Consider a long cylindrical solid rod – total length L – mass density p; – radius R1 = R2 = R; - suspended from the top and hung freely downward along the (-1)Ź direction;B - subjected to gravitational pull from the ground in the (-1)Ź direction. R2 Z = h R, Z = 0 a) Calculate the longitudinal stress o within the rod as a function of h; b) Repeat the calculation if R1 = 0 and R2 = R (i.e., when the cylindrical rod turns into a cone); c) Based on your answers to a) and b), provide some reasoning as to why a space elevator would not work [hint: think about the rod as the load bearing element of the space elevator].

Consider a long cylindrical solid rod – total length L – mass density p; – radius R1 = R2 = R; - suspended from the top and hung freely downward along the (-1)Ź direction;B - subjected to gravitational pull from the ground in the (-1)Ź direction. R2 Z = h R, Z = 0 a) Calculate the longitudinal stress o within the rod as a function of h; b) Repeat the calculation if R1 = 0 and R2 = R (i.e., when the cylindrical rod turns into a cone); c) Based on your answers to a) and b), provide some reasoning as to why a space elevator would not work [hint: think about the rod as the load bearing element of the space elevator].

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Concept explainers

Axial Load

When a bar of the uniform cross-section is acted by a load, such that the load acts along the longitudinal axis of the bar, such kinds of loadings are known as axial loadings. In general terms, a load is an external force acting on a component. An axial load acts perpendicular to the geometric cross-section of the component. Based on the direction of the application of the load, an axial load can be tensile or compressive. The analysis of bodies under the action of axial loads is generally studied under the branch of mechanics, known as statics.

Question

**Problem Description:**

Consider a long cylindrical solid rod with the following properties:
- Total length: \( L \)
- Mass density: \( \rho \)
- Radius: \( R_1 = R_2 = R \)
- Suspended from the top and freely hung downward along the \((-1)\hat{Z}\) direction
- Subjected to gravitational pull from the ground in the \((-1)\hat{Z}\) direction

**Tasks:**

a) Calculate the longitudinal stress \( \sigma \) within the rod as a function of height \( h \).

b) Repeat the calculation if \( R_1 = 0 \) and \( R_2 = R \) (i.e., when the cylindrical rod becomes a cone).

c) Based on your answers to a) and b), provide some reasoning as to why a space elevator would not work. [Hint: Consider the rod as the load-bearing element of the space elevator.]

**Diagram Explanation:**

The diagram on the right illustrates a cylindrical solid rod suspended from a surface:

- The rod extends from \( Z = 0 \) to \( Z = L \).
- The radius of the rod is \( R \) at both the top and bottom in the cylindrical case.
- In the conical case (part b), \( R_1 \) becomes 0, making the bottom radius taper to a point.

The diagram depicts the rod's orientation along the \( Z \)-axis, with the suspension at the top and the gravitational force acting downward.

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