Given Measurement Uncertainties
The problem provides the following measurements:
- Density ($\rho$): $(20 \pm 4)\text{ gm/cm}^3$. This means the nominal density is $\rho = 20\text{ gm/cm}^3$ and the absolute error in density is $\Delta\rho = 4\text{ gm/cm}^3$.
- Volume ($V$): $(10 \pm 1)\text{ cm}^3$. This means the nominal volume is $V = 10\text{ cm}^3$ and the absolute error in volume is $\Delta V = 1\text{ cm}^3$.
Calculating Nominal Mass
Mass ($M$) is calculated using the formula $M = \rho \times V$. We first find the nominal mass using the nominal values of density and volume:
$ M = \rho \times V = (20\text{ gm/cm}^3) \times (10\text{ cm}^3) = 200\text{ gm} $
Error Propagation Rule for Multiplication
When two quantities are multiplied (like density and volume to find mass), their relative errors add up. The formula for the relative error in mass ($\Delta M / M$) is:
$ \frac{\Delta M}{M} = \frac{\Delta \rho}{\rho} + \frac{\Delta V}{V} $
Calculating Relative Errors
Calculate the relative error for density and volume:
- Relative error in density: $ \frac{\Delta \rho}{\rho} = \frac{4\text{ gm/cm}^3}{20\text{ gm/cm}^3} = 0.2 $
- Relative error in volume: $ \frac{\Delta V}{V} = \frac{1\text{ cm}^3}{10\text{ cm}^3} = 0.1 $
Calculating Absolute Error in Mass
Now, find the total relative error in mass:
$ \frac{\Delta M}{M} = 0.2 + 0.1 = 0.3 $
To find the absolute error in mass ($\Delta M$), multiply the nominal mass ($M$) by the total relative error:
$ \Delta M = M \times \left( \frac{\Delta M}{M} \right) = (200\text{ gm}) \times (0.3) = 60\text{ gm} $
Final Answer
The absolute error in the measurement of mass is 60 gm.
Similar Questions
- The velocity $v$ of a particle at time $t$ is given by $v = at + \frac{b}{t+c}$, where $a$, $b$ and $c$ are constants. The dimension of $a$, $b$ and $c$ are, respectively
Important Questions from General Physics
- When the position vector $\overrightarrow{r} = x\hat{i} + y\hat{j} + z\hat{k}$ changes sign as $-\overrightarrow{r}$, which one of the following vector will not flip under sign change ?
- Match List - I with List - II.
List - I List - II A. Meter (L) I. $\sqrt{\frac{hc}{G}}$ B. Second (S) II. $\sqrt{\frac{Gh}{c^{5}}}$ C. Kilogram (M) III. $\sqrt{\frac{K^{2}L^{2}c^{3}}{Gh}}$ D. Kelvin (K) IV. $\sqrt{\frac{Gh}{c^{3}}}$
where h (Planck's constant), G (gravitational constant) and c (speed of light in vacuum) as fundamental units.
Choose the correct answer from the options given below : - In an experiment to determine the resistance of a given wire using Ohm's law, the voltmeter and ammeter readings are noted as $10 \text{ V}$ and $5 \text{ A}$, respectively. The least counts of voltmeter and ammeter are $500 \text{ mV}$ and $200 \text{ mA}$, respectively. The estimated error in the resistance measurement is ___________ $\Omega$
- The velocity $v$ of a particle at time $t$ is given by $v = at + \frac{b}{t+c}$, where $a$, $b$ and $c$ are constants. The dimension of $a$, $b$ and $c$ are, respectively
- The density $\rho$ of a uniform cylinder is determined by measuring its mass $m$, length $l$ and diameter $d$. The measured values of $m, l$ and $d$ are $97.42 \pm 0.02 \text{ g}$, $8.35 \pm 0.05 \text{ mm}$ and $20.20 \pm 0.02 \text{ mm}$, respectively. Calculated percentage fractional error in $\rho$ is _________.
