Question

The speed of a particle executing motion changes with time according to the equation \(\mathrm{y}=\mathrm{a} \sin \omega \mathrm{t}+\mathrm{b} \cos \omega \mathrm{t}\), then \(\ldots \ldots \ldots\) (A) Motion is periodic but not a S.H.M. (B) It is a S.H.M. with amplitude equal to \(\mathrm{a}+\mathrm{b}\) (C) It is a S.H.M. with amplitude equal to \(\mathrm{a}^{2}+\mathrm{b}^{2}\)

Short answer

The motion of the particle is a Simple Harmonic Motion with amplitude equal to \(\sqrt{a^2 + b^2}\). Option (C) is correct.

Step-by-step solution

1. Identify the equation of the motion

First, let's observe that the given equation describes the speed of the particle, not its displacement. Simple Harmonic Motion is mainly defined by the displacement equation, so we need to find the equation for displacement. To do that, we will integrate the speed equation with respect to time.

2. Integrate the speed equation

The given speed equation is: \(y= a \sin(\omega t) + b \cos(\omega t)\). Now, to find the displacement equation, integrate this expression with respect to time (t): Integrating, we get: \(x(t) = -\frac{a}{\omega} \cos(\omega t) + \frac{b}{\omega} \sin(\omega t) + C\), where x(t) is the displacement and C is the integration constant.

3. Analyzing the displacement equation and comparing with S.H.M.

The general displacement equation for Simple Harmonic Motion is: \(x(t) = A \cos(\omega t + \phi)\) or \(x(t) = A \sin(\omega t + \phi)\), where A is the amplitude, \(\omega\) is the angular frequency, and \(\phi\) is the phase angle. Our integrated displacement equation is: \(x(t) = -\frac{a}{\omega} \cos(\omega t) + \frac{b}{\omega} \sin(\omega t) + C\). If we apply the trigonometric identity \(\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta)\), we obtain: \(x(t) = A \cos(\omega t + \phi)\), where \(A = \sqrt{(\frac{a}{\omega})^2 + (\frac{b}{\omega})^2} = \frac{\sqrt{a^2 + b^2}}{\omega}\) and \(\phi = -\arctan(\frac{a}{b})\).

4. Conclusion

Since the displacement equation of the particle can be expressed in the form of a Simple Harmonic Motion equation, we can conclude that the motion is a Simple Harmonic Motion. The amplitude A can be calculated as follows: \(A = \frac{\sqrt{a^2 + b^2}}{\omega}\) Therefore, the correct option is (C) It is an S.H.M. with amplitude equal to \(\sqrt{a^2 + b^2}\).

Key concepts

displacement equation

In simple harmonic motion (SHM), the displacement equation plays a crucial role in describing the motion of a particle. The displacement equation specifies the location of a particle at any given time. For SHM, this equation generally takes the form \( x(t) = A \sin(\omega t + \phi) \) or \( x(t) = A \cos(\omega t + \phi) \), where:

• \( A \) is the amplitude, representing the maximum displacement from the mean position.
• \( \omega \) is the angular frequency, indicating how quickly the particle moves through its cycle.
• \( \phi \) is the phase angle, which determines the initial position of the particle.

To arrive at the displacement equation from a given velocity equation as we see in the original exercise, integration with respect to time is necessary. By identifying the velocity components, \( a \sin(\omega t) + b \cos(\omega t) \), and integrating them, we deduce:
\[ x(t) = -\frac{a}{\omega} \cos(\omega t) + \frac{b}{\omega} \sin(\omega t) + C \]
Here, \( C \) is a constant that represents the initial displacement. By comparing this with the general form of SHM, we recognize the motion's nature and can further analyze its properties.

amplitude calculation

Calculating the amplitude in the context of simple harmonic motion is fundamental for understanding the scale of the motion. From the displacement equation obtained—expressed as a combination of sine and cosine components—it becomes apparent that the amplitude \( A \) needs to combine these components into a singular representative value.
To achieve this, we employ the Pythagorean identity, allowing us to express amplitude quantitatively:

• First, identify the coefficients of the sine and cosine terms: \(-\frac{a}{\omega}\) and \(\frac{b}{\omega}\).
• Then, compute the amplitude using the formula \( A = \sqrt{ \left(\frac{a}{\omega}\right)^2 + \left(\frac{b}{\omega}\right)^2 } \).

Thus, the amplitude of the motion, which indicates the maximum extent of oscillation, is \( \frac{\sqrt{a^2 + b^2}}{\omega} \). This approach allows for a streamlined understanding of the motion's intensity, distinguishable by the presence of both sine and cosine contributions.

trigonometric identities

Trigonometric identities are indispensable tools in simplifying and expressing equations related to simple harmonic motion. These identities help manipulate expressions involving sine and cosine, providing a pathway to rewrite an equation to reveal its underlying SHM form.

• The cosine angle subtraction identity, \( \cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + \sin(\alpha) \sin(\beta) \), is pivotal to rearranging the displacement equation derived from speed.
• Using this identity allows expressing the combined terms \(-\frac{a}{\omega} \cos(\omega t) + \frac{b}{\omega} \sin(\omega t)\) as a singular cosine function that matches the standard SHM form.

Implementing these identities, we achieve a clearer expression:
\[ x(t) = A \cos(\omega t + \phi) \]
where \( A \) and \( \phi \) are determined by the components derived from the original expressions. Trigonometric identities thus simplify the complex expressions into their harmonic motion, showcasing these identities' value in physics and mathematics. They help bridge complex trigonometric functions back to understandable physical phenomena like SHM.