Name: finding cable tensions in 3d equilibrium
Uploaded: 2026-06-08T09:38:41.985Z
Duration: 1 min 26 s
Channel: JoVE

A 1000-newton chandelier is proposed for an exhibition hall, supported by three cables fixed to ceiling anchor points. The goal is to find the tension in each cable that keeps the chandelier in equilibrium.The setup uses a 3D coordinate system, with the chandelier at the origin, and each anchor point assigned coordinates.The position vectors from the origin to each anchor point give the cable directions. Dividing each position vector by its magnitude gives a unit vector without changing its direction. Since tension acts along the cable direction, each unit vector is then multiplied by an unknown scalar to give the corresponding tension vector.The chandelier&#x2019;s weight acts vertically downward and is shown as a force vector.Since the chandelier remains stationary, it is in static equilibrium. So, the net force is zero, and the tension vectors balance the weight.This gives a system of three equations by balancing the forces in the x, y, and z directions. Solving this system step by step gives the values of the scalar multipliers. Substituting these into the tension vectors gives the tension in each cable needed to hold the chandelier in place.

vectors in space: problem solving

A chandelier suspended by multiple cables can be analyzed using principles of three-dimensional static equilibrium. In this setup, a chandelier weighing 1000 N is positioned at the origin of a three-dimensional coordinate system, while three ceiling anchor points are fixed at known locations above it. Each cable connects the chandelier to one anchor point and transmits a tensile force along its length.To find out the forces in the cables, the spatial direction of each cable must first be identified. This is done by defining position vectors that extend from the chandelier at the origin to each ceiling anchor point. These vectors describe the orientation of the cables in space. Because the magnitude of a tension force depends only on direction and not on the actual cable length, each position vector is normalized to produce a unit direction vector. These unit vectors represent the direction along which each cable exerts force on the chandelier.The tension force in each cable is then expressed as a vector pointing along the corresponding unit direction vector, with an unknown magnitude. The chandelier is also subject to its weight, which acts vertically downward due to gravity. Since the chandelier remains stationary, it is in static equilibrium. This means that the vector sum of all forces acting on it must be zero. The three cable tension vectors together must exactly balance the downward weight. Resolving this condition into horizontal and vertical components yields a system of three independent equations, corresponding to the three spatial dimensions.By solving this system step by step, the magnitudes of the three cable tensions are obtained. These values represent the exact forces required in each cable to keep the chandelier fixed in position without motion or rotation.

Vectors in Space: Problem Solving

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Education

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Calculus

Cálculo

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Unidad

Vectors in Space

Vectores en el espacio

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Vectores en el espacio: resolución de problemas

finding cable tensions in 3d equilibrium

Three-dimensional static equilibrium can be used to find the tension in cables that support a chandelier. In this setup, the chandelier weighs 1000 N and sits at the origin of a 3D coordinate system. Three ceiling anchor points are located above it, and each cable runs from the chandelier to one anchor point.

To analyze the forces, the direction of each cable must first be described with a position vector. A position vector is a vector that points from the chandelier to an anchor point. These vectors show how each cable is oriented in space.

The length of a cable does not change the direction of the tension force. For that reason, each position vector is turned into a unit vector, which has length 1 and shows direction only. The tension in each cable is then written as a vector along its unit direction, with an unknown magnitude.

The chandelier also has a downward weight force because of gravity. Since the chandelier stays still, it is in static equilibrium. That means the vector sum of all forces must be zero. The three cable tension vectors must balance the 1000 N weight exactly.

To solve the problem, the force balance is split into horizontal and vertical components. This gives three independent equations, one for each spatial dimension. Solving the system step by step gives the tension in each cable. These values show the exact forces needed to keep the chandelier fixed in place without motion or rotation.

Finding Cable Tensions in 3D Equilibrium

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Three-dimensional static equilibrium can be used to find the tension in cables that support a chandelier. In this setup, the chandelier weighs 1000 N and sits at the origin of a 3D coordinate system. Three ceiling anchor points are located above it, and each cable runs from the chandelier to one anchor point. To analyze the forces, the direction of each cable must first be described with a position vector. A position vector is a vector that points from the chandelier to an anchor point. These...

key terms and concepts

Use 3D vectors and static equilibrium to calculate the tension in three cables supporting a 1000 N chandelier at the origin.

Finding Cable Tensions in 3D Equilibrium | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Cable Tensions in 3D Equilibrium

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