The equation of motion for an oscillating particle is given by `x = 3 sin 4pi t + 4 cos pi t` where x is in mm and t is in second. The particle

Options

A. starts its motion from rest
B. starts its motion with an initial velocity `u = 12 pi mm//s`
C. has its maximum acceleration equal to `80 pi^(2) mm//s^(2)`
D. has its maximum velocity equal to `20pi mm//s`

Explanation

To solve the problem, we need to analyze the motion of the oscillating particle given by the equation:

\[ x = 3 \sin(4\pi t) + 4 \cos(\pi t) \]

where \( x \) is in mm and \( t \) is in seconds. We will find the amplitude, initial velocity, maximum velocity, and maximum acceleration.

### Step 1: Rewrite the Displacement Equation
The displacement equation is given in both sine and cosine forms. We can convert it into a single sine function using the following identity:

\[ R \sin(\omega t + \phi) = A \sin(\omega t) + B \cos(\omega t) \]

where \( A = 3 \) and \( B = 4 \).

### Step 2: Calculate the Amplitude
The amplitude \( R \) can be calculated using:

\[ R = \sqrt{A^2 + B^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{mm} \]

### Step 3: Determine the Phase Angle
The phase angle \( \phi \) can be found using:

\[ \tan(\phi) = \frac{B}{A} = \frac{4}{3} \]

Thus,

\[ \phi = \tan^{-1}\left(\frac{4}{3}\right) \]

### Step 4: Find the Velocity
The velocity \( v(t) \) is the derivative of the displacement \( x(t) \):

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(3 \sin(4\pi t) + 4 \cos(\pi t)) \]

Using the chain rule:

\[ v(t) = 3 \cdot 4\pi \cos(4\pi t) - 4\pi \sin(\pi t) \]

### Step 5: Calculate Initial Velocity
To find the initial velocity \( v(0) \):

\[ v(0) = 12\pi \cos(0) - 4\pi \sin(0) = 12\pi \]

### Step 6: Find Maximum Velocity
The maximum velocity occurs when \( \cos(4\pi t) = 1 \):

\[ v_{\text{max}} = 12\pi \, \text{mm/s} \]

### Step 7: Find Acceleration
The acceleration \( a(t) \) is the derivative of the velocity:

\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(12\pi \cos(4\pi t) - 4\pi \sin(\pi t)) \]

Using the chain rule:

\[ a(t) = -12\pi \cdot 4\pi \sin(4\pi t) - 4\pi^2 \cos(\pi t) \]

### Step 8: Calculate Maximum Acceleration
The maximum acceleration occurs when \( \sin(4\pi t) = 1 \):

\[ a_{\text{max}} = 48\pi^2 + 4\pi^2 = 52\pi^2 \, \text{mm/s}^2 \]

### Summary of Results
- Amplitude: \( 5 \, \text{mm} \)
- Initial Velocity: \( 12\pi \, \text{mm/s} \)
- Maximum Velocity: \( 12\pi \, \text{mm/s} \)
- Maximum Acceleration: \( 52\pi^2 \, \text{mm/s}^2 \)