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Chapter 1: Problem 7

Verify the dimensions, in both the \(F L T\) system and the \(M L T\) system, of the following quantities which appear in Table 1.1 (a) acceleration. (b) stress, (c) moment of a force. (d) volume, and (e) Work.

Short Answer

Expert verified
The dimensions of the physical quantities in the \(F L T\) system are as follows: acceleration (\(L T^{-2}\)), stress (\(F L^{-2}\)), moment of a force (\(F L\)), volume (\(L^3\)), and work (\(F L\)). And in the \(M L T\) system, they are: acceleration (\(M^0 L T^{-2}\)), stress (\(M L^{-1} T^{-2}\)), moment of a force (\(M L^2 T^{-2}\)), volume (\(L^3\)), and work (\(M L^2 T^{-2}\)).

Step by step solution

01

Dimension Analysis of Acceleration

In both systems, acceleration is derived from length and time. It's the rate of change of velocity with time. Hence, its dimension is \(L T^{-2}\). In the \(F L T\) system, it would still remain \(L T^{-2}\) without any F, and in the \(M L T\) system, it would be \(M^0 L T^{-2}\) as there is no mass component in acceleration.
02

Dimension Analysis of Stress

Stress is defined as force per unit area. Hence, in the \(F L T\) system, its dimension would be \(F L^{-2} T^{0}\) and in the \(M L T\) system, since force is given by mass times acceleration, the dimension of stress would be \(M L^{-1} T^{-2}\) as the force is 'M times LT^{-2}' divided by \(L^2\).
03

Dimension Analysis of Moment of a Force

Moment of a force is force times distance. Therefore, its dimension in the \(F L T\) system would be \(F L T^{0}\). In the \(M L T\) system, since force is given by mass times acceleration, its dimension would be \(M L^2 T^{-2}\).
04

Dimension Analysis of Volume

Volume is derived from length only. It's the amount of 3D space occupied by an object. Thus, its dimension in both systems would be \(L^3\).
05

Dimension Analysis of Work

Work is defined as force times distance. Hence, in the \(F L T\) system, its dimension would be \(F L T^{0}\). In the \(M L T\) system, since force is given by mass times acceleration, its dimension would be \(M L^2 T^{-2}\).

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