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Chapter 13: Q1P (page 378)

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

Expert verified

The magnitude of the gravitational force between the parts is maximized at12

Step by step solution

01

The given data

A mass Mis split into two parts: m andM-m

02

Understanding the concept of Newton’s law of gravitation

The problem is based on the concept of Newton鈥檚 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=GMmr2 (i)

03

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=G脳m脳M-mr2=Gr2脳mM-m2

Differentiating with respect m:

dfdm=Gr2脳M-2m

For the force to be maximum, we should have,

dfdm=0

Gr2脳M-2m=0

Hence,

M-2m=0M=2m

Therefore, the gravitational force is maximum at12

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