The specialized plumbing wrench shown is used in confined...

The specialized plumbing wrench shown is used in confined areas (e.g., under a basin or sink). It consists essentially of a jaw $B C$ pinned at $B$ to a long rod. Knowing that the forces exerted on the nut are equivalent to a clockwise (when viewed from above) couple with a magnitude of 135 Ib in, determine ( $a$ ) the magnitude of the force exerted by pin $B$ on jaw $B C,(b)$ the couple $M_{0}$ that is applied to the wrench.

Key Concepts

Free-Body Diagram

A free-body diagram is a graphical representation of a body isolated from its surroundings, illustrating all external forces and moments acting upon it. This tool is essential in the analysis of mechanics problems, as it simplifies the process of setting up equations for equilibrium and solving for unknown forces or torques.

Equilibrium in Rigid Bodies

Equilibrium in rigid body mechanics occurs when the sum of all forces and the sum of all moments (or torques) acting on a body equal zero. Ensuring equilibrium is crucial for the analysis of static systems, as it allows us to solve for unknown forces and moments acting within structural or mechanical systems.

Pin Joint Reactions

Pin joint reactions refer to the forces transmitted through a pin connection, which ideally allows rotation without transmitting any moment. Analyzing these reactions is important in understanding how forces are distributed in mechanisms and ensuring that joints operate within their designed load capacities.

Torque (Couple)

Torque (or a couple) is the rotational equivalent of force, representing a pair of equal and opposite forces whose effect is to produce rotation without causing any net linear force. It is calculated by multiplying the magnitude of the force by the perpendicular distance from the point about which the moment is taken, emphasizing the lever arm's role in rotational dynamics.

Lever Arm (Moment Arm)

The lever arm, or moment arm, is the perpendicular distance from the line of action of a force to the pivot or axis of rotation. This concept is central in determining the effectiveness of a force in generating a moment, highlighting how distance amplifies the impact of a given force in rotational systems.

Transcript

00:01 We need to find the value of the force that the point b exerts on the jaw bc.

00:08 And we also need to find the value of this moment about the rod.

00:14 So i drew three different free body diagrams.

00:18 One for the jaw, one for the nut, and one for the rod.

00:23 And notice that i added in a point d.

00:27 This is where, in the given figure, this is where the nut makes contact.

00:34 With the rod.

00:36 So these two points here are both connecting.

00:42 So to start solving this problem, we're going to first look at this free body diagram to the left and take the sum of moments about the point b.

00:53 So the sum of moments about point b is equal to 0, which you can write as the x component of the 4c times 1 .5 minus the y component of the 4c times 5 over 8, is equal to 0.

01:18 And solving this expression for c of y, we can get a value of c of y that is equal to 2 .4 times c of x.

01:35 And now we're going to say that the sum of forces in the y direction is equal to zero.

01:42 And from the free body diagram, we can say that b of y minus c of y is equal to zero, which we can write as b of y is equal to zero, which we can write as b of y is is equal to c of y.

02:00 And we can do the same thing in the x direction.

02:03 So we take the sum of forces, the x direction is equal to zero, write that c of x minus b of x is equal to zero.

02:15 And we can also say that b of x will be equal to the c of x.

02:24 So we're now going to look at the three body diagram of the nut.

02:31 So from the nut, we can say that the sum of forces in the x direction is equal to 0.

02:40 And we can write it as c of x minus the x component of the force at d is equal to 0...

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