Question

The amplitude for a S.H.M. given by the equation \(\mathrm{x}=3 \sin 3 \mathrm{pt}+4 \cos 3 \mathrm{pt}\) is \(\ldots \ldots \ldots \ldots \mathrm{m}\) (A) 5 (B) 7 (C) 4 (D) \(3 .\)

Short answer

The amplitude for the given SHM equation is \(A = 5\,m\).

Step-by-step solution

Write down the given equation in general SHM form

The given equation for the SHM is \(x = 3\sin(3pt) + 4\cos(3pt)\). We aim to rewrite this equation in the general SHM form: \(x(t) = A\sin(\omega t + \phi)\).

Apply the sum-to-product trigonometric identity

We use the sum-to-product trigonometric identity to convert the sum of sine and cosine functions into a product form: \[\sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B)\] Using this identity, we rewrite our given equation as follows: \[x = R\sin(3pt + \phi)\] where \(R = \sqrt{3^2 + 4^2}\) and \(\tan{\phi} = \frac{3}{4}\).

Calculate R

Calculate the value of R using the Pythagorean theorem: \[R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]

Identify the amplitude

The amplitude, A, is equal to the value of R in our transformed equation: \[A = R = 5\]

Find the correct option

The correct option for the amplitude is: (A) 5

Key concepts

Amplitude Calculation

In simple harmonic motion (SHM), the amplitude represents the maximum displacement from the equilibrium position. In other words, it's how far, at most, the object moves away from the central rest position. When given an equation like \( x = 3\sin(3pt) + 4\cos(3pt) \), we need to find out the amplitude to determine this maximum extent of motion.

To find the amplitude, we convert the given equation into the form \( x(t) = A\sin(\omega t + \phi) \), where \( A \) is the amplitude. Through this process, we identify how both terms, 3 being the coefficient of sine and 4 of cosine, combine mathematically using the Pythagorean Theorem to form the amplitude.

The method involves combining these orthogonal components, much like finding the resultant vector magnitude in physics. The calculated result, \( R = \sqrt{3^2 + 4^2} = 5 \), tells us the amplitude is 5 meters, which indicates the solution to the problem.

Trigonometric Identities

Trigonometric identities are essential tools in solving equations involving trigonometric functions. In our SHM problem, we need to simplify the equation \( 3\sin(3pt) + 4\cos(3pt) \) into a single sinusoidal function. This is where the sum-to-product identities help.

Specifically, the identity \( \sin(A) \cos(B) + \cos(A) \sin(B) = \sin(A + B) \) is utilized to combine the sine and cosine terms into a single sine function with an amplitude and phase shift.

• This transformation is crucial because it allows us to express the equation in the familiar form \( x(t) = R\sin(3pt + \phi) \).
• It transforms our initial equation with separate trigonometric terms into a more straightforward and physically meaningful expression where \( R \) is the amplitude.
• The angle \( \phi \), found from \( \tan(\phi) = \frac{3}{4} \), represents the phase shift and helps in understanding the starting position of the motion relative to time.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, but it also finds vital applications in calculating amplitudes in SHM when dealing with trigonometric functions. Given the terms \( 3 \) and \( 4 \) from the original equation, these are viewed akin to the legs of a right triangle. The theorem helps us find the hypotenuse, which, in this context, is the resultant amplitude.

• Using the formula \( R = \sqrt{3^2 + 4^2} \), we compute the amplitude by considering each trigonometric component as contributing to a resultant vector's magnitude.
• The computed result, \( R = 5 \), shows us the hypotenuse, which stands for the maximum extent of motion or amplitude.
• Knowing that the amplitude directly results from these components helps visualize how orthogonal sine and cosine waves combine to describe SHM fully.