Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

A vertical trapezoidal gate that is used as an automatic valve is heldshut by two springs attached to hinges located along edge $A B .$ Knowing that each spring exerts a couple of magnitude $1470 \mathrm{N} \cdot \mathrm{m}$, determine the depth $d$ of water for which the gate will open.

Physics 101 Mechanics

Chapter 9

Distributed Forces: Moments of Inertia

Section 2

Parallel-Axis Theorem and Composite Areas

Moment, Impulse, and Collisions

University of Washington

Hope College

University of Sheffield

McMaster University

Lectures

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

03:30

In physics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Given a force, F, applied for a time, t, the resulting change in momentum, p, is equal to the impulse, I. Impulse applied to a mass, m, is also equal to the change in the object's kinetic energy, T, as a result of the force acting on it.

11:03

A $4 \times 2$ -ft gate is…

04:47

A prismatically shaped gat…

02:25

Gate $A B C$ in Fig. $\mat…

05:41

A square flap gate is show…

01:45

04:41

we're told. A vertical trapezoidal gate that is used as an automatic valve is held shut by two springs attached to hinges located Longley along the edge, A B knowing that each spring exerts a couple of magnitude 1470 Newton meters determine the depth of water for which the gate open. Okay, so I've reproduced, uh, drawing here a little bit. I'm So here's our valve gate and some springs up here holding it shut. I'm obviously there's a wall here. And then there is, ah, water behind that wall. And if it gets deep enough and if the pressure gets great enough than this, the force of the pressure will overcome the force and the springs and the gate will open and we'll let water out. So we need to figure out a few things. We know that the depth. So we're looking for the depth of the water that will give this no open this. We know how far from the bottom the bottom of the gate is. We know thes dimensions, the wits, and we know the height of the gate. So the area I'm gonna break it into three pieces a two triangles and the rectangle. You could do it in terms of to try and goes here and here, too. But I thought it's easier for this. So, um, the area of this region is B two times each, and the area of one of these regions is 1/2 age times be one minus B 2/2. So this distance here is is this value Now we confined the centrally, and I've measured why, down from the level of the water and x from the center of the gate. So a times the central eight of the trap is oId is, um, a the area of the square times a century of the square, plus two times the area of this triangle times a century of this triangle. Because we got two of them here. No, we can figure out what why one and why to our in terms of the dimensions were given. So we know that the central right of the square is right at the center of the square. So we need to go down a distance. D back up it instance a and then back up a distance h over to, and that would locate p central to the square. Likewise, the century of each of these triangles. We have to go down d up a and then up 2/3 of H to locate the central right of each of the triangles. We can plug everything in and we wind up getting that. Why? Bar equals 0.55 plus D. So, um, let's see here. Yeah, so about 0.55 up from here. So, you know, somewhere in this region here now he can again if we calculate the area which we had to do to get this. Anyway, on Ben, we can calculate area moments about Theo X one axis. So this about the x axis. So this is the X axis here? Um, And for the square, its central It'll, um, area moment is this value. And then we have the area of the square times the distance from this X from the surface of the water to the central to the square squared and likewise for the triangles. We have the, um said Troy, it'll area moment about this entitle x axis is this. And then the area of the triangle is here, and then the distance from the surface of the water Top of the water to the central point of the triangle is this distance and we square. We can then get, um, the the total the total area moment about this axis for this trappers, right? Is this plus two times this. And if we do that, we can plug in all the numbers. We have everything in terms of D. So if we plot all that in, we get this expression here. 0.170 minus 0.572 times D plus 0.25202 times the square. And then we know the center of pressure is the area moment about the X axis, the area of the gate, then times the central, the central it'll the distance of the centrally of the gate. And we know that we can. We have all these things in terms of D, so we find that the senator of pressure is 0.3264 of minus 1.1 D plus B squared over minus 0.55 plus d. All right. So we have we have this expression now in terms of D and Now we need to do some, um, some mechanics to see free by diagram of the gate. So the gate is pivoted about this point and there's a couple of springs holding it shut with moments, and then we have the result in force from the pressure in the water acting at some distance l from this pivot point. Now we can figure out that if we go back and look at this picture here, we see that, um de equals, um, Let's see here. No, said Senator of Pressure. So the center of pressure is that, um d minus a minus h plus l s. So that's de minus a minus h plus L. So you know, in here. So that's we get the center of pressure in terms of D. And so then in terms of the senator pressure in terms of L. Because that's what we have in this. And so we know our egos. A plus h plus the center of pressure minus d. It's a now we got l in terms of d. Now for this thing to be an equilibrium to stay shut, Um, we need that that this ah the result in force here has to be less than two times the moment. So at the critical angle, the critical value of the critical depth, this will be inequality two times. The moment is 2940. And then we know the, um, resultant force from the pressure is the density of water times G, which is this value times the area of the trap, the trap door or the third gate and the sometimes the center of pressure. Now we can we have everything we need. All right, So we have this, we can plug that into here. We can plug this into here. We have that in terms of the we complied this into here, and we have that in terms of D. And so we got a function of D over here equals this and then we consol for D, and we get the d equals 2.8 six meters. Right? So, again, this is function of D here. Yes. Um, here we have a function of the from here, and so this is a function of the entity called some constant. And so then we can just sell for D. And I didn't go through all the algebra because it's kind of a long and tedious, but you should be able to do that right now and get this answer.

View More Answers From This Book

Find Another Textbook

02:55

Using Mohr's circle, determine for the area indicated the orientation o…

08:48

A series of small packages, each with a mass of $0.5 \mathrm{kg}$, are disch…

05:04

Determine the force $\mathbf{P}$ that must be applied to the toggle $C D E$ …

18:06

Knowing that at the instant shown the velocity of collar $A$ is$900 \mat…

03:52

Determine the components of the reactions at $A$ and $E$ if a $750-N$ force …

11:31

People with mobility impairments can gain great health and social benefits f…

03:09

A projectile is fired from point $A$ with an initial velocity $\mathbf{v}_{0…

09:27

The subway train shown is traveling at a speed of 30 mi/h when the brakes ar…

06:35

Knowing that $\mu_{k}=0.30$, determine the acceleration of each blockwhe…

04:27

A trailer weighing 2400 lb is attached to a 2900 -lb pickup truck by a ball-…