Three hikers are shown crossing a footbridge. Knowing that...

Three hikers are shown crossing a footbridge. Knowing that the wrights of the hikers at points $C, D$, and $E$ are $800 \mathrm{~N}, 700 \mathrm{~N}$, and 540 N , mepectively, determine ( $a$ ) the horizontal distance from $A$ to the line of action of the resultant of the three weights when $a=1.1 \mathrm{~m},(b)$ the value of $a$ so that the loads on the bridge supports at $A$ and $B$ are equal.

Key Concepts

Moment Calculation

A moment (or torque) is the product of a force and the perpendicular distance from a pivot or reference point to the line of action of the force. This concept is critical for analyzing how different forces cause rotation or bending in structures. Calculating moments about a specific point allows one to determine the location of the resultant force and assess the balance of the system.

Resultant Force and Center of Gravity

The resultant force is a single force that represents the combined effect of multiple forces acting on a body. Its point of application, often referred to as the center of gravity or centroid, is found by taking the weighted average of the positions of the individual forces based on their magnitudes. This concept is used to simplify complex force systems into one equivalent force acting at a specific location.

Static Equilibrium

Static equilibrium is the state in which the sum of all the forces and the sum of all the moments (torques) acting on a system are zero. This concept is fundamental in structural analysis and mechanics, ensuring that an object or system remains at rest. It underpins the method used to calculate reaction forces and determine effective load distributions in structures like bridges.

Load Distribution and Reaction Forces

Load distribution involves determining how applied forces are shared among the supports of a structure. By analyzing moments and ensuring static equilibrium, engineers can design supports to share the load appropriately, such as achieving equal load distribution in a bridge. This process involves setting up moment balance equations to solve for unknown variables that equalize the support reactions.

Transcript

00:01 Okay, in this problem we see a hiker going across a bridge.

00:05 And the hiker is stopping a fifth of the length across the bridge.

00:10 The bridge is supported on each end by two forces.

00:16 Those are f1 and f2.

00:18 These forces are counteracting the weight of the hiker and the weight of the bridge.

00:23 On the right side of the screen here, you'll see that i've written the weight of the bridge as wb and the weight of the hiker as wh.

00:31 For part a of this problem, we want to figure out what f2 or the force of the support on the near end is equal to.

00:41 And we know, or at least we would hope, that this bridge is not rotating.

00:47 It's staying still.

00:49 It's stable.

00:50 That means our torques have to all add up to zero.

00:57 And we're going to call the far end our center of rotation.

01:10 And since the bridge is not rotating, the torque at that far end adds up to zero.

01:21 And torque is the force applied times the lever arm.

01:26 So we have three forces that would be creating a torque around that far end.

01:32 The first is the weight of the bridge times our lever arm.

01:37 We're going to assume this bridge is approximately uniform and that the center of gravity of the bridge is at the middle.

01:47 So we're going to call that one half of the length of the bridge.

01:52 This would create a negative torque as it would push the bridge clockwise around that center of rotation.

02:01 And we have the weight of the hiker.

02:02 Again, this is a negative torque pushing the bridge clockwise.

02:10 And the hiker is four -fifths of the length from that far end...

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