A beam is supported at its ends by supports which are 12 metres apart. Since the load is connected at its centre, there is a deflection of e centre and the deflected beam is in the shape of a parabola. How far from the centre is the deflection 1 cm?

Options

A. 2`sqrt(2)`
B. 3`sqrt(2)`
C. 2`sqrt(3)`
D. 2`sqrt(6)`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Problem
We have a beam supported at its ends, 12 meters apart, with a load applied at the center causing a deflection that takes the shape of a parabola. The maximum deflection at the center is 3 cm, and we need to find out how far from the center the deflection is 1 cm.

**Hint:** Visualize the beam and the parabola formed due to the deflection.

### Step 2: Set Up the Coordinate System
Let’s set up a coordinate system where the vertex of the parabola is at the origin (0,0). The x-axis will run horizontally along the beam, and the y-axis will represent the deflection.

**Hint:** The center of the beam corresponds to the origin of our coordinate system.

### Step 3: Identify Key Points
The beam is 12 meters long, so the supports are at (-6, 0) and (6, 0). The maximum deflection occurs at the center (0, 3 cm).

**Hint:** Remember that the coordinates of the points on the beam will help us define the parabola.

### Step 4: Write the Equation of the Parabola
The general form of a parabola that opens upwards is given by the equation:
\[ x^2 = 4ay \]
Here, \( a \) is the distance from the vertex to the focus.

### Step 5: Find the Value of \( a \)
At the maximum deflection point (6, 3 cm), we substitute into the parabola equation:
\[ 6^2 = 4a \cdot 3 \]
\[ 36 = 12a \]
\[ a = \frac{36}{12} = 3 \]

Thus, the equation of the parabola becomes:
\[ x^2 = 12y \]

**Hint:** This equation describes the shape of the beam's deflection.

### Step 6: Find the Coordinates for 1 cm Deflection
We need to find the points where the deflection is 1 cm. The deflection at these points will be:
\[ y = 3 \text{ cm} - 1 \text{ cm} = 2 \text{ cm} \]

So, we need to find \( x \) when \( y = 2 \):
\[ x^2 = 12 \cdot 2 \]
\[ x^2 = 24 \]
\[ x = \pm \sqrt{24} = \pm 2\sqrt{6} \]

**Hint:** The symmetry of the parabola means that the deflection will be the same on both sides of the center.

### Step 7: Conclusion
The distance from the center (0,0) to the points where the deflection is 1 cm is \( 2\sqrt{6} \) meters.

**Final Answer:** The distance from the center where the deflection is 1 cm is \( 2\sqrt{6} \) meters.