Sphere Volume Formula
The volume ($V$) of a sphere is given by the formula: $V = \frac{4}{3}\pi r^3$, where $r$ is the radius.
Volume Calculation Using Diameter
Since the radius ($r$) is half the diameter ($d$), $r = \frac{d}{2}$. Substituting this into the volume formula gives the volume in terms of diameter:
$V = \frac{4}{3}\pi \left(\frac{d}{2}\right)^3 = \frac{4}{3}\pi \frac{d^3}{8} = \frac{\pi}{6}d^3$.
This shows that the volume ($V$) is proportional to the cube of the diameter ($d^3$), meaning $V \propto d^3$.
Error Propagation Rule
When a quantity depends on a power of a variable, the percentage error propagates accordingly. If $V \propto d^n$, the relationship between the percentage errors is:
$ \frac{\Delta V}{V} \times 100\% = n \left( \frac{\Delta d}{d} \times 100\% \right) $
Percentage Error in Sphere Volume
In this problem, the volume depends on the cube of the diameter ($V \propto d^3$), so the exponent $n=3$. We are given the percentage error in the diameter measurement:
$ \frac{\Delta d}{d} \times 100\% = 2\% $
Now, we apply the error propagation rule:
$ \text{Percentage Error in Volume} = 3 \times (\text{Percentage Error in Diameter}) $
$ \frac{\Delta V}{V} \times 100\% = 3 \times (2\%) $
$ \frac{\Delta V}{V} \times 100\% = 6\% $
Final Volume Error
Therefore, the percentage error in the calculated volume of the sphere is $6\%$.
Similar Questions
- 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 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 _________.
- Let PQR be a triangle such that $\vec{PQ} = -2\hat{i} - \hat{j} + 2\hat{k}$ and $\vec{PR} = a\hat{i} + b\hat{j} - 4\hat{k}$, $a, b \in \mathbb{Z}$. Let S be the point on QR, which is equidistant from the lines PQ and PR. If $|\vec{PR}| = 9$ and $\vec{PS} = \hat{i} - 7\hat{j} + 2\hat{k}$, then the value of $3a - 4b$ is ________
- For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2 = 9$ and $|2\vec{a}+k\vec{b}+k\vec{c}| = 3$, the positive value of k is
- Keeping the significant figures in view, the sum of the physical quantities $52.01\text{ m}$, $153.2\text{ m}$ and $0.123\text{ m}$ is :
- The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants G, $\hbar$ and c. Which of the following correctly gives the Planck length ?
A man in a car at location Q on a straight highway is moving with speed v. He decides to reach a point P in qa field at a distance d from the highway (point m) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach P is minimum ?
- Three particles P, Q and R are moving along the vectors $\vec{A} = \hat{i} + \hat{j}$, $\vec{B} = \hat{j} + \hat{k}$ and $\vec{C} = -\hat{i} + \hat{j}$ respectively. They strike on a point and start to move in different directions.<br>Now particle P is moving normal to the plane which contains vector $\vec{A}$ and $\vec{B}$. Similarly particle Q is moving normal to the plane which contains vector $\vec{A}$ and $\vec{C}$. The angle between the direction of motion of P and Q is $\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)$. Then the value of x is __________.
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 ___________ $\Omega$
- Density and volume of a body are given as $(20 \pm 4)\text{ gm/cm}^3$ and $(10 \pm 1)\text{ cm}^3$ respectively. The absolute error in measurement of mass is
- 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
