Concept explainers
Review. A student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. He cuts out various shapes in various sizes, calculates their areas, measures their masses, and prepares the graph of Figure PI.51. (a) Consider the fourth experimental point from the top. How far is it from the best-lit straight line? Express your answer as a difference in vertical-axis coordinate, (b) Express your answer as a percentage, (c) Calculate the slope of the line, (d) State what the graph demonstrates, referring to the shape of the graph and the results of parts (b) and (c). (e) Describe whether this result should be expected theoretically, (f) Describe the physical meaning of the slope.
(a)
The distance of the fourth experimental point from the top from the best –fit straight line.
Answer to Problem 1.51P
The distance of the fourth experimental point from the top from the best fit straight line is
Explanation of Solution
Given data: The student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. A graph mass versus area is plotted for different sizes of the paper.
Consider the following figure.
Figure (1)
Figure indicates the graph plotted area of pieces versus mass of the pieces of the paper.
The fourth experimental point from the top is a circle. It lies slightly above the best fit line.
From figure (1), the vertical coordinate for the forth experiment is
So the difference in the vertical axis coordinate is,
Here,
Substitute
Conclusion:
Therefore the distance of the fourth experimental point from the top from the best fit straight line is
(b)
The answer in the form of percentage.
Answer to Problem 1.51P
The answer in the form of percentage is
Explanation of Solution
Given data: The student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. A graph mass versus area is plotted for different sizes of the paper.
The expression for the percentage is,
Substitute
Conclusion:
Therefore the answer in the form of percentage is
(c)
The slope of the line.
Answer to Problem 1.51P
The slope of the line is
Explanation of Solution
Given data: The student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. A graph mass versus area is plotted for different sizes of the paper.
The expression for the slope is,
Here,
Substitute
Conclusion:
Therefore the slope of the line is
(d)
The demonstration from the graph referring to the shape of the graph and the results of part (b) and (c).
Answer to Problem 1.51P
The graph demonstrates that the mass of the cutout is proportional to its area for the shape cuts from this copy paper and the proportionality constant is
Explanation of Solution
Given data: The student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. A graph mass versus area is plotted for different sizes of the paper.
The graph given figure (1) demonstrates that the mass of the cutout in each shape is proportional to its area for the shape cuts from this copy paper and the proportionality constant is
The value of the slope from part (c) of the question and from the part (b) question the percentage uncertainty is
Conclusion:
Therefore the graph demonstrates that the mass of the cutout is proportional to its area for the shape cuts from this copy paper and the proportionality constant is
(e)
Whether this result should be expected theoretically or not.
Answer to Problem 1.51P
This result is to be expected theoretically if the paper has thickness and density that are uniform within the experimental uncertainty.
Explanation of Solution
Given data: The student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. A graph mass versus area is plotted for different sizes of the paper.
The graph given figure (1) demonstrates that the mass of the cutout in each shape is proportional to its area for the shape cuts from this copy paper and the proportionality constant is
The value of the slope from part (c) of the question is
Thus this result is expected theoretically when the object is having same aerial mass density and the paper has thickness and density that are uniform within the experimental uncertainty.
Conclusion:
Therefore this result is to be expected theoretically if the paper has thickness and density that are uniform within the experimental uncertainty.
(f)
The physical meaning of the slope.
Answer to Problem 1.51P
The physical meaning of the slope is the aerial density of the paper that represents the mass per unit area.
Explanation of Solution
Given data: The student is supplied with a stack of copy paper, ruler, compass, scissors, and a sensitive balance. A graph mass versus area is plotted for different sizes of the paper.
The graph given figure (1) demonstrates that the mass of the cutout in each shape is proportional to its area for the shape cuts from this copy paper and the proportionality constant is
The value of the slope from part (c) of the question is
The physical meaning of the slope is the aerial density of the paper that represent the mass per unit area.
Conclusion:
Therefore the physical meaning of the slope is the aerial density of the paper that represents the mass per unit area.
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