정역학을 공부하려는 학생들에게

이 책을 펴내게 된 배경

원저자들은, ‘요즈음 쓰이고 있는 정역학 교재들이 학생들이 갖추고 있는 컴퓨터 수준에 전혀 미치지 못하고 있어, 정역학 문제 해결방식에 한계가 있다고 느꼈기 때문’이라고 이 책의 발간 동기를 저자 서문에서 분명히 밝히고 있습니다. 한 마디로, 정역학 기본 교재는 대부분이 1960년대 이래로 그 내용이 거의 바뀐게 없다는 말이죠.

이 번역서 또한 이 말에 공감하여, 공학 계열 학생들이 역학 분야에 첫 발을 들여놓으려면 반드시 이수해야 하는 필수 선행학습 과목으로서 정역학 과목을 학습하는 데 큰 도움이 되길 바라는 마음으로 발간하게 되었습니다.

역학이란?

역학은 물리학 가운데에서 가장 오래된 학문 분야로서, 기원전 400년 무렵의 그리스 문화에 기원을 두고 있습니다. 당시의 대표적인 인물로는 아리스토텔레스나 아르키메데스를 들 수 있죠. 그 후, 역학 원리의 공식적인 표현은 뉴턴에 의해 이루어졌습니다. 그러니까, 현대의 양자역학 이전의 고전역학은 뉴턴에 와서야 바야흐로 꽃을 피우게 된 것이죠.

역학은 글자 그대로 표현하자면 ‘力學’으로, ‘힘에 관한 학문’입니다. 당연히 힘이 작용하는 대상체에 따라 해석 방법도 달라지고, 그에 따라 합목적적으로 여러 학과목이 정립이 되어 있고요. 그렇게 해서, 역학은 기계설계, 구조해석, 응력해석, 진동, 전자기장 이론, 유체 및 고체 역학 등의 토대를 쌓은 기초응용과학(engineering science)으로 발전하게 되었습니다.

간단히 말하자면, 역학은 운동을 다루고 기체, 액체 및 고체에 작용하는 힘의 효과를 다루는 물리 분야인 것입니다.

정역학이란?

정역학은 이러한 역학에서, ‘정적인 물체와 힘과의 관계’를 다루는 분야로 분화한 것입니다. 여기에서 ‘정적인 물체’라는 표현은 힘이 작용하는 대상체가 ‘정지 상태’이거나 ‘등속 운동’을 하고 있음을 의미합니다.

학생들은 정역학에 이 ‘등속 운동’을 하고 있는 물체가 포함된다는 사실에 매우 의아해하고, 또 흔히 잊어버리기도 합니다!

이 물체들은 가속도가 0인 물체라는 이야기입니다. 또한, 이 물체들은 외부에서 힘을 받고 있는데, 이동하거나 회전 운동을 하지 않는 힘의 평형 상태에 놓여 있는 물체를 의미하기도 한다는 점을 꼭 기억하길 바랍니다.

정역학 문제와 그 해결방식 사이에는?

그 동안 정역학 교재는 저자 서문에도 소개되어 있듯이, 1960년을 기점으로 하여 Beer and Johnson (벡터 판) → J. L. Meriam (SI 판의 효시) 등으로 출판이 되면서, 정역학 문제를 풀 때에도 일반 대수학 → 벡터 대수학 등이 적용되었습니다. 이에 따라 계산자 및 삼각함수표 → 계산기 등의 도구를 사용하여 문제를 주로 ‘손으로’ 풀어 오다가, 1990년대 초부터는 컴퓨터를 사용하여 푸는 문제들이 일부 책에 실리기 시작하면서, 비로소 ‘컴퓨터로’ 푸는 방식이 조금씩 적용되었으므로, 그때까지는 미스 매치가 일어나지는 않았습니다.

그러나 컴퓨터 사용이 늘어가고 그에 따라 학생들의 컴퓨터 수준이 높아져가면서, 상업용 소프트웨어에 적합한 새로운 해법인 행렬 대수학 등을 사용해야 할 필요성 또한 매우 높아지게 된 것이죠.

