| List I | List II |
| a) Amplitude | 1.) Maximum displacement on either side |
| b) Frequency | 2.) 1/time period |
| c) Wave form | 3.) Presence of other frequencies |
| d) Loudness | 4.) Amplitude |
The question asks to match terms related to wave characteristics (Amplitude, Frequency, Waveform, Loudness) with their correct definitions or related concepts. We need to go through each term and find its accurate description from the given list.
Combining these matches, we get a-1, b-2, c-3, d-4.
A) a-1,b-2,c-3,d-4 is the correct option as it accurately matches all the terms with their respective definitions or related concepts.
The question asks for the term that defines the number of oscillations per unit time. This is a fundamental definition in physics related to wave phenomena and periodic motion. We need to recall the definitions of the given options to identify the correct one.
A) Frequency is the number of oscillations or cycles that occur per unit of time. Mathematically, if \(N\) is the number of oscillations and \(T\) is the total time taken, then frequency \(f\) is given by:
\[ f = \frac{N}{T} \]When \(N=1\), the time taken is the time period \(T_p\), so \(f = \frac{1}{T_p}\).
The question asks to identify the term 'ω' in the standard equation for simple harmonic motion (SHM), which is given as \(x = A \cos(\omega t + \phi)\). We need to recall the definitions of each parameter in this equation.
Correct Option: C) Angular frequency
To find the length of a simple pendulum that "ticks seconds", we need to understand what a "seconds pendulum" is. A seconds pendulum is defined as a simple pendulum whose period of oscillation is exactly two seconds. This means it takes one second to swing in one direction (e.g., from one extreme to the other) and another second to swing back, completing one full oscillation in two seconds. We will use the formula for the period of a simple pendulum to calculate its length.
Define a seconds pendulum: A seconds pendulum has a time period \(T = 2\) seconds.
Recall the formula for the period of a simple pendulum: The period \(T\) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]where:
Substitute the known values into the formula: We know \(T = 2 \, \text{s}\) and \(g \approx 9.8 \, \text{m/s}^2\). We need to find \(L\).
\[ 2 = 2\pi \sqrt{\frac{L}{9.8}} \]Solve for \(L\):
First, divide both sides by \(2\):
\[ 1 = \pi \sqrt{\frac{L}{9.8}} \]Next, divide both sides by \(\pi\):
\[ \frac{1}{\pi} = \sqrt{\frac{L}{9.8}} \]Square both sides:
\[ \left(\frac{1}{\pi}\right)^2 = \frac{L}{9.8} \]\[ \frac{1}{\pi^2} = \frac{L}{9.8} \]Now, solve for \(L\):
\[ L = \frac{9.8}{\pi^2} \]Using the approximation \(\pi \approx 3.14159\), \(\pi^2 \approx (3.14159)^2 \approx 9.8696\).
\[ L = \frac{9.8}{9.8696} \]\[ L \approx 0.992 \, \text{m} \]This value is very close to \(1 \, \text{m}\).
A) 1 m — The calculated length of a seconds pendulum is approximately \(0.992 \, \text{m}\), which is very close to \(1 \, \text{m}\). In competitive exams, this value is typically rounded to \(1 \, \text{m}\) for practical purposes.
The question asks us to identify the correct statements regarding the time period (T) of an oscillating pendulum. We need to evaluate each statement based on the definition and properties of the time period in simple harmonic motion.
B) (ii), (iii) and (iv) only
Based on our analysis, statements (ii), (iii), and (iv) are correct. Statement (i) is incorrect because the motion repeats not just once, but infinitely many times at integer multiples of T.
The question asks about the position around which a body oscillates in mechanical oscillation. This fundamental concept relates to the state where the net force on the body is zero, and it tends to return to this state after displacement.
Correct Option: D) Equilibrium position — In mechanical oscillation, a body oscillates about its mean position, which is also its equilibrium position. At this position, the net force acting on the body is zero, and it represents the stable state around which the periodic motion occurs.