To find the amplitude of the particle performing simple harmonic motion (SHM) described by the equation \( x = 8 \sin(\omega t) + 6 \cos(\omega t) \), we can use a method that combines the sine and cosine terms into a single sine function. This will help us determine the amplitude more easily.
Understanding the Components
The displacement equation consists of two parts: one involving the sine function and the other involving the cosine function. In SHM, the general form can be expressed as:
x = A \sin(\omega t + \phi)
where \( A \) is the amplitude and \( \phi \) is the phase angle. Our goal is to rewrite the given equation in this form.
Combining the Terms
We can express the equation \( x = 8 \sin(\omega t) + 6 \cos(\omega t) \) in the form of a single sine function. To do this, we can use the following relationship:
A = \sqrt{a^2 + b^2}
where \( a \) and \( b \) are the coefficients of the sine and cosine terms, respectively. Here, \( a = 8 \) and \( b = 6 \).
Calculating the Amplitude
Now, let's calculate the amplitude:
- First, square the coefficients:
- \( a^2 = 8^2 = 64 \)
- \( b^2 = 6^2 = 36 \)
- Next, add these squares together:
- \( a^2 + b^2 = 64 + 36 = 100 \)
- Finally, take the square root to find the amplitude:
- \( A = \sqrt{100} = 10 \) cm
Final Result
The amplitude of the motion is therefore 10 cm. This corresponds to option (a). In simple harmonic motion, the amplitude represents the maximum displacement from the equilibrium position, which in this case is 10 cm. This means that the particle will oscillate between +10 cm and -10 cm from its central position.
Understanding how to combine sine and cosine terms is crucial in solving problems related to SHM, as it allows for a clearer interpretation of the motion's characteristics. If you have any further questions or need clarification on any part of this process, feel free to ask!
