Chapter 1 Mastering Physics - Dimensions of Physical...

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Dimensions of Physical Quantities Learning Goal: To introduce the idea of physical dimensions and to learn how to find them. Physical quantities are generally not purely numerical: They have a particular dimension or combination of dimensions associated with them. Thus, your height is not 74, but rather 74 inches, often expressed as 6 feet 2 inches. Although feet and inches are different units they have the same dimension -- length . Part A In classical mechanics there are three base dimensions. Length is one of them. What are the other two? Hint A.1 MKS system Hint not displayed ANSWER: acceleration and mass acceleration and time acceleration and charge mass and time mass and charge time and charge Correct There are three dimensions used in mechanics: length ( ), mass ( ), and time ( ). A combination of these three dimensions suffices to express any physical quantity, because when a new physical quantity is needed (e.g., velocity), it always obeys an equation that permits it to be expressed in terms of the units used for these three dimensions. One then derives a unit to measure the new physical quantity from that equation, and often its unit is given a special name. Such new dimensions are called derived dimensions and the units they are measured in are called derived units. For example, area has derived dimensions . (Note that "dimensions of variable " is symbolized as .) You can find these dimensions by looking at the formula for the area of a square , where is the length of a side of the square. Clearly . Plugging this into the equation gives .
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Dimensions of Physical Quantities Part B Part C Find the dimensions of speed. Hint C.1 Equation for speed Hint not displayed Hint C.2 Familiar units for speed Hint not displayed Express your answer as powers of length ( ), mass ( ), and time ( ). ANSWER: = Correct The dimensions of a quantity are not changed by addition or subtraction of another quantity with the same dimensions. This means that , which comes from subtracting two speeds, has the same dimensions as speed. It does not make physical sense to add or subtract two quanitites that have different dimensions, like length plus time. You can add quantities that have different units, like miles per hour and kilometers per hour, as long as you convert both quantities to the same set of units before you actually compute the sum. You can use this rule to check your answers to any physics problem you work. If the answer involves the sum or difference of two quantities with different dimensions, then it must be incorrect. This rule also ensures that the dimensions of any physical quantity will never involve sums or differences of the base dimensions. (As in the preceeding example, is not a valid dimension for a physical quantitiy.) A valid dimension will only involve the product or ratio of powers of the base dimensions (e.g. ).
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Part D Find the dimensions of acceleration.
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This note was uploaded on 10/16/2010 for the course MASTERING physics taught by Professor Michael during the Spring '10 term at Virtual University of Tunisia.

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Chapter 1 Mastering Physics - Dimensions of Physical...

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