Name: cable tension and force balance
Uploaded: 2026-06-08T09:38:24.138Z
Duration: 1 min 30 s
Channel: JoVE

A five-thousand-newton steel beam is lifted by two identical, weightless cables. The goal is to find the tension in each cable that keeps the beam in equilibrium.Equilibrium means the beam remains at rest, with zero net horizontal and vertical force. The setup is symmetric, and each cable makes a sixty-degree angle with the horizontal beam.The tension in each cable is a vector that acts along the cable.Since each cable pulls in the x and y directions, each tension is resolved into horizontal and vertical components using the unit vectors i and j.The beam&#x2019;s weight acts downward in the negative j direction. For equilibrium, the two cable tensions must balance this weight, so their vector sum equals the negative weight vector.Comparing the horizontal components shows that the positive contribution from one cable cancels the negative contribution from the other. This means both cables have the same tension magnitude.Comparing the vertical components shows that the sum of the upward components equals five thousand newtons.Solving this equation gives the tension magnitude. Substituting it into the component expressions gives the full tension vector for each cable, which keeps the beam stable and at rest.

vectors in engineering applications

A steel beam supported by two identical cables provides a practical example of static equilibrium. The beam has a downward weight of 5000 N, while the two cables support it from opposite sides. Because the arrangement is symmetric, each cable makes the same angle of 60&#xB0; with the horizontal beam and carries the same tension.In equilibrium, the beam remains completely at rest. This means that the total horizontal and vertical forces must both be zero. Each cable pulls along its own direction, so the tension force in each cable can be separated into horizontal and vertical parts. The horizontal parts act in opposite directions, canceling one another because the cables are identical and symmetrically positioned.The vertical components of the cable tensions act upward and, together, balance the beam&#x27;s downward weight. Since the beam weighs 5000 N, each cable must provide half the required upward support, accounting for the cable angle. Solving the force balance shows that the tension in each cable is approximately 2887 N.The tension vectors have equal upward components and horizontal components that act in opposite directions. One cable pulls upward and to the left, while the other pulls upward and to the right. Their combined effect exactly balances the beam&#x27;s weight and prevents any motion.This example shows how vector components and equilibrium conditions are applied in mechanics to analyze forces acting on suspended structures such as beams, bridges, and support systems.

Vectors in Engineering Applications

https://cloudfront.jove.com/files/ftp_upload/40629/40629_article_thumb_40629.png

Education

JoVE Core

Calculus

Unit

Vectors in Space

KEY TERMS AND CONCEPTS

cable tension and force balance

Cable tension and force balance are shown with a steel beam held by two identical cables. The beam has a downward weight of 5000 N. The cables support it from opposite sides. Because the setup is symmetric, each cable makes a 60° angle with the horizontal beam and carries the same tension.

The beam stays completely at rest, so it is in static equilibrium. Static equilibrium means the total horizontal force and the total vertical force are both zero. Each cable pulls in its own direction, so its tension force can be split into horizontal and vertical components. The horizontal parts point in opposite directions and cancel because the cables are identical and placed symmetrically.

The vertical components of the two cable tensions act upward. Together, they balance the beam’s weight. Since the beam weighs 5000 N, each cable provides half of the needed upward support when the cable angle is taken into account. Solving the force balance gives a tension of about 2887 N in each cable.

The tension vectors have equal upward components. Their horizontal components act in opposite directions. One cable pulls upward and to the left, while the other pulls upward and to the right. Together, they exactly balance the beam’s weight and keep it from moving. This example shows how vector components and equilibrium are used to analyze forces in suspended structures such as beams, bridges, and support systems.

Cable Tension and Force Balance

introduction to vectors

Vectors in Three-Dimensional Motion

Vectors describe motion in three-dimensional space with both size and direction. They are useful for quantities such as displacement and velocity, since those measurements need more than just a number. A vector is shown as an arrow from one point to another. The arrow’s length gives the magnitude, and its direction shows the way the motion points. In three dimensions, a vector is written in terms of its x-, y-, and z-components. These components are often shown with angle brackets, such as ⟨x,...

