A 20 -lb force is applied to the control rod A B as shown. Knowing that the length of the rod is

A 20 -lb force is applied to the control rod A B as shown....

Key Concepts

