(II) A 75-kg adult sits at one end of a 9.0-m-long board. His 25-kg child sits on the other end.

step 1 Chapter 9, problem 11. A 75 -kilogram adult sits at one end of a 9 -meter -long board, his 25 -k

(II) A 75-kg adult sits at one end of a 9.0-m-long board....

Key Concepts

Center of Mass

The center of mass is the point at which the entire mass of a system can be considered to be concentrated when analyzing translational and rotational motion. In balance problems, determining the center of mass of composite systems (like a board with additional objects on it) is crucial for locating the point where support or pivot forces should act to maintain equilibrium.

Torque Equilibrium

This principle states that for a system to be in rotational equilibrium, the sum of all torques (the product of a force and its perpendicular distance from the pivot point) acting on the system must equal zero. In problems involving balancing, setting the counterclockwise torques equal to the clockwise torques allows one to solve for the unknown pivot point or force magnitudes.

Uniform Mass Distribution

A uniformly distributed mass, such as that of a homogeneous board, has its mass evenly spread along its length, so its center of mass is located at its geometric center. This concept simplifies balance calculations because it allows the weight of the board to be treated as if it were concentrated at its midpoint.

Transcript

00:01 Chapter 9, problem 11.

00:03 A 75 -kilogram adult sits at one end of a 9 -meter -long board, his 25 -kilogram trouts sits on the other end.

00:09 A, where should the pivot be placed so that the board is balanced, ignoring the board's mass? and b, where should it be placed if it has a mass of 15 kilograms and its uniform? so, here is our free body diagram.

00:25 And if we look, we can see that we have the 75 -kilogram adult sitting at the left end.

00:37 So let's call big m75 kgs here, little m 25 kgs, kilograms.

00:48 All right.

00:54 And then we also have the mass of the board, right? but this is all, this is changing.

01:00 Oops, i'm going to put that somewhere else.

01:02 Just put it here.

01:05 Mb.

01:05 So it could be zero or it could be 15.

01:09 This is basically the difference between part a and part b, right? let's talk about the free body diagram real quick.

01:17 So the full length here, this is the full length of the long board, the 9 meter long board.

01:27 So l is going to be 9 .0 meters.

01:32 And the distance between, well, we don't really know where the pivot point is quite yet, right? because that's the question.

01:41 Where should the pivot be placed if it's balanced? that's why we have this thing here that i put, the variable pivot point.

01:48 So because we don't know what it is, we kind of have to guess a little bit first.

01:53 And then with our guessing and with our educated guesses, we can find where exactly this place is, depending on the masses.

02:03 And so this is a very important feature.

02:08 So i want you guys to note this is the variable pivot point, this x, it depends on the mass of, well, it depends on everybody, all the, every mass in the problem.

02:18 And you'll see that coming soon.

02:21 But we have a variable map.

02:22 We have a variable pivot point.

02:23 And what that means is we can talk about torques around this pivot point.

02:29 And then we can use these descriptions of the distance, like x and then l minus x...

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