Cable tension and force balance are shown with a steel beam held by two identical cables. The beam has a downward weight of 5000 N. The cables support it from opposite sides. Because the setup is symmetric, each cable makes a 60° angle with the horizontal beam and carries the same tension. The beam stays completely at rest, so it is in static equilibrium. Static equilibrium means the total horizontal force and the total vertical force are both zero. Each cable pulls in its own direction, so...

dot product of two vectors

Vector Projection and the Dot Product

The dot product connects vectors, angles, and projections in vector algebra. It combines two vectors to produce a scalar, which is a single number. This makes it useful in math and in physics, especially for work. For vectors a = ⟨a1, a2, a3⟩ and b = ⟨b1, b2, b3⟩, the dot product is found by multiplying matching components and adding them: a · b = a1b1 + a2b2 + a3b3. The operation is commutative, so the order does not matter. It is also distributive over vector addition and works with scalar...

cross product and its geometry

Cross Product, Area, and Torque

The cross product connects vector direction, area, and rotation in three-dimensional space. When two non-zero vectors are not parallel, they define a unique plane and form a parallelogram. Their cross product gives a third vector that is perpendicular to that plane. The direction of this new vector is found with the Right-Hand Rule. Point the fingers of your right hand along the first vector and curl them toward the second vector. Your thumb shows the direction of the result. This convention...

lines in space

Vector Form for a Line in Space

A line in three-dimensional space can be described with a vector equation. This method uses a known point on the line and a direction vector. It is a useful way to model straight paths in space. The transcript uses a surveying laser beam as an example. A laser beam from a surveying instrument can be treated as a line across a construction site. The beam starts at a known point, called P₀, whose position vector is r₀. The beam also has a direction vector v, which is parallel to its path. Any...

planes in space

Plane Equations from a Point and Normal

A plane in three-dimensional space can be described by one point on the plane and a normal vector. The normal vector is perpendicular to the surface, and it sets the plane’s orientation. In geometry, this gives a clear way to model a flat surface in space. The transcript uses a sloped glass panel on a building façade as a real-world example. Let r₀ be the position vector of a known point on the panel, such as the base mounting point. Let n be the normal vector to the panel. Any other point r...

cylinders in three-dimensional space

Modeling Cylindrical Surfaces in 3D

Cylindrical surfaces in three-dimensional space come from moving a two-dimensional curve in a straight line. As the curve slides, it leaves behind translated copies that build a surface. These copies form parallel rulings, which are straight lines in the same fixed direction. This idea helps describe some 3D shapes with simple equations in Cartesian coordinates. A cylindrical surface is often shown by an equation that leaves out one of the three variables. For example, y = x^2 does not include...

quadric surfaces

Shapes of Quadric Surfaces in 3D

Quadric surfaces are smooth three-dimensional shapes described by second-degree equations in x, y, and z. The way the squared and linear terms are arranged determines the surface. The main quadric surfaces are ellipsoids, cones, paraboloids, and hyperboloids. An ellipsoid is a closed surface that appears when all three variables are squared with the same sign. It is bounded and symmetric about its principal axes. Cross-sections taken along those axes produce circles or ellipses. A standard...

vectors in 2d: problem solving

Crosswind Vector Navigation in 2D

Crosswind vector navigation in 2D shows how wind changes an airplane’s path. A plane flew due north at 180 km/h in still air, but after 30 minutes it was 80 km off course. Its track was about 5 degrees east of north, which shows that a crosswind pushed it away from the intended route. The plane’s ground path was diagonal, not straight north. That means its effective ground speed was reduced to 160 km/h and shifted slightly east. By studying the displacement from the planned path, the wind’s...

vectors in space: problem solving

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...

key terms and concepts

Use vector components to analyze a steel beam in static equilibrium. See how two identical cables share the load and balance a 5000 N weight.

Cable Tension and Force Balance | 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

Cable Tension and Force Balance

Description

Transcript

Tags