Chapter 13: Q1P (page 378)

A massMis split into two parts, $m$ and role="math" localid="1657189921712" $M - m$ , which are0 then separated by a certain distance. What ratio $m - M$ maximizes the magnitude of the gravitational force between the parts?

Short Answer

Expert verified

The magnitude of the gravitational force between the parts is maximized at $\frac{1}{2}$

Step by step solution

The given data

A mass Mis split into two parts: m and $M - m$

Understanding the concept of Newton’s law of gravitation

The problem is based on the concept of Newton’s law of Gravitation. If the magnitude of the force is to be maximum, its derivative to massmwill be zero. Hence this law is used to find the ratiom / M.

Formula:

Gravitational Force, $F = \frac{G M m}{r^{2}}$ (i)

Calculating the value of m / M that maximizes the force

Using equation (i), the gravitational force between the two parts m and $M - m$ of the mass M can be given as:

$F = \frac{G \times m \times (M - m)}{r^{2}} = \frac{G}{r^{2}} \times (m M - m^{2})$

Differentiating with respect m:

$\frac{d f}{d m} = \frac{G}{r^{2}} \times (M - 2 m)$

For the force to be maximum, we should have,

$\frac{d f}{d m} = 0$

$\frac{G}{r^{2}} \times (M - 2 m) = 0$

Hence,

$M - 2 m = 0 M = 2 m$

Therefore, the gravitational force is maximum at $\frac{1}{2}$

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A massMis split into two parts, $m$ and role="math" localid="1657189921712" $M - m$ , which are0 then separated by a certain distance. What ratio $m - M$ maximizes the magnitude of the gravitational force between the parts?

Short Answer

Step by step solution

The given data

Understanding the concept of Newton’s law of gravitation

Calculating the value of m / M that maximizes the force

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A massMis split into two parts,mand role="math" localid="1657189921712" M-m, which are0 then separated by a certain distance. What ratiom-Mmaximizes the magnitude of the gravitational force between the parts?

Short Answer

Step by step solution

The given data

Understanding the concept of Newton’s law of gravitation

Calculating the value of m / M that maximizes the force

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Measurements

Torque and Rotational Motion

Classical Mechanics

Energy Physics

Engineering Physics

Solid State Physics

Study anywhere. Anytime. Across all devices.

A massMis split into two parts, $m$ and role="math" localid="1657189921712" $M - m$ , which are0 then separated by a certain distance. What ratio $m - M$ maximizes the magnitude of the gravitational force between the parts?