When a wood shelf of mass 6.6 kg is fastened inside a slot in a vertical support as shown in Fig

step 1 So once again, welcome to a new problem. This time we're given a shelf. It's a wood shelf. And i

When a wood shelf of mass 6.6 kg is fastened inside a slot...

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical illustration used to visualize the forces, moments, and reactions acting on a single body isolated from its surroundings. It is essential in statics to identify all external forces for applying the equilibrium equations. In the context of this problem, drawing a free-body diagram involves representing the gravitational force acting at the center of mass and the two points of contact between the shelf and the support slot.

Static Equilibrium

Static equilibrium is the condition in which a body is at rest or moving with constant velocity, meaning that the sum of all forces and the sum of all moments (torques) acting on the body are zero. This principle is used to set up equations that allow for the determination of unknown forces and torques, as seen in the problem involving the shelf.

Reaction Forces

Reaction forces are the forces exerted by supports, connections, or surfaces that prevent movement of a body under load. In the problem, the support slot provides two vertical reaction forces at different contact points, which, together with the gravitational force, ensure that the shelf remains in static equilibrium.

Torque (Moment of Force)

Torque, or moment of force, is the measure of the tendency of a force to rotate an object about some axis or pivot, calculated as the product of the force and its lever arm. In this context, the torque exerted by the support is calculated about the left end of the shelf, accounting for the distance from the force's line of action to the pivot point.

Lever Arm

The lever arm is the perpendicular distance between the line of action of a force and the axis or point about which the moment is calculated. Correctly identifying lever arms is crucial for solving problems involving torque in static equilibrium, as it directly influences the magnitude of the calculated torque.

Transcript

00:01 So once again, welcome to a new problem.

00:04 This time we're given a shelf.

00:06 It's a wood shelf.

00:08 And it happens to have a groove like that.

00:13 And so the groove is holding the shelf, and it goes all the way up until inside.

00:23 So you can assume this is like a wall.

00:32 Holding it.

00:34 The thickness of the shelf happens to be given us three centimeters and also the entire length of the shelf from the wall up until the right edge is given us 32 centimeter.

00:55 And then also the depth of the shelf going into the wall is equivalent to 0 .0 centimeters.

01:06 So this is the information that's given.

01:09 It's fastened inside the slot.

01:13 So this is a slot that holds the shelf.

01:17 You can think of it like a slot like that that holds the shelf.

01:23 The first thing you want to do is on top of that.

01:28 Well, on top of that, we're given that the mass of the shelf is equivalent to 6 .6 kilograms.

01:37 We want to find the, we need a free body diagram.

01:47 You know, that's the first thing we need.

01:51 Off the shelf, assuming that you're going to have two vertical forces at this point.

01:58 This is a special point where the groove is, where the slot is.

02:03 The second thing about the problem is that we need to find this.

02:12 Size of all the forces.

02:18 So the size of all the forces involved in the problem in keeping equilibrium.

02:29 That's one thing we want to do.

02:31 And also the next thing we want to deal with is we want to find the talk due to the slot supports.

02:45 Remember the shelf is being supported by slots, well it's a single slot.

02:50 So we want to see what's the talk so the first thing we'll do is go ahead and have a free body diagram there's a groove like that and there's a wall a shelf popping out from the groove all the way up until right side and we do have the weight in the middle of the shelf acting downwards m .g this distance is still 32 centimeters doesn't change.

03:31 Just like the distance that the shelf moves into the slot is given us two centimeters.

03:44 These are force acting downwards that's supposed to balance out the shelf.

03:53 So, you know, first of all, this force is pushing upwards.

03:59 So there's a force pushing upwards right here on the right of the slot so i'm gonna call that f r and then there's another force pushing downwards to balance out that force and i'm gonna call it fl for the left left force that's your free body diagram the other thing that happens is the the right force the right force this one right here fr is responsible for the talk that balances out, balances out the weight.

04:54 Okay, so that's what f .r is responsible for.

04:58 The second part, the other thing we know is that fl has no talk, but gives equilibrium to f .r.

05:23 Okay, it has no talk, but it gives equilibrium to fr.

05:28 If we're going to take the talk around this point right here, if we're going to take the talk around this point right here, that's what's going to happen.

05:43 Fl is not going to have a talk, but then we'll have a clockwise talk due to mg going that way, and that's a negative talk.

05:54 And then we're going to have a counterclockwise.

05:58 Talk due to fr and that one is going that way.

06:04 So that's going to be a positive talk.

06:06 Those are the pieces of information that we're dealing with at this point.

06:11 The sum of forces in this system in the wide direction is zero and then also the sum of talks is also going to be zero.

06:25 So since the sum of talks is zero, we take the force to the right and multiply that by 2 centimeters.

06:34 It's a positive talk and then the weight is going to give a negative torque.

06:40 So if you go back, you see that the distance from this point up until the end of the slot is equivalent to 32 centimeters plus 2 centimeters.

06:59 But then the weight acts in the middle.

07:03 So it splits this distance in the middle.

07:05 So that has to be divided by two.

07:08 That gives the 17 centimeters.

07:10 Going back, the perpendicular distance of the weight is 17 centimeters.

07:16 And since the system is in equilibrium, we make all of that equals to zero.

07:21 Our purpose is to solve for this right force.

07:25 So then we have fr being equivalent to mg, 17 centimeters, over 2 centimeters so f r becomes equals to the mass which is 6 .6 kilograms times the acceleration due to gravity which is 9 .81 meters second squared and then we do have these units canceling out 17 over 2 that's equivalent to 8 .5 it's just a unit so nothing changes so the force we're getting due to the reaction on the right becomes equivalent to 549 .8 newtons.

08:18 That's the force we're dealing with right...

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