To loosen a frocen value, a force 𝐅 of magnitude 70 lb is aple to the handle of the valve. Knowi

To loosen a frocen value, a force 𝐅 of magnitude 70 lb is...

Key Concepts

Torque (Moment) Calculation

Torque, also known as moment, is a measure of the rotational effect of a force acting about a point or axis. It is calculated as the cross product of the position (or lever arm) vector and the force vector, meaning its magnitude is the product of the force and the perpendicular distance from the line of action of the force to the pivot. Breaking down the moment into its components allows for solving problems involving equilibrium and rotation.

Lever Arm

The lever arm is the shortest (perpendicular) distance from the axis of rotation or the point of interest to the line along which the force acts. It is a fundamental concept in mechanics because the moment is directly proportional to both the force applied and the lever arm distance. Correct identification of the lever arm is essential for accurate moment or torque calculations.

Force Decomposition in Components

When a force is applied at an angle, it is often necessary to decompose it into its horizontal and vertical (or other orthogonal) components to simplify analysis. This process helps in relating the applied force to its effects in specific directions, especially when calculating moments about particular axes. Understanding the resolution of a force using angles allows for accurate determination of its contribution to rotational and translational motion.

Angular Relationships in Forces

Angles such as those between the direction of the applied force and reference axes or lever arms are critical in bridging geometric and algebraic representations of mechanics. They facilitate the transformation of the force's magnitude into its directional components and aid in determining the effectiveness of the force in generating a moment around an axis. Recognizing these angular relationships is key to solving problems that involve forces applied at various orientations.

Transcript

00:01 The goal of this problem is to find our angle d and to find our angle phi.

00:08 So to start with, i'm going to write out our components of the force vector.

00:16 So it's going to be 70 times a cosine of 25 degrees times the cosine of phi, i plus negative sign of 25 degrees, j, plus the cosine of the cosign of 5, j, plus the cosign of 5.

00:52 Of 25 degrees times the sign of phi.

01:03 Okay, and we can simplify this down 63 .4 .4 .1 cosine of phi i.

01:22 Minus 29 .583 j plus 63.

01:37 441 times the sign of sine of phi, k, and this is our force vector.

01:51 Next we're going to find our position vector, our r vector.

01:55 So from looking at the figure, we can figure out that our r vector, our position vector, is going to be negative 4i plus 11j minus d, d, d, k, where k is the value that we're trying to find.

02:21 Now to find our moment, our moment about the origin, we can do our vector cross our force vector.

02:37 And that's going to look like, it's going to look like this.

02:42 So we write our force vector, write our force vector at the bottom here, the components of our force vector.

02:48 So 63 .4 .1 times cosine of five.

02:55 And then we have negative 29 .583, and then we're going to have 63 .441 times the sign of phi.

03:22 And in our middle row, we're going to have our precision factor, which is negative 4, 11, and negative d.

03:39 And then we'll have our i, j, and k components in the top row.

03:51 And solving for the determinant of this, solving for the cross product of this, of this, we'll end up with 697 .83 times the sign of phi minus 295 minus 295 .5 .5 times the sign of five, minus 29 .5 .5 .5...

Please give Ace some feedback