The displacement of the point of a wheel initially in contact with the ground when the wheel rolls forward quarter revolution where perimeter of the wheel is `4pi m`, is (Assume the forward direction as x-axis)

Options

A. `sqrt((pi+2)^(2)+4) "along tan"^(-1) 2/pi` with x-axis
B. `sqrt((pi+2)^(2)+4) "along tan"^(-1)(2)/(pi-2)` with x-axis
C. `sqrt((pi+2)^(2)+4) "along tan"^(-1)(2)/(pi)` with x-axis
D. `sqrt((pi+2)^(2)+4) "along tan"^(-1)(2)/(pi-2)` with x-axis

Explanation

To solve the problem, we need to determine the displacement of a point on the wheel that is initially in contact with the ground after the wheel rolls forward by a quarter revolution. Given that the perimeter of the wheel is \(4\pi\) meters, we can follow these steps:

### Step 1: Calculate the radius of the wheel
The perimeter (circumference) of the wheel is given by the formula:
\[ \text{Perimeter} = 2\pi R \]
where \(R\) is the radius of the wheel. We know the perimeter is \(4\pi\) meters, so we can set up the equation:
\[ 2\pi R = 4\pi \]
Dividing both sides by \(2\pi\):
\[ R = 2 \text{ meters} \]

### Step 2: Determine the distance traveled during a quarter revolution
A quarter revolution corresponds to \(90^\circ\) of rotation. The distance traveled by the wheel in this case is:
\[ \text{Distance} = \frac{1}{4} \times \text{Perimeter} = \frac{1}{4} \times 4\pi = \pi \text{ meters} \]

### Step 3: Analyze the position of the point after rolling
Initially, the point in contact with the ground is at the bottom of the wheel. After rolling forward by a quarter revolution, this point will move to the right by \(\pi\) meters along the x-axis and will also rise to a height equal to the diameter of the wheel (which is \(2R = 4\) meters) because it moves from the bottom to the top of the wheel.

### Step 4: Determine the coordinates of the point after rolling
Initially, the point is at the coordinates \((0, 0)\). After rolling forward by \(\pi\) meters, the new coordinates of the point will be:
\[ \text{New position} = (\pi, 4) \]

### Step 5: Calculate the displacement
Displacement is the straight-line distance from the initial position to the final position. The initial position is \((0, 0)\) and the final position is \((\pi, 4)\). The displacement can be calculated using the Pythagorean theorem:
\[ \text{Displacement} = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(\pi - 0)^2 + (4 - 0)^2} = \sqrt{\pi^2 + 16} \]

### Final Answer
The displacement of the point of the wheel after rolling forward by a quarter revolution is:
\[ \sqrt{\pi^2 + 16} \text{ meters} \]

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