Formula for the Time Period of a Simple Pendulum
The time period \( T \) of a simple pendulum is given by:
$$ T = 2\pi \sqrt{\frac{L}{g}} $$
Where:
\( L \) = Length of the pendulum
\( g \) = Acceleration due to gravity
Step 1: Calculate the Original Time Period
For a pendulum with mass \( m \) and length \( L_0 \), the time period is:
$$ T_0 = 2\pi \sqrt{\frac{L_0}{g}} $$
Step 2: Analyze the Changes
According to the problem:
Step 3: Compute the New Time Period
The time period of the new pendulum is:
$$ T_{\text{new}} = 2\pi \sqrt{\frac{L_0/2}{g}} $$
Rewriting:
$$ T_{\text{new}} = 2\pi \sqrt{\frac{L_0}{2g}} $$
Comparing with \( T_0 \):
$$ T_{\text{new}} = T_0 \frac{1}{\sqrt{2}} $$
Step 4: Solve for \( x \)
According to the question:
$$ \frac{x}{2} = \frac{1}{\sqrt{2}} $$
Solving for \( x \):
$$ x = \sqrt{2} $$
Conclusion
The value of \( x \) is \(\sqrt{2}\).
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The potential (V) at any axial point, at 2 m distance(r) from the centre of the dipole of dipole moment vector
\(\vec{P}\) of magnitude, 4 × 10-6 C m, is ± 9 × 103 V.
(Take \(\frac{1}{4\pi\epsilon_0}=9\times10^9\) SI units)
Reason R : \(V=±\frac{2P}{4\pi \epsilon_0r^2}\), where r is the distance of any axial point, situated at 2 m from the centre of the dipole.
In the light of the above statements, choose the correct answer from the options given below :
The output (Y) of the given logic gate is similar to the output of an/a :