A student determined Young’s Modulus of elasticity using the formula Y = The value of g is taken to be 9.8 m/s², without any significant error, his observation are as follows:

Physical Quantity

Least count of the Equipment used for measurement

Observed value

Mass (M)

1 g

2 kg

Length of bar (L)

1 mm

1 m

Breadth of bar (b)

0.1 mm

4 cm

Thickness of bar (d)

0.01 mm

0.4 cm

Depression

0.01 mm

5 mm

Then the fractional error in the measurement of Y is:

1 View | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

alok kumar singh | Contributor-Level 10

2 months ago

$Y = \frac{M g L^{3}}{4 b d^{3} ?}$

For significant error in Y

$= \frac{? M}{M} + \frac{3 ? L}{L} + \frac{? b}{b} + 3 \frac{? d}{d} + \frac{? ?}{?}$

$= \frac{1 * 1 0^{? 3}}{2} + \frac{3 * 1 0^{? 3}}{1} + \frac{1 0^{? 2}}{4} + 3 * \frac{0.01 * 1 0^{? 1}}{0.4} + \frac{1 0^{? 2}}{5}$

= 0.0155

<math> <mrow> <mi>Y</mi> <mo>=</mo> <mfrac> <mrow> <mi>M</mi> <mi>g</mi> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mi>b</mi> <msup> <mrow> <mi>d</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mi>? </mi> </mrow> </mfrac> </mrow> </math>               <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1816001564"> </o:OLEObject></xml><! [endif]->For significant error in Y <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mi>? </mi> <mi>M</mi> </mrow> <mrow> <mi>M</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <mi>? </mi> <mi>L</mi> </mrow> <mrow> <mi>L</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>? </mi> <mi>b</mi> </mrow> <mrow> <mi>b</mi> </mrow> </mfrac> <mo>+</mo> <mn>3</mn> <mfrac> <mrow> <mi>? </mi> <mi>d</mi> </mrow> <mrow> <mi>d</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mi>? </mi> <mi>? </mi> </mrow> <mrow> <mi>? </mi> </mrow> </mfrac> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1816001565"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>*</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>? </mo> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <mo>*</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>? </mo> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>? </mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> <mo>+</mo> <mn>3</mn> <mo>*</mo> <mfrac> <mrow> <mn>0</mn> <mo>.</mo> <mn>0</mn> <mn>1</mn> <mo>*</mo> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>? </mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mn>0</mn> <mo>.</mo> <mn>4</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <msup> <mrow> <mn>0</mn> </mrow> <mrow> <mo>? </mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>5</mn> </mrow> </mfrac> </mrow> </math>              <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1816001566"> </o:OLEObject></xml><! [endif]->= 0.0155

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Physical Quantity	Least count of the Equipment used for measurement	Observed value
Mass (M)	1 g	2 kg
Length of bar (L)	1 mm	1 m
Breadth of bar (b)	0.1 mm	4 cm
Thickness of bar (d)	0.01 mm	0.4 cm
Depression	0.01 mm	5 mm