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.