Question

The equation for displacement of a particle at time \(\mathrm{t}\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). The amplitude of oscillation is \(\ldots \ldots \ldots . \mathrm{cm}\). (A) 1 (B) 3 (C) 5 (D) 7

Short answer

The amplitude of oscillation is 5 cm. The correct answer is (C) 5.

Step-by-step solution

Identify the sinusoidal function

The displacement equation is given as \(y = 3 \cos 2t + 4 \sin 2t\). This equation is a sum of a cosine and a sine function, which, in general, can be expressed as \(y = A \cos(2t - \delta)\), where \(A\) represents the amplitude and \(\delta\) represents the phase difference.

Find the amplitude of the oscillation

To find the amplitude, we use the formula \(A = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the coefficients of the cosine and sine functions in the given equation, respectively. In our case, \(a = 3\) and \(b = 4\). So, the amplitude will be: \[A = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\] The amplitude of oscillation is 5 cm. The correct answer is (C) 5.

Key concepts

Amplitude of Oscillation

The amplitude of oscillation in a sinusoidal function reflects the maximum extent of movement from the equilibrium or central position. It can be visualized as the height from the middle value to the peak of the wave.
In the given equation for displacement, which is a combination of sine and cosine functions, the amplitude is determined by a mathematical method.
To find the amplitude of the oscillation described by the equation \( y = 3 \cos 2t + 4 \sin 2t \), we use the Pythagorean identity and the resultant vector approach:

• Identify constants from both sine and cosine terms; here, they are 3 and 4 respectively.
• Utilize the amplitude formula, \( A = \sqrt{a^2 + b^2} \).
• Substitute in the values: \( A = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

The amplitude is 5 cm, indicating the wave moves 5 cm above and below the midpoint during each oscillation.

Sinusoidal Functions

Sinusoidal functions form the foundation of trigonometric oscillations and can be expressed using sine or cosine wave equations that describe repetitive motions. Such functions illustrate how oscillations behave over time.
The displacement equation \( y = 3 \cos 2t + 4 \sin 2t \) combines cosine and sine components:

• Cosine function, \( 3 \cos 2t \), which influences the shape of the wave by its coefficient.
• Sine function, \( 4 \sin 2t \), which also contributes to it in a similar way.

These sinusoidal functions can be transformed into a single cosine or sine equation with a phase shift, showcasing their interchangeability and their role in modeling cyclic behaviors found in nature and technology.

Phase Difference

Phase difference, often denoted by \( \delta \), is crucial in describing how two sinusoidal functions with the same frequency relate in time. It can be thought of as the shift necessary for two waves to align or synchronize.
When combining different trigonometric terms like in our function \( y = 3 \cos 2t + 4 \sin 2t \), determining the phase difference helps express it neatly as one sinusoidal function. Typically:

• The formulas \( A \cos(2t - \delta) \) or \( A \sin(2t + \delta) \) can be used.
• The specific phase difference reflects how far one wave lags or leads relative to another.

Understanding phase difference provides insight into wave superposition, interference, and resonance, key phenomena in physics and engineering related to oscillatory motion.