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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?
(a)
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
(b)
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
Conclusion
Option (b) is no; that is there is no need of the quantities to have the same dimensions.
(c)
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
Conclusion
Option (c) is yes; because the quantities must have the same dimensions.
(d)
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
Conclusion
Option (d) is no, that is there is no need of the quantities to have the same dimensions
(e)
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,
Conclusion
Option (e) is also yes because the quantities should have same dimension to equate.
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Chapter 1 Solutions
Principles of Physics
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