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Question

The following observations were taken for determining surface tension T of water by the capillary method: diameter of the capillary, D=1.25x10-2m rise of water, h=1.45x10-2m. Using g=9.80m/s2and the simplified relation T=rhg2x103N/m, the possible error in surface tension is closest to:


  1. 0.15%

  2. 1.5%

  3. 2.4%

  4. 10%

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Solution

The correct option is B

1.5%


Step 1: Given data:

Diameter of capillary, D=1.25×102m
Rise of water, h=1.45×102m
The acceleration due to gravity, g=9.8m/s2

Given, the formula for surface tension is,

T=rhg2×103N/m

Step 2: Formula for surface tension:

The formula for surface tension can be given as:

T=rhg2×103N/m
Also,r=D2
where r stands for the radius of the capillary tube and D stands for the diameter of the capillary tube.

Step 3: Calculation of surface tension:
Substituting this value of r in the given equation we can calculate surface tension as:
T=Dhg4×103N/m

Step 4: Expression of the possible error in surface tension:
Taking logarithm on both sides of the above equation, we get
logT=logD+logh+logglog4+log103
Here, log103,log4, and logg have constant value; hence, there will be no error in them, so we can neglect them in the expression.
Hence, the expression will become,
logT=logD+logh …..(1)
Thus, here the error in T can be only due to the error in Dand h

Step 5: Expressing the final expression:
By differentiating, the equation (1) can be written as,
dlogT=dlogD+dlogh

(dlogxdx=1xdlogx=dxx)
ΔTT=ΔDD+Δhh
The maximum error in D can be written as,

As the number of significant figures in the given value of D is 3 therefore,
ΔD=0.01×10-2m
The maximum error in h can be written as,

Since the number of significant figures in the given value of h is 3 therefore,
Δh=0.01×10-2m
Step 6: Calculation of the error in surface tension:
Substituting the values in the given expression, we can calculate the surface tension as:
ΔTT=0.01×10-21.25×10-2+0.01×10-21.45×10-2=0.015

Step 7: Calculation of the percentage error in surface tension:
The percentage error of surface tension can be written as,
ΔTT×100=0.015×100=1.5%
Thus, the possible error in surface tension is the closest to 1.5%.

Hence, option B is correct.


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