Tag: measurement of length

Questions Related to measurement of length

If $x=10.0 \pm 0.1$ and $y=10.0 \pm 0.1$, then $2x-2y$ is equal to

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. $(0.0 \pm 0.1)$

  2. $Zero$

  3. $(0.0 \pm 0.4)$

  4. $(20 \pm 0.2)$

Show answer
Correct Option: B
Explanation:

Apply formula

$(A\pm \Delta A)-(B\pm \Delta B)=(A-B)\pm (\Delta A-\Delta B)$

Similarly

  $ 2x-2y=2\left( 10.0\pm 0.1 \right)-2\left( 10\pm 0.1 \right) $

 $ =\left( 2\times 10-2\times 10 \right)\pm (2\times 0.1-2\times 0.1) $

 $ =0 $

Hence, $2x-2y=ZERO$ 

A physical quantity $S$ is given by $S = \dfrac {a^{2}b^{3}}{c\sqrt {d}}$.
If errors of measurements in $a, b, c, d$ are $4\%, 2\%, 3\%, 1\%$ respectively, find the percentage error in the value of $S$.

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. $7.5\%$.

  2. $17.5\%$.

  3. $27.5\%$.

  4. $10.5\%$.

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Correct Option: B

In an experiment four quantities a, b, c and d are measured with percentage error $1$ %, $2$%, $3$% and $4$% respectively. Quantity P is calculated as follows:
$P=\frac{ab^2}{\sqrt{cd^3}}$
Percentage error is P is

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. $4$%

  2. $7$%

  3. $9$%

  4. $10$%

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Correct Option: D
Explanation:

Quantity $P$ is calculated as follow $P = \frac{{a{b^2}}}{{\sqrt {c{d^3}} }}$

Formula of % error should be,
% error in $P=3 \times$%error in $a+2 \times $% error in $b+$%error in $c+$% error in d
Given,
% error in $a=1$%
% error in $b=2$%
% error in $c=3$%
% error in $d=4$%
Hence,
% error in $P=3 \times 1+2 \times 2+3+4$
$=3$%-$4$%+$3$%+$4$%
$=10$%
$\therefore $ Option $D$ is correct answer.

In a relation $ S=\dfrac{b}{b-c}$, where b, c,s are physical quantities, b is $(5.0 \pm 0.1)$ N and c is $(2.0 \pm 0.2)N$ then the percentage error in S is

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. $12\%$

  2. $2\%$

  3. $24\%$

  4. $6\%$

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Correct Option: A

The radius of a sphere is measured as  $ (10 \pm 0.02) $ cm. The error in the measurement of its volume is:

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. 251 cc

  2. 25.1cc

  3. 2.51 cc

  4. 251.2cc

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Correct Option: B
Explanation:

Given that,

$r = 10$

$\Delta r=0.02$


 We know that,

The volume of sphere is

  $ V=\dfrac{4}{3}\pi {{r}^{3}} $

 $ V=\dfrac{4}{3}\times 3.14\times {{\left( 10 \right)}^{3}} $

 $ V=4186.7\,cc $

Now, taking log

$\log V=\log \dfrac{4}{3}+3\log r$

Differentiating on both sides

$\dfrac{\Delta V}{V}=0+3\dfrac{\Delta r}{r}$

Now, the error is

  $ \dfrac{\Delta V}{V}=3\times \dfrac{\Delta r}{r} $

 $ \Delta V=V\times 3\times \dfrac{\Delta r}{r} $

 $ \Delta V=4186.7\times 3\times \dfrac{0.02}{10} $

 $ \Delta V=25.1\,cc $

Hence, the error in the measurement of its volume is $25.1$ cc

The radius and height of a cone are measured as $6cms$ each by scale in which there is an error of $0.01cm$ in each cm. then the approximate error in its volume is.

