Units And Measurements
If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle.
Mathematically [LHS] = [RHS]
Applications of Dimensions
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To check the accuracy of physical equations.
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To change a physical quantity from one system of units to another system of units.
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To obtain a relation between different physical quantities.
Significant Figures
In the measured value of a physical quantity, the number of digits about the correctness of which we are sure plus the next doubtful digit, are called the significant figures.
Rules for Finding Significant Figures
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All non-zeros digits are significant figures, e.g., 4362 m has 4 significant figures.
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All zeros occuring between non-zero digits are significant figures, e.g., 1005 has 4 significant figures.
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All zeros to the right of the last non-zero digit are not significant, e.g., 6250 has only 3 significant figures.
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In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00325 has only 3 significant figures.
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All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.4750 has 5 significant figures.
Significant Figures in Algebric Operations
(i) In Addition or Subtraction In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. e.g.,
If l1= 4.326 m and l2 = 1.50 m
Then, l1 + l2 = (4.326 + 1.50) m = 5.826 m
As l2 has measured upto two decimal places, therefore
l1 + l2 = 5.83 m
(ii) In Multiplication or Division In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values. e.g., If length 1= 12.5 m and breadth b = 4.125 m.
Then, area A = l x b = 12.5 x 4.125 = 51.5625 m2
As l has only 3 significant figures, therefore
A= 51.6 m2
Rules of Rounding Off Significant Figures
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If the digit to be dropped is less than 5, then the preceding digit is left unchanged. e.g., 1.54 is rounded off to 1.5.
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If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g., 2.49 is rounded off to 2.5.
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If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g., 3.55 is rounded off to 3.6.
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If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g., 3.750 is rounded off to 3.8 and 4.650 is rounded off to 4.6.
Error
The lack in accuracy in the measurement due to the limit of accuracy of the instrument or due to any other cause is called an error.
1. Absolute Error
The difference between the true value and the measured value of a quantity is called absolute error.
If a1 , a2, a3 ,…, an are the measured values of any quantity a in an experiment performed n times, then the arithmetic mean of these values is called the true value (am) of the quantity.
The absolute error in measured values is given by
Δa1 = am – a1
Δa2 = am – a1
………….
Δam = Δam – Δan
2. Mean Absolute Error
The arithmetic mean of the magnitude of absolute errors in all the measurement is called mean absolute error.
3. Relative Error The ratio of mean absolute error to the true value is called relative
4. Percentage Error The relative error expressed in percentage is called percentage error.
Propagation of Error
(i) Error in Addition or Subtraction Let x = a + b or x = a – b
If the measured values of two quantities a and b are (a ± Δa and (b ± Δb), then maximum absolute error in their addition or subtraction.
Δx = ±(Δa + Δb)
(ii) Error in Multiplication or Division Let x = a x b or x = (a/b).
If the measured values of a and b are (a ± Δa) and (b ± Δb), then maximum relative error