Question

A wire in a circular shape has \(10 \Omega\) resistance. The resistance per one meter is \(1 \Omega\) The resultant between \(A \& B\) is equal to \(2.4 \Omega\), then the length of the chord \(\mathrm{AB}\) will be equal to (A) \(2.4\) (B) 4 (C) \(4.8\) (D) 6

Short answer

The length of the chord AB is approximately equal to 2.19 meters. The closest option to this value is (A) 2.4.

Step-by-step solution

Find the radius of the circular wire

The total resistance of the circular wire is 10Ω, and the resistance per meter is 1Ω. If we let the length of the wire be L, we can calculate the length of the wire by this formula: Total resistance = Resistance per meter * Length of wire 10Ω = 1Ω * L L = 10 meters Now, we know the length of the wire is 10 meters. Since the wire forms a circle, we have: Circumference = 2πr 10 = 2πr Divide both sides by 2π: r = 10 / 2π

Find the length of the chord

Now that we have the radius of the circular wire, we can start finding the length of the chord AB which has a resistance of 2.4Ω between points A and B. Let's first find the length of the arc AB using the resistance formula: Resistance of AB = Resistance per meter * Length of arc AB 2.4Ω = 1Ω * Length of arc AB Length of arc AB = 2.4 meters Now, let's find the central angle θ formed by the arc AB (in radians): θ = (Length of arc AB) / r θ = 2.4/ (10 / 2π) θ = 0.48π Now that we have the central angle, we can find the length of the chord AB using the Law of Cosines. Considering a triangle formed by points O (center of the circle), A, and B: \(AB^2 = OA^2 + OB^2 - 2(OA)(OB)cos(θ)\) \(AB^2 = r^2 + r^2 - 2r^2 cos(0.48π)\) Plug in the value we obtained for r (10/2π meters) into the formula: \(AB^2 = (10 / 2π)^2 + (10 / 2π)^2 - 2(10 / 2π)^2 cos(0.48π)\) Calculate the value of AB²: \(AB^2 ≈ 4.8\) Now, obtain the value of AB by taking the square root: \(AB ≈ \sqrt{4.8}\) AB is approximately equal to 2.19 meters. Since 2.19 is not among the given options, the closest option is (A) 2.4.

Key concepts

Resistance per unit length

In the context of the exercise, resistance per unit length is a crucial concept. It tells us how much resistance a piece of wire exhibits for each unit of length. In our problem, this value is given as \(1 \, \Omega/\text{m}\). This means every meter of the wire offers a resistance of \(1 \, \Omega\). To find the total length of a wire when the total resistance is known, simply divide the total resistance of the wire by the resistance per unit length.

For example, given a total resistance of \(10 \, \Omega\), and knowing that each meter adds \(1 \, \Omega\) of resistance, we calculate:

• Total Length \(L = \frac{10 \, \Omega}{1 \, \Omega/\text{m}} = 10\, \text{m}\)

Understanding this will aid students in evaluating resistance factors for various wire configurations.

Chord length calculation

The task of determining the chord length involves several mathematical steps. First, we need to know the arc length, which correlates with the resistance between two points on the circle. In our exercise, the resistance between points A and B equates to \(2.4 \, \Omega\). Given the resistance per meter is \(1 \, \Omega/\text{m}\), the arc length between A and B is \(2.4 \, \text{m}\).

The length of the chord can then be derived using the Law of Cosines, which connects the chord and arc length with the circle's radius and central angle. Converting our solved problem using these relationships, we utilize both the calculated central angle and the radius to ultimately find the chord length.

Law of Cosines in circles

The Law of Cosines is often used in triangles, but it is equally applicable in circular segments, especially when calculating chord lengths. Once the central angle is known, and by treating radii as sides of a triangle, this method can compute the chord length within a circle.

In the problem, we form a triangle with vertices at the circle’s center (O) and the chord endpoints (A and B). Each of these sides is equal to the radius of the circle. The formula \[ AB^2 = r^2 + r^2 - 2r^2 \cos(\theta) \]relates the radius \(r\) and central angle \(\theta\), allowing for the computation of the chord \(AB\). Solving provides the square of the chord's length that can further be transformed to find the actual chord length.

Central angle in radians

When working with circular arcs, expressing angles in radians is often preferred due to its direct relationship with arc length and radius. Radians offer a simplified approach to angular measurement by equating them directly with arc sections of the circle.

In our exercise, the central angle \(\theta\) is calculated by the formula:\[ \theta = \frac{\text{Arc Length}}{\text{Radius}} \]By substituting \(\text{Arc Length} = 2.4 \, \text{m}\) and \(r = \frac{10}{2\pi} \, \text{m}\), we derive \(\theta = 0.48\pi\). Understanding this calculation is pivotal, as the central angle provides the missing link in equations relating to arc and chord lengths, completing the circle's geometrical puzzle.