of the bowling ball)}, \ r_2 = 0.028 \text{ m (radius of the billiard ball)}, \ G = 6.674 \times 10^{-11} \text{ N m}^2/\text{kg}^2 \text{ (Gravitational constant)}.\]
03

Calculate the Distance Between the Two Objects

The maximum gravitational force occurs when the objects are just touching each other. The distance between the centers of the two balls when they touch is the sum of their radii:\[d = r_1 + r_2 = 0.11 \text{ m} + 0.028 \text{ m} = 0.138 \text{ m}.\]
04

Use Newton's Law of Universal Gravitation

Newton's law of universal gravitation gives the force as:\[F = \frac{G \cdot m_1 \cdot m_2}{d^2}.\]Substitute the known values:\[F = \frac{6.674 \times 10^{-11} \cdot 7.2 \cdot 0.38}{(0.138)^2}.\]
05

Calculate the Gravitational Force

Perform the calculation:\[F = \frac{6.674 \times 10^{-11} \cdot 2.736}{0.019044} \approx 9.59 \times 10^{-9} \text{ N}.\]Thus, the magnitude of the maximum gravitational force between the two spheres is approximately \(9.59 \times 10^{-9}\) Newtons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gravitational Force
Gravitational force is the attractive force that acts between any two masses, no matter how small or large. It's a fundamental force in nature and is described by Newton's Law of Universal Gravitation. The formula for this force is:\[F = \frac{G \cdot m_1 \cdot m_2}{d^2},\]where:
  • \(F\) is the gravitational force,
  • \(G\) is the gravitational constant,
  • \(m_1\) and \(m_2\) are the masses of the objects, and
  • \(d\) is the distance between the centers of the two masses.
This force is always attractive and acts along the line joining the centers of the two objects. Gravitational force plays an essential role not only in holding planets, stars, and galaxies together but also in everyday experiences like the fall of an apple or the motion of the moon around the Earth.
When dealing with celestial bodies or any object with size, we often approximate them as point masses located at their centers, easing our calculations.
Uniform Spheres
The term 'uniform spheres' refers to objects whose mass is evenly distributed throughout their volume, much like a perfect marble. For objects that are spherical and uniform, mass can be considered concentrated at the center, simplifying gravitational calculations.
For example, in our problem, both the bowling ball and billiard ball are treated as uniform spheres. This assumption allows us to apply the law of universal gravitation directly using their total masses and the distance between their centers.
The gravitational force is calculated as if all the mass were concentrated at the center of the sphere. This simplification is valid for spherical objects that have a consistent density throughout.
Gravitational Constant
The gravitational constant, denoted by \(G\), is a key factor in Newton's Law of Universal Gravitation. Its value is a small but crucial component in calculating gravitational forces.The constant \(G\) is equal to \(6.674 \times 10^{-11} \) N m\(^2\) kg\(^{-2}\), and it provides the proportionality needed for the gravitational force formula:\[F = \frac{G \cdot m_1 \cdot m_2}{d^2}.\]Here, \(G\) allows us to calculate the force in newtons (N) using masses in kilograms and distance in meters. Without \(G\), we wouldn't be able to quantify the gravitational force between objects of a given mass.
It universally applies to every pair of masses in the universe, making it one of the fundamental constants of physics and reinforcing the universality of gravitational interactions.

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