A wrecking ball (weight =4800 N ) is supported by a boom, which may be assumed to be uniform and

A wrecking ball (weight =4800 N ) is supported by a boom,...

A wrecking ball (weight $=4800 \mathrm{N}$ ) is supported by a boom, which may be assumed to be uniform and has a weight of $3600 \mathrm{N}$. As the drawing shows, a support cable runs from the top of the boom to the tractor. The angle between the support cable and the horizontal is $32^{\circ},$ and the angle between the boom and the horizontal is $48^{\circ} .$ Find (a) the tension in the support cable and (b) the magnitude of the force exerted on the lower end of the boom by the hinge at point $P$

Key Concepts

Center of Mass of Uniform Bodies

For uniform bodies, the weight is effectively concentrated at the geometric center, or center of mass. This simplification allows for easier calculation of moments by treating the weight as a point force acting at the midpoint of the body.

Static Equilibrium

This concept refers to the condition where a body at rest has no net force or net moment acting on it. In problems like this, the system must satisfy both translational and rotational equilibrium, meaning that the sum of all forces in any direction and the sum of all moments about any point must be zero.

Force Decomposition

In mechanics, it is essential to resolve forces into their horizontal and vertical components, especially when forces are applied at angles. This allows for the application of Newton’s laws of equilibrium to accurately account for each directional influence in the analysis.

Moment (Torque) Equilibrium

This principle involves calculating the turning effect (moment or torque) of forces about a pivot point. For a system in equilibrium, the sum of the clockwise moments must equal the sum of the counterclockwise moments. This is critical when analyzing beams or levers, such as the boom in this problem.

Structural Hinge Reaction Forces

Hinge joints in structures transmit both vertical and horizontal reaction forces. Analyzing these forces involves considering the moments about the hinge and the equilibrium of the forces acting on the structure, which is crucial when determining the forces at connection points such as point P in the problem.

Transcript

00:01 Okay, so for part a, we can say that equation for the network about 0.8.

00:07 So if you consider the counterclockwise torque is positive memories, the khaleq wise torque is connected.

00:14 So have satchel question here.

00:16 So we know that will be here is the weight of a bone and and our here ihsaa length off the cable.

00:25 And we know wwor here is the weight of the wrecking ball.

00:30 And two years is the attention.

00:32 So if it was on a random here, we'll have, uh, such a quitting here.

00:39 So you can say that attention is able to cause i'm 48 degree turns that will be over two plus wwor over science 16 degree.

00:50 So we know the weight off boom is 3600 newton and the weight of a racquetball is 4800 newton.

01:01 So not as plugging better the equation to determine attention, which is you goingto because i'm 48 degree motivated by 3600 u turn do you want to plus for yea under newton and then over sign 16 degree and this will give us the attention is about ah, one point six kinds.

01:41 10 to the power of four you turn.

01:49 So now you also determine the magnitsky off the force extraordinary on the lower end when to determine the x squad components over such force.

01:59 So now let's take a look at an air force in the x direction, and this is able to f ax man's tico san 30 32 degree and we know allow me x that russian on then the net force tribunal zero because the system didn't move to anywhere...

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