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
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 __________.
Three students $S_1$, $S_2$ and $S_3$ perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.
(least count of length =0.1 cm
least count for time =0.1s )
If $E_1$, $E_2$ and $E_3$ are the percentage errors in 'g' for students 1, 2 and 3 respectively, then the minimum percentage error is obtained by student no. __________.
- 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 ________
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A quantity $Q$ is formulated as $X^{-2} Y^{\frac{3}{2}} Z^{\frac{-2}{5}}$. $X, Y$ and $Z$ are independent parameters which have fractional errors of $0.1, 0.2$ and $0.5$, respectively in measurement. The maximum fractional error of $Q$ is
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Important Questions from General Physics
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 __________.
Three students $S_1$, $S_2$ and $S_3$ perform an experiment for determining the acceleration due to gravity (g) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.
(least count of length =0.1 cm
least count for time =0.1s )
If $E_1$, $E_2$ and $E_3$ are the percentage errors in 'g' for students 1, 2 and 3 respectively, then the minimum percentage error is obtained by student no. __________.
- 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 ________
- Given a charge $q$, current $I$ and permeability of vacuum $\mu_o$. Which of the following quantity has the dimension of momentum ?
