Chapter 8: Problem 13

Two identical solid spheres each of radius 'r' are in contact with each other. If the gravitati between them is $\mathrm{F}$, then which of the following relations is correct? (1) $\mathrm{F} \propto \frac{1}{\mathrm{r}}$ (2) $\mathrm{F} \propto \frac{1}{\mathrm{r}^{2}}$ (3) $\mathrm{F} \propto \frac{1}{\mathrm{r}^{3}}$ (4) $\mathrm{F} \propto \frac{1}{\mathrm{r}^{4}}$

Short Answer

Expert verified

Short Answer: The gravitational force (F) is proportional to the radius (r) raised to the power of 4, which is represented by the formula $$F \propto \frac{1}{r^4}$$. Therefore, option (4) is the correct relationship between the gravitational force and the radius of the identical solid spheres.

Step by step solution

Write down Newton's law of universal gravitation

The formula for the gravitational force between two masses is given by Newton's law of universal gravitation: $$F = G \frac{m_1 m_2}{d^2}$$ where F is the gravitational force, G is the gravitational constant, $m_1$ and $m_2$ are the masses of the two spheres, and $d$ is the distance between their centers.

Determine the distance between the centers of the spheres

Since the two spheres are in contact with each other, the distance between their centers is equal to the sum of their radii. Because they are identical, we can say: $$d = r_\mathrm{1} + r_\mathrm{2} = 2r$$

Write an expression for the masses of the spheres

The two spheres are identical, so their masses are equal. We can write the mass of each sphere as: $$m = \rho V$$ where $\rho$ is the density and $V$ is the volume. For a sphere, the volume is: $$V = \frac{4}{3}\pi r^3$$ So the mass of each sphere becomes: $$m = \rho\frac{4}{3}\pi r^3$$

Substitute expressions in the gravitation formula

Now that we have expressions for the masses of the spheres and the distance between their centers, we can substitute them into the formula for gravitational force: $$F = G \frac{(\rho\frac{4}{3}\pi r^3)^2}{(2r)^2}$$

Simplify the expression

Now we will simplify the expression for the gravitational force: $$F = G \frac{(\rho^2 (\frac{4}{3}\pi)^2 r^6}{4r^2}$$ $$F = G \frac{\rho^2 (\frac{4}{3}\pi)^2 r^4}{4}$$

Determine the proportionality

Looking at our simplified expression, we can see that the gravitational force F is proportional to the radius r raised to the power of 4: $$F \propto \frac{1}{r^4}$$ Thus, the correct relation is given by option (4).

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Write down Newton's law of universal gravitation

Determine the distance between the centers of the spheres

Write an expression for the masses of the spheres

Substitute expressions in the gravitation formula

Simplify the expression

Determine the proportionality

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Translational Dynamics

Mechanics and Materials

Electromagnetism

Electricity

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.