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. $.14$

  2. $.12$

  3. $.36$

  4. $0.16$

Show answer
Correct Option: A
Explanation:
Formula,

$V=\pi r^2 \dfrac{h}{3}$

$=\pi\times 6^2 \dfrac{6}{3}=226.19$

$\dfrac{\Delta V}{V}=\dfrac{\pi}{3}6 \times 0.01 \times 0.01=0.000628$

The change in volume is,

$V=0.000628\times 226.19=0.14$%

In a resonant column method, resonance occurs at two successive levels of $l _1=30.7 cm$ and $l _2=63.2 cm$ using a tuning fork of $f=512 Hz$. What is the maximum error in measuring speed of sound using the relations $v=f\lambda$ and $\lambda =2(l _2-l _1)$?

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. 256 cm/sec

  2. 92 cm/sec

  3. 128 cm/sec

  4. 102.4 cm/sec

Show answer
Correct Option: D
Explanation:

The maximum error is given by :


$c=f\lambda = 512 \times 2(l _2-l _1)=512 \times 2 \times (63.2-30.7) = 332.8 \ m/s$

$\dfrac{\Delta c}{c} = \dfrac{\Delta {f}}{f}  + \dfrac{ {\Delta \lambda}}{\lambda}$

Least count of scale $=0.1 cm$ 

total error in measurement of $\lambda$ 

$\Rightarrow 2\times L.C. =0.2 cm$

Total error in measurement of $f$  $\Rightarrow 0$

$\dfrac{\Delta c}{332.8} = \dfrac{0}{f}  + \dfrac{ {0 .2}}{65}$

$\Delta c =  \dfrac{ {0.2}}{65}\times 332.8 =1.024 \ m/s =102.4 \ cm/s$

Given $x=\dfrac {ab^2}{c^3}$, if the percentage errors in a, b and c are $\pm$ 1%, $\pm$ 3% and $\pm$ 2% respectively, the percentage error in $x$ can be:

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. $\pm$ 13%

  2. $\pm$ 7%

  3. $\pm$ 18%

  4. $\pm$ 19%

Show answer
Correct Option: A
Explanation:

The given quantity is   $x = \dfrac{ab^2}{c^3}$
Maximum percentage error in $x$ is   $\dfrac{\Delta x}{x}\times 100 = [\dfrac{\Delta a}{a}+2\dfrac{\Delta b}{b}+3\dfrac{\Delta c}{c}]\times 100$
$\dfrac{\Delta x}{x}\times 100 = [1+2\times 3+3\times 2] = \pm 13$ %

The pressure on  square plate is measured by measuring the force on the plate and length of sides of plate. If the maximum error in the measurement of force and length are respectively $4$% and $2$%, the maximum error in measurement of pressure is..........

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. 1%

  2. 2%

  3. 6%

  4. 8%

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Correct Option: D

A physical quantity P is repleted to four observed a,b,c,d as follows: $P=\dfrac{a^3b^2}{\left(\sqrt c.d\right)}$ The percentage errors in the measurement of a,b,c and d are $1\%3\%,4\%$ and $2\%$ respectively. The percentage error in the quantity P is

physics measurements and experimentation vernier calliper and screw gauge least count of vernier calliper and screw gauge measurement of length

  1. $7\%$

  2. $9\%$

  3. $11\%$

  4. $13\%$

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Correct Option: D
Explanation:
$P=\dfrac { { a }^{ 3 }{ b }^{ 2 } }{ \sqrt { cd }  } $

$\dfrac { \Delta P }{ P } =\dfrac { 3\Delta a }{ a } +\dfrac { 2\Delta b }{ b } +\dfrac { 1 }{ 2 } \dfrac { \Delta c }{ c } +\dfrac { \Delta d }{ d } $

$\left( \dfrac { \Delta P }{ P } \times 100 \right) $% $=\left( 3\times \dfrac { \Delta a }{ a } \times 100+2\times \dfrac { \Delta b }{ b } \times 100+\dfrac { 1 }{ 2 } \times \dfrac { \Delta c }{ c } \times 100+\dfrac { \Delta d }{ d } \times 100 \right) $%
                              $=3\times 1+2\times 3+\dfrac { 1 }{ 2 } \times 4+2$
                              $=3+6+2+2=13$%
Percentage error $=13$%