그런데도, 현실은 그렇지가 않다는 것이 원저자들의 주장하는 바입니다.

맺는 말

사실, 정역학은 놀랍게도 많지 않은 수의 기본 원리들을 기반으로 하고 있고, 그 동안은 벡터 대수학이라는 수학 분야를 도입하고 있었습니다. 이제 저자들은 행렬 대수학이라는 또 다른 수학 분야를 도입하고자 하는데, 이렇게 하게 되면 통상적인 컴퓨터 소프트웨어로 작업을 할 때에 유용하게 될 것입니다.

결론적으로 말하자면, 요즘에는 Mathematica®, MATLAB®, Mathcad®, Maple®, 그리고 다른 패키지들의 학생판이 나와 있고, 또한 많은 컴퓨터 해법을 휴대용 계산기에서도 사용할 수 있습니다. 이러한 소프트웨어는 대부분 행렬 표기법을 사용하고 있으므로 이제는 ‘행렬 대수학’을 기반으로 하는 정역학 문제 해결방식으로 공부를 할 때가 되었다는 의미이기도 합니다.

현실적인 면을 고려한다면, 우리가 정역학 문제를 풀 때, 특히 시험에서는 아직까지 컴퓨터나 스마트폰의 앱을 자유로이 사용할 수 있다고는 할 수 없을 것입니다. 그러므로 반드시 전통적인 해법으로 계산기를 사용하여 ‘손으로 푸는’ 연습을 하는 한편, 시대의 흐름에 맞춰 이 책에서 새로이 도입하고 있는 행렬 표기법을 사용하는 정역학 문제 해법의 이론을 철저히 습득하여, 문제를 해결할 수 있는 수준에 이르도록 연습해야 할 것입니다.

Chapter 01 서론

1.1 역학 ·························································································································· 2

1.2 기본 개념 ·················································································································· 2

1.3 단위 ·························································································································· 5

1.4 수치 계산 ··············································································································· 11

1.5 문제 풀이 전략 ······································································································· 13

1.6 컴퓨터 소프트웨어 ·································································································· 14

Chapter 02 벡터 해석

2.1 서론 ························································································································ 18

2.2 벡터 ························································································································ 20

2.3 힘과 그 특성 ·········································································································· 36

2.4 3차원 직교 좌표와 단위 기본 벡터 ······································································ 42

2.5 벡터 연산의 계산 ··································································································· 51

2.6 벡터의 비직교 방향 성분 ······················································································· 58

2.7 선형 연립 방정식 ··································································································· 63

2.8 두 벡터의 스칼라 곱 ······························································································ 71

2.9 벡터 곱 / 크로스 곱 ······························································································· 78

2.10 직접 벡터 해 ········································································································ 88

Chapter 03 질점 평형

3.1 질점의 자유 물체도 ································································································ 94

3.2 질점의 평형 ············································································································ 99

3.3 스프링 ··················································································································· 113

3.4 부정정 문제 ·········································································································· 117

3.5 특별 주제 ············································································································· 121

Chapter 04 강체: 등가 힘 계

4.1 강체 ······················································································································ 136

4.2 강체의 모델링과 힘의 모멘트 ············································································· 137

4.3 공간에 있는 점에 관한 힘의 모멘트 ··································································· 141

4.4 바리뇽 정리 ·········································································································· 146

4.5 축에 관한 힘 모멘트 ··························································································· 156

4.6 우력 모멘트 ·········································································································· 160

4.7 등가 힘 계 ··········································································································· 164

4.8 특별한 등가 힘 계 ······························································································· 168

4.9 일반적인 등가 힘 계 ··························································································· 175

Chapter 05 분포력: 도형 중심과 무게 중심

5.1 서론 ······················································································································ 188

5.2 질량 중심과 무게 중심 ························································································ 191

