Chapter 1, Problem 1OQ

Answer each question yes or no. Must two quantities have the same dimensions (a) if you are adding them? (b) If you are multiplying them? (c) If you are subtracting them? (d) If you are dividing them? (e) If you are equating them?

To determine

The equivalence in dimensionality upon addition

Answer to Problem 1OQ

Yes.

Explanation of Solution

For the four basic operations that is addition, subtraction, multiplication, and division will have different conditions to for performing these operations under the conditions of dimensions.

Considering each operations separately in the case of addition or subtraction, the dimensions of the quantities must have the same units or dimensions whereas for multiplication and division, the dimensions of the quantities need not be of same units or dimensions.

Conclusion

For example, in the case of addition, one cannot add $1\text{metre}$ and $1\mathrm{second}$ as both are completely different quantities. Therefore, the quantities should have the same dimensions to perform the operation of addition.

To determine

The equivalence in dimensionality upon multiplication

Answer to Problem 1OQ

No.

Explanation of Solution

The dimensions of the quantities need not be of same units or dimensions for operations such as multiplication.

Take an example, to obtain the area of a rectangle of dimension $1\text{metre}\times \text{50}\text{centimetre}$, the solution can be obtained,

 $\begin{array}{c}A=\left(1\times \text{100}\right)\text{centimetre}\times \left(50\text{centimetre}\right)\\ =5000\text{sq}\text{centimetre}\end{array}$ .

Conclusion

Option (b) is no; that is there is no need of the quantities to have the same dimensions.

To determine

The equivalence in dimensionality upon subtraction.

Answer to Problem 1OQ

Yes.

Explanation of Solution

The dimensions of the quantities should have same units or dimensions for subtraction.

For example, in the case of subtraction, one cannot subtract $\text{1}\text{kg}$ with $5\text{hr}$ as both are completely different quantities. Therefore, the quantities should have the same dimensions to perform the operation of subtraction.

Conclusion

Option (c) is yes; because the quantities must have the same dimensions.

To determine

The equivalence in dimensionality upon division.

Answer to Problem 1OQ

No.

Explanation of Solution

There is no need of the quantities to have same dimensions to operate the division operation.

Take an example, to obtain the density of a system whose mass is $\text{275}\text{g}$ and volume is $250\text{mL}$ , the solution can be obtained,

 $\begin{array}{c}density=\frac{mass}{volume}\\ =\frac{\text{275}\text{g}}{250\text{mL}}\\ =1.1\text{g/mL}\end{array}$ .

Conclusion

Option (d) is no,  that is there is no need of the quantities to have the same dimensions

To determine

The equivalence in dimensionality upon equating two quantities.

Answer to Problem 1OQ

Yes.

Explanation of Solution

For equating two quantities, the dimensions have to be same because what is in one side should be the same on the other side.

Take an example, to equate the velocity of a system, the distance and the time should the same dimension that of velocity that is,

 $\text{velocity}\left(\text{m}\text{/s}\right)\text{=}\frac{\text{distance}\left(\text{m}\right)}{\text{time}\left(\text{s}\right)}$ .

Conclusion

Option (e) is also yes because the quantities should have same dimension to equate.