Problem

Some miners wish to remove water from a mine shaf…

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Problem 83 Hard Difficulty

The hydraulic system of a backhoe is used to lift a load as shown in Figure 11.48 . (a) Calculate the force $F$ the slave cylinder must exert to support the 400 -kg load and the $150-\mathrm{kg}$ brace and shovel. (b) What is the pressure in the hydraulic fluid if the slave cylinder is 2.50 $\mathrm{cm}$ in diameter? (c) What force would you have to exert on a lever with a mechanical advantage of 5.00 acting on a master cylinder 0.800 $\mathrm{cm}$ in diameter to create this pressure?

Answer

(a) $F_{\text {slave }}=1.38 \times 10^{4} \mathrm{N}$
(b) $P=2.81 \times 10^{7} \mathrm{N} \mathrm{m}^{-2}$
(c) $F=283 \mathrm{N}$

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Physics 101 Mechanics

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Chapter 11

Fluid Statics

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Video Transcript

told. The hydraulic system of backhoe is used to lift a load as shown in figure 11.48 So I've reproduced the backhoe here. There's a force from the hydraulic system acting here. That's a distance L one from our pivot point at a There's the weight of the arm, which, actually with center of mass. That's a distance l two from a. And then there's the way of the payload in the bucket. That accident distance L three from the pivot point. No, the point of a is fixed, so there's some of the moments about that. Point is zero. And if we take counterclockwise is positive, we wind up with our moment. Balance is the force from the hydraulic system times L and the force from the hydraulic system X perpendicular to the moment arm for the weight and the load. Do not at public a killer, so we need to take the side beginning to get the component that is perpendicular by taking a sign of this angle. So we get the weight of the arm times. A sign of that angle is the component in this direction. Kind 02 It's in the clap eyes direction. So we have a minus sign, likewise, for out for the weight of the payload. But now we have all three. And again we have a minus. Sign here because it's acting to rotate the bucket, the arm clockwise so we can solve for elf. And we get just this simple equation for our the force acting on hydraulic cylinder. We can start plugging in some numbers, so we have the have the length ratio l one over to over one. The weight of the arm is its mass times gravity and then sign the angle that the arm is acting, um, relative to the article is 30 degrees. And so we have our three over l one times the weight of the payload, which is 400 kilograms times t and then again, sign of 30 degrees. Now, if you plug everything in, we wind up with a force of 13.8 killings. If we figure out what the pressure in the slaves under is, well, that's just the force acting on the bucket arm divided by the area of the slave so under. So we have given that the the radius of the diameter of the slave cylinder is 2.5 centimeters, so it's one over its area. Is this value here multiplying it times the force and we get 28.1 killer pass scales. Now we're told that we have a a lever arm pushing on the master cylinder and that this lever arm has a mechanical advantage of five, meaning that this distance here is gonna be 1/5 of this distance here. You know the pressure in our hydraulic fluid In here? There's just 28.1 killer Pascal's so we can figure out what forces acting on our are master cylinder. And that is simply the pressure in the hydraulic fluid times the area of the master cylinder were given that the diameter of the master cylinder it is 2.8 centimeters. And so you plug that in and we get that their force acting on the master cylinder is 14. Point is one is 1413 Newtons, And so then, if we have a mechanical advantage of five, the force acting at our at the other end of the lever arm is 1/5 of that, and that is 283 Newtons and in fact this guy should be drawn the other way

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