5.3 평균 위치: 1차 모멘트 ························································································ 195

5.4 파푸스굴디누스 정리 ························································································ 206

5.5 합성체의 도형 중심 ····························································································· 209

5.6 보에 작용하는 분포 하중 ···················································································· 218

5.7 잠긴 표면에 작용하는 유체 압력으로 인한 힘 ··················································· 221

Chapter 06 강체의 평형

6.1 서론 ······················································································································ 228

6.2 2차원 모델의 지지 ······························································································· 230

6.3 3차원 모델의 지지 ······························································································· 233

6.4 자유 물체도 ·········································································································· 236

6.5 2차원 강체의 평형 ······························································································· 241

6.6 3차원 강체의 평형 ······························································································· 255

6.7 부정정 반력과 부적절한 구속 ············································································· 262

Chapter 07 구조물 해석

7.1 서론 ······················································································································ 276

7.2 평면 트러스 ·········································································································· 278

7.3 단순 트러스 ·········································································································· 280

7.4 조인트 법 ············································································································· 281

7.5 행렬 해법을 사용한 조인트 법 ············································································ 286

7.6 분할법 ··················································································································· 294

7.7 공간 트러스 ·········································································································· 299

7.8 복합 트러스 ·········································································································· 305

7.9 프레임과 기계 ······································································································ 306

Chapter 08 구조 부재의 내력

8.1 서론 ······················································································································ 318

8.2 부재의 내력 ·········································································································· 318

8.3 보에서의 하중 및 지지 유형 ··············································································· 325

8.4 보의 전단 모멘트와 굽힘 모멘트 ········································································ 326

8.5 보 식에 사용되는 불연속 함수 ············································································ 337

8.6 케이블 ··················································································································· 345

Chapter 09 마찰

9.1 서론 ······················································································································ 358

9.2 쿨롱 마찰 ············································································································· 361

9.3 쐐기 ······················································································································ 376

9.4 사각 단면 나사 ···································································································· 382

9.5 벨트 마찰 ············································································································· 388

9.6 베어링 ··················································································································· 391

9.7 쓰러스트 베어링, 칼라 및 클러치 ······································································· 394

9.8 구름 저항 ············································································································· 395

Chapter 10 관성 모멘트

10.1 서론 ···················································································································· 402

10.2 2차 면적 모멘트 ································································································ 402

10.3 극 관성 모멘트 ·································································································· 404

10.4 특정한 면적의 도형 중심 축에 관한 2차 면적 모멘트 ··································· 405

10.5 2차 면적 모멘트의 평행 축 정리 ····································································· 407

10.6 면적의 회전 반경 ······························································································· 409

10.7 합성 면적의 2차 면적 모멘트 ··········································································· 411

10.8 주 2차 면적 모멘트 ··························································································· 415

10.9 모어 원을 사용하여 주 2차 면적 모멘트 구하기 ············································ 423

10.10 아이겐밸류 문제 ······························································································· 427

10.11 질량 관성 모멘트 ····························································································· 430

Chapter 11 가상 일

11.1 서론 ···················································································································· 442

11.2 가상 일 ··············································································································· 443

11.3 연결된 강체 계의 가상 일 원리 ······································································· 444

11.4 힘과 모멘트의 유한 일 ······················································································ 452

11.5 보존력과 퍼텐셜 에너지 ···················································································· 456

11.6 퍼텐셜 에너지와 평형 ························································································ 460

11.7 평형의 안정성 ···································································································· 461

Appendix 01 선형 연립 방정식의 해법 ····························································································· 471

• GaussJordan 소거법 _ 471

• 행렬의 역행렬 _ 474

• 벡터 방정식의 해법 _ 476

Appendix 02 벡터 대수학 ··················································································································· 480

Appendix 03 선형 대수학 ··················································································································· 483

Appendix 04 정역학 용어집 ··············································································································· 486

• 연습문제 해답 ············································································································ 499

• 찾아보기 ····················································································································· 515