Chapter 15, Problem 10P

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) A lamina has constant density *ρ* and takes the shape of a disk with center the origin and radius *R.* Use Newton’s Law of Gravitation (see Section 13.4) to show that the magnitude of the force of attraction that the lamina exerts on a body with mass *m* located at the point (0, 0, *d*) on the positive *z*-axis is

[*Hint:* Divide the disk as in Figure 15.3.4 and first compute the vertical component of the force exerted by the polar subrectangle *R _{ij}*.]

(b) Show that the magnitude of the force of attraction of a lamina with density *ρ* that occupies an entire plane on an object with mass *m* located at a distance *d* from the plane is
*d.*

(a)

To determine

**To show:** The magnitude of the force of attraction is

Explanation

**Given:**

The lamina has the constant density and takes the shape of the disk with center at origin and radius *R.*

**Formula used:**

The Newton’s law of gravitation is given by,

If

**Calculation:**

Since the given region is the disk centered at origin and radius *R,* use of polar coordinates is a wise choice. So separate the given region into small sub regions by split the *r* into *i* denotes the value of *r* *j* denotes the position of

The distance of *m* is denoted by

Substitute the value of *M* and **u** and rewrite the above mentioned formula as follows.

The total force of the lamina present only in the direction of *z-*axis as mentioned in the problem. Thus, consider only the components in the vertical direction. So, the force value becomes, *m.* Hence,

Thus, the total force of attraction can be mentioned by the Riemann sum for double integrals as given below

(b)

To determine

**To show:** The magnitude of the force attraction of the given lamina that occupies an entire plane is given by,

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