Position-time relationship of a particle executing simple harmonic motion is given by equation
`x=2sin(50pit+(2pi)/(3))` where x is in meters and time t is in seconds.
What is the position of particle at t=1s ?

Options

A. `sqrt(2)m`
B. `sqrt(3)m`
C. 1m
D. 2 m

Answer

To find the position of the particle at \( t = 1 \) second, we will substitute \( t \) into the given equation of motion. The equation provided is:

\[ x = 2 \sin(50 \pi t + \frac{2\pi}{3}) \]

### Step 1: Substitute \( t = 1 \) into the equation

We start by substituting \( t = 1 \):

\[ x = 2 \sin(50 \pi \cdot 1 + \frac{2\pi}{3}) \]

This simplifies to:

\[ x = 2 \sin(50 \pi + \frac{2\pi}{3}) \]

### Step 2: Simplify the argument of the sine function

Next, we simplify \( 50 \pi + \frac{2\pi}{3} \). We can express \( 50 \pi \) in terms of a common denominator:

\[ 50 \pi = \frac{150\pi}{3} \]

So we have:

\[ 50 \pi + \frac{2\pi}{3} = \frac{150\pi}{3} + \frac{2\pi}{3} = \frac{152\pi}{3} \]

### Step 3: Use the periodic property of the sine function

The sine function has a periodicity of \( 2\pi \). We can reduce \( \frac{152\pi}{3} \) by subtracting \( 2\pi \) multiples until it falls within the range of \( 0 \) to \( 2\pi \):

\[ \frac{152\pi}{3} - 50\pi = \frac{152\pi}{3} - \frac{150\pi}{3} = \frac{2\pi}{3} \]

Thus:

\[ \sin\left(\frac{152\pi}{3}\right) = \sin\left(\frac{2\pi}{3}\right) \]

### Step 4: Calculate \( \sin\left(\frac{2\pi}{3}\right) \)

Now we need to find \( \sin\left(\frac{2\pi}{3}\right) \). The angle \( \frac{2\pi}{3} \) is in the second quadrant, where sine is positive. We can express it as:

\[ \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) \]

From trigonometric values, we know:

\[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

### Step 5: Substitute back to find \( x \)

Now substituting back into our equation for \( x \):

\[ x = 2 \cdot \sin\left(\frac{2\pi}{3}\right) = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} \]

### Final Answer

Thus, the position of the particle at \( t = 1 \) second is:

\[ x = \sqrt{3} \text{ meters} \]

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