To determine the factors affecting the time period of a simple pendulum, we need to recall the formula for the time period of a simple pendulum and analyze each variable in the formula.
A) Length of the pendulum — As per the formula \(T = 2\pi \sqrt{\frac{L}{g}}\), the time period \(T\) is directly proportional to the square root of the length \(L\) of the pendulum. Thus, the time period depends on the length of the pendulum.
To determine how the time period of a simple pendulum changes when its length is doubled, we need to recall the formula for the time period of a simple pendulum and analyze its relationship with length.
The time period \(T\) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]where:
From the formula, we can see that \(T\) is directly proportional to the square root of \(L\), assuming \(g\) is constant:
\[ T \propto \sqrt{L} \]Let the initial length be \(L_1 = L\), and the initial time period be \(T_1\).
The new length is \(L_2 = 2L\).
We need to find the new time period \(T_2\).
Using the proportionality, we can write:
\[ \frac{T_2}{T_1} = \frac{\sqrt{L_2}}{\sqrt{L_1}} \]Substitute the values:
\[ \frac{T_2}{T_1} = \frac{\sqrt{2L}}{\sqrt{L}} \]\[ \frac{T_2}{T_1} = \sqrt{\frac{2L}{L}} \]\[ \frac{T_2}{T_1} = \sqrt{2} \]Therefore, the new time period \(T_2\) is:
\[ T_2 = T_1 \sqrt{2} \]This means the time period increases by a factor of \(\sqrt{2}\).
Correct Option: D) Increases by a factor of \(\sqrt{2}\)
To identify the restoring force in a simple pendulum, we need to analyze the forces acting on the pendulum bob when it is displaced from its equilibrium position. The restoring force is the force component that always acts to bring the system back to its equilibrium.
Correct Option: A) Gravity. The component of gravitational force (\(mg \sin\theta\)) acting tangential to the arc of motion provides the restoring force that pulls the bob back towards the equilibrium position.
The question asks for the correct formula for the time period of a simple pendulum. This is a fundamental concept in physics, specifically in oscillations and waves. The time period of a simple pendulum depends on its length and the acceleration due to gravity.
Correct Option: A) The formula \(T = 2\pi\sqrt{L/g}\) correctly represents the time period of a simple pendulum, where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity.
The question asks for the primary factor affecting the period of a simple pendulum. We need to recall the formula for the period of a simple pendulum and analyze which variables in the formula significantly influence its value.
Correct Option: B) The formula for the period of a simple pendulum is \(T = 2\pi \sqrt{\frac{L}{g}}\). This formula clearly shows that the period \(T\) is directly proportional to the square root of the length of the string \(L\). Therefore, the length of the string is the primary factor affecting the period.
To determine how the period of a simple pendulum changes when its length is quadrupled, we need to recall the formula for the period of a simple pendulum and analyze its relationship with the length of the string.
The period \(T\) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]where \(L\) is the length of the pendulum string and \(g\) is the acceleration due to gravity.
From the formula, we can see that the period \(T\) is directly proportional to the square root of the length \(L\). That is, \(T \propto \sqrt{L}\).
The problem states that the length of the string is quadrupled. Let the initial length be \(L_1\) and the new length be \(L_2\). So, \(L_2 = 4L_1\).
Let the initial period be \(T_1\) and the new period be \(T_2\). Using the proportionality:
\[ \frac{T_2}{T_1} = \frac{\sqrt{L_2}}{\sqrt{L_1}} \]Substitute \(L_2 = 4L_1\):
\[ \frac{T_2}{T_1} = \frac{\sqrt{4L_1}}{\sqrt{L_1}} \]\[ \frac{T_2}{T_1} = \frac{2\sqrt{L_1}}{\sqrt{L_1}} \]\[ \frac{T_2}{T_1} = 2 \]Therefore, \(T_2 = 2T_1\).
When the length of the string is quadrupled, the period of the pendulum doubles.
Correct Option: B) Double