A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is 4 m, then the time period of small oscillations will be _____ s. [take \( g = \pi^2 \, m/s^2 \)]
A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is 4 m, then the time period of small oscillations will be _____ s. [take \( g = \pi^2 \, m/s^2 \)]
Correct Answer: 8
Solution and Explanation
The time period of a pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}}, \] where \( L \) is the length of the pendulum, and \( g \) is the acceleration due to gravity.
For the given scenario, the effective acceleration due to gravity is:
\[ g' = \frac{g}{4}. \]
Substituting this into the formula for the time period:
\[ T = 2\pi \sqrt{\frac{L}{g'}} = 2\pi \sqrt{\frac{L}{\frac{g}{4}}} = 2\pi \sqrt{\frac{4L}{g}}. \]
Given that \( L = 4 \, m \), the formula becomes:
\[ T = 2\pi \sqrt{\frac{4 \times 4}{g}}. \]
Simplifying further:
\[ T = 2\pi \sqrt{\frac{16}{g}}. \]
Given \( g = \pi^2 \, m/s^2 \), substitute into the equation:
\[ T = 2\pi \sqrt{\frac{16}{\pi^2}} = 2\pi \times \frac{4}{\pi}. \]
Simplify:
\[ T = 2 \times 4 = 8 \, s. \]
Final Answer:
\[ T = 8 \, s. \]
Top Questions on Electrostatics
- Two point charges of \(1\,\text{nC}\) and \(2\,\text{nC}\) are placed at two corners of an equilateral triangle of side \(3\) cm. The work done in bringing a charge of \(3\,\text{nC}\) from infinity to the third corner of the triangle is ________ \(\mu\text{J}\). \[ \left(\frac{1}{4\pi\varepsilon_0}=9\times10^9\,\text{N m}^2\text{C}^{-2}\right) \]
- JEE Main - 2026
- Physics
- Electrostatics
Match List-I with List-II.
Choose the correct answer from the options given below :}- JEE Main - 2026
- Physics
- Electrostatics
There are three co-centric conducting spherical shells $A$, $B$ and $C$ of radii $a$, $b$ and $c$ respectively $(c>b>a)$ and they are charged with charges $q_1$, $q_2$ and $q_3$ respectively. The potentials of the spheres $A$, $B$ and $C$ respectively are:
- JEE Main - 2026
- Physics
- Electrostatics
Two resistors $2\,\Omega$ and $3\,\Omega$ are connected in the gaps of a bridge as shown in the figure. The null point is obtained with the contact of jockey at some point on wire $XY$. When an unknown resistor is connected in parallel with $3\,\Omega$ resistor, the null point is shifted by $22.5\,\text{cm}$ towards $Y$. The resistance of unknown resistor is ___ $\Omega$.
- JEE Main - 2026
- Physics
- Electrostatics
- The electrostatic potential in a charged spherical region of radius $r$ varies as $V = ar^{3} + b$, where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $a \times \pi \varepsilon_{0}$. The value of $a$ is ___.
(Permittivity of vacuum is $\varepsilon_{0}$)
- JEE Main - 2026
- Physics
- Electrostatics
Questions Asked in JEE Main exam
- The number of numbers greater than $5000$, less than $9000$ and divisible by $3$, that can be formed using the digits $0,1,2,5,9$, if repetition of digits is allowed, is
- JEE Main - 2026
- Permutations
- Evaluate: \[ \frac{6}{3^{26}}+\frac{10\cdot1}{3^{25}}+\frac{10\cdot2}{3^{24}}+\frac{10\cdot2^{2}}{3^{23}}+\cdots+\frac{10\cdot2^{24}}{3}. \]
- JEE Main - 2026
- Integral Calculus
In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$)
- JEE Main - 2026
- Rotational Mechanics
- If the coefficient of \( x \) in the expansion of \[ (ax^2 + bx + c)(1 - 2x)^{26} \] is \(-56\) and the coefficients of \( x^2 \) and \( x^3 \) are both zero, then \( a + b + c \) is equal to
- JEE Main - 2026
- Binomial theorem
The equivalent resistance between the points \(A\) and \(B\) in the given circuit is \[ \frac{x}{5}\,\Omega. \] Find the value of \(x\).
- JEE Main - 2026
- Current electricity