Question

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\) (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) d Statement \(-1:\) For a particle executing S.H.M. with an amplitude of \(0.01 \mathrm{~m}\) and frequency \(30 \mathrm{hz}\), the maximum acceleration is \(36 \pi^{2} \mathrm{~m} / \mathrm{s}^{2}\). Statement \(-2:\) The maximum acceleration for the above particle is \(\pm \omega 2 \mathrm{~A}\), where \(\mathrm{A}\) is amplitude. (A) a (B) \(\mathrm{b}\) (C) \(c\) (D) \(\mathrm{d}\)

Short answer

The correct answer is (A) Both statement 1 and statement 2 are true, and statement 2 is the correct explanation of statement 1.

Step-by-step solution

Calculate the maximum acceleration for statement 1

Given the amplitude A = 0.01 m and frequency f = 30 Hz, we need to find the maximum acceleration. Since the maximum acceleration is given by the formula: \[a_{max} = \omega^2 A\] where \(\omega = 2\pi f\) is the angular frequency, we can calculate the maximum acceleration using the given values. Step 2: Calculate the angular frequency

Calculate the angular frequency

With f = 30 Hz, we can calculate the angular frequency as follows: \[\omega = 2\pi f = 2\pi(30) = 60\pi\ rad/s\] Step 3: Calculate the maximum acceleration

Calculate the maximum acceleration using the angular frequency

Now we can calculate the maximum acceleration: \[a_{max} = \omega^2 A = (60\pi)^2 (0.01 \mathrm{m}) = 36\pi^2\ \mathrm{m/s^2}\] Step 4: Analyze statement 2

Check the validity of the statement 2

We already calculated the maximum acceleration for the given particle using the formula \(a_{max} = \omega^2 A\). As Statement 2 suggests, this is indeed the correct formula. Therefore, statement 2 is true. Step 5: Comparison of statements and options

Compare statements and options

Both statement 1 and statement 2 are true. Additionally, statement 2 is indeed the correct explanation for statement 1. Therefore, the correct option is: (a) Statement 1 is true and statement 2 is true. Statement 2 is the correct explanation of statement 1.

Key concepts

Amplitude

In simple harmonic motion (SHM), the amplitude is a crucial concept to understand. It represents the maximum extent of oscillation from the equilibrium position. Think of it as the furthest point the object reaches from the center of its path during its motion. In this exercise, the amplitude is given as 0.01 meters.

Amplitude is constant for a given system under constant energy conditions, and it determines the total energy of the system. The larger the amplitude, the more energy is stored in the motion, since energy in SHM is proportional to the square of the amplitude. If you're dealing with problems or exercises involving SHM, knowing the amplitude can help you compute other significant quantities, like potential and kinetic energy, using energy concepts for SHM.

Angular Frequency

Angular frequency, denoted by \(\omega\), plays a central role in analyzing systems that execute simple harmonic motion. It represents how quickly the object oscillates through its cycle. Angular frequency links directly to the frequency, \(f\), through the equation \(\omega = 2\pi f\).

This gives us a quantifiable measure of how "fast" an oscillation is. In our problem, the given frequency is 30 Hz, leading to an angular frequency of \(60\pi\) rad/s. This means the object completes \(60\pi\) radians of its cycle per second, emphasizing how SHM is rooted deeply in rotational kinematics.

In terms of units, remember that angular frequency is measured in radians per second. It's crucial for calculating other properties of the system or ensuring equations balance correctly. With angular frequency, one can further explore other properties such as phase angles and time periods, pivotal in understanding oscillatory motion.

Maximum Acceleration

Maximum acceleration in a simple harmonic motion system is an essential aspect because it tells us the greatest change in velocity an object will experience. This is achieved at the extremities of the motion, where the velocity changes direction.

The formula for computing this is \(a_{max} = \omega^2 A\), where \(A\) is the amplitude, and \(\omega\) is the angular frequency. Applying this to our study, given \(A = 0.01 \, \mathrm{m}\) and \(\omega = 60\pi\) rad/s, the maximum acceleration is calculated to be \(36\pi^2\, \mathrm{m/s^2}\).

This concept is useful because it helps you understand the limits of force that the system—or anything attached to it—experiences. Recognizing this can be crucial for applications in engineering, where constraints need to be managed so that structures or mechanisms remain stable and safe.

Frequency

Frequency, a familiar term in everyday physics, is defined as the number of oscillations or cycles a wave completes in one second. It is measured in hertz (Hz). In any SHM system, knowing the frequency gives a quick insight into how fast the oscillations occur.

In this specific example, the frequency is 30 Hz. This means the object completes 30 full back-and-forth vibrations every second. Frequency is linked with the angular frequency by the formula \(\omega = 2\pi f\), where \(\omega\) is in radians per second.

Understanding frequency helps you predict how often an event happens over time, which is crucial not only in physics but also in areas like signal processing, acoustics, and electrical engineering. It also assists you in calculating other attributes of SHM such as the time period, which is the reciprocal of frequency, giving you a comprehensive idea of the motion dynamics.