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. __________.
The formula for the acceleration due to gravity (g) using a simple pendulum is given by:
\( g = \frac{4\pi^2L}{T^2} \)
where \( L \) is the length of the pendulum and \( T \) is the time period for one oscillation.
The percentage error in g, \( E \), can be calculated by the formula:
\( E = \left(\frac{\Delta L}{L} + 2\frac{\Delta T}{T}\right) \times 100\% \)
where \( \Delta L \) and \( \Delta T \) are the least counts for L and T respectively.
- For Student 1:
- \( L = 64.0 \, \text{cm}, \, n = 8, \, \text{Total time} = 128.0 \, \text{s} \)
- \( T = \frac{128.0}{8} = 16.0 \, \text{s} \)
- \( \Delta L = 0.1 \, \text{cm}, \, \Delta T = 0.1 \, \text{s} \)
- \( E_1 = \left(\frac{0.1}{64.0} + 2\frac{0.1}{16.0}\right) \times 100\% = 0.15625\% + 1.25\% = 1.40625\% \)
- For Student 2:
- \( L = 64.0 \, \text{cm}, \, n = 4, \, \text{Total time} = 64.0 \, \text{s} \)
- \( T = \frac{64.0}{4} = 16.0 \, \text{s} \)
- \( E_2 = \left(\frac{0.1}{64.0} + 2\frac{0.1}{16.0}\right) \times 100\% = 1.40625\% \)
- For Student 3:
- \( L = 20.0 \, \text{cm}, \, n = 4, \, \text{Total time} = 36.0 \, \text{s} \)
- \( T = \frac{36.0}{4} = 9.0 \, \text{s} \)
- \( E_3 = \left(\frac{0.1}{20.0} + 2\frac{0.1}{9.0}\right) \times 100\% = 0.5\% + 2.22\% = 2.72\% \)
Comparing \( E_1, E_2, \) and \( E_3 \), we find that the minimum percentage error is for Students 1 and 2, both at \( 1\% \).
Therefore, the minimum percentage error is obtained by student no. 1
Similar Questions
- 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
- 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 ________
- 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 :
- 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 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 __________.
- Given a charge $q$, current $I$ and permeability of vacuum $\mu_o$. Which of the following quantity has the dimension of momentum ?
Important Questions from General Physics
- 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
- 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 ________
- 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 :
- 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 :
