For the plate of Prob. 2.89 , determine the tensions in cables A B and A D knowing that the tens

For the plate of Prob. 2.89 , determine the tensions in...

Key Concepts

Static Equilibrium

This concept refers to a state where all forces acting on a body sum to zero, meaning there is no net force causing acceleration. In problems involving cables or supports, ensuring the structure is in static equilibrium requires that both the horizontal and vertical components of all forces cancel out.

Vector Decomposition

Vector decomposition involves breaking a force into its horizontal and vertical components. This method is essential when analyzing forces acting at angles, as it allows the application of equilibrium conditions along specific directions.

Resultant Force Consideration

When multiple forces act at a point, their sum—the resultant force—must follow the conditions of equilibrium. In cases where the resultant force is specified to be vertical, the horizontal components of all individual forces must cancel each other out, guiding the analysis of unknown tensions or forces.

Tension Analysis in Cable Systems

Cables support structures by transmitting tension forces along their lengths. Understanding how to calculate the individual tensions in cables involves applying equilibrium conditions and vector decomposition to determine how the force distribution sustains the overall balance of the system.

Transcript

00:03 In problem 96, we are trying to find the tension in cables ab and ad, given that the resultant force is only in the y direction.

00:14 We're also given the tension in ac as 54 newtons.

00:21 And so to solve this problem, we're going to have to create some equations looking at the components of the resultant force in the x, y, and z direction, and making ratios out of the lengths of each, tension vector so ad, ab, and ac, and that's where we'll start.

00:55 So length ab is negative 320 millimeters traveled in the i direction.

01:08 This is all from the diagram, negative 480 millimeters in the j direction, and 360 millimeters in the k direction.

01:24 We can solve for the total length of ab to be the square root of the sum of each component squared and we get it to be 680 millimeters.

01:52 We can use the ratio of each component of ab over its total length is going to be the same ratio as the force vector for the tension in the cable ab.

02:08 And so we'll do this for each cable ac is 450 millimeters in the eye direction and it travels naked 480 millimeters in the j direction and 360 millimeters in the k direction.

02:33 Using the same formula of the square root of the sum of the squares we can find the total length of ac and it is 750 millimeters.

03:07 Length ad is 250 millimeters in the i direction, also known as the x negative 480 millimeters in the j or y direction and negative 360 millimeters in the k direction using the same formula we find that ad is equal to 650 millimeters now we know that the resultant force in the x direction is equal to zero and the resultant force in the z direction is equal to zero because it's stated in the problem that the resultant force is only in the direction.

04:23 We can use this to set up equations to find the tension in ab and ad.

04:31 So in the x direction, we have the tension in ab's x component plus the tension in ac's x component and the tension in ad's x component, and we can set that all equal to zero.

04:49 Now we know the tension in a, b's component is going to be equal to the overall resultant tension in a, b times the ratio of the length of the segment ab, x component over its resultant length...

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