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
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The diameter of a wire measured by a screw gauge of least count $0.001\text{ cm}$ is $0.08\text{ cm}$. The length measured by a scale of least count $0.1\text{ cm}$ is $150\text{ cm}$. When a weight of $100\text{ N}$ is applied to the wire, the extension in length is $0.5\text{ cm}$, measured by a micrometer of least count $0.001\text{ cm}$. The error in the measured Young's modulus is $\alpha \times 10^9\text{ N/m}^2$. The value of $\alpha$ is ________.
(Ignore the contribution of the load to Young's modulus error calculation)
Important Questions from General Physics
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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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 ?
- Given a charge $q$, current $I$ and permeability of vacuum $\mu_o$. Which of the following quantity has the dimension of momentum ?
- 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 __________.
