A 39 -ft length of railroad rail of weight 44 lbft is lifted by the tongs shown. Determine the f

A 39 -ft length of railroad rail of weight 44 lbft is lifted...

Key Concepts

Static Equilibrium

In mechanics, static equilibrium refers to the state of an object or system in which there are no net forces or moments acting on it. This ensures that the object remains at rest or moves with constant velocity. The concept is fundamental in analyzing structures and mechanisms to determine how forces balance to maintain stability.

Moment (Torque) Analysis

The moment (or torque) is the rotational effect produced by a force acting at a distance from a pivot point. Understanding moments is crucial in mechanics because it allows for the calculation of turning effects about points, ensuring that all moments balance around any pivot for equilibrium.

Distributed Load and Center of Gravity

A distributed load, such as a uniformly weighted rail, can be modeled by a single equivalent force acting at the center of gravity of the distribution. This simplification enables the analysis of the load’s impact on a structure by reducing a continuous load into a single point force for calculating moments and reaction forces.

Reaction Forces

Reaction forces are the forces exerted by supports or connections in response to applied loads. They are critical for understanding how structures and mechanisms like tongs transmit and balance loads, ensuring that all forces in a system are in equilibrium.

Transcript

00:01 We need to find the forces exerted at the point d and the point f whenever this rail is lifted by the tongs.

00:13 So we have three different free body diagrams to represent the different points on the machine.

00:21 So we have a free body diagram for the rail, a diagram for the a junction, and also a diagram for the bdf tongue.

00:34 So to start solving this problem, we're going to first find our weight w of the rail.

00:42 And we're given the length of the rail and also its unit weight.

00:50 So to find the weight w, w is going to be equal to 39 times 44.

01:11 And that'll give us a weight of 17, 16 pounds.

01:22 So now that we have the weight, we can now take the moment, about this point e on this free body diagram.

01:38 So you can say that the sum of the moments about the point e is equal to zero, and the formula for that is going to be negative w times 0 .8 plus f of y, the y component at the point of the force, times 1 .6 equal to 0.

02:15 Solving this for f of y and plugging in w, we get f of y is equal to 1716 times 0 .8 divided by 1 .6, and this will give us a force of 8 and 58 pounds.

02:47 So next we're going to calculate an angle theta.

02:52 So we're going to calculate angle theta from this free body diagram and this angle theta also appears on the tonne free body diagram.

03:07 So we can solve for that angle theta by taking the tangent inverse of 6 over 9 .6 and this will go as an angle of 32 degrees.

03:33 So we can now say that using the free body diagram in the middle we can say that the sum of all forces in the x direction is equal to zero.

03:46 So we have that sum of all forces in the x direction is equal to zero, which in terms of that free body diagram, we can write as force of ac times a cosine of theta minus the force ab times the cosine of theta is equal to zero.

04:22 And from this equation, we can rewrite this equation as f of ac is equal.

04:32 To f of ab.

04:41 And next we're going to use the same free body diagram.

04:46 We're now going to say that some of all forces in the y direction is equal to zero, which we can write as w minus the force ac times the sine of theta minus the force av times the sign of theta is equal to zero.

05:23 So we can substitute the known terms.

05:28 And solve for the unknown f of ab.

05:41 Since in this equation we can replace the f of ac with f of a b from the previous expression.

05:53 So after plugging in the known terms, we get 1716 minus f of ab times 2 times the sine of 32 degrees is equal to 0.

06:18 And solving for the unknown f of a b, we get a force that is equal to 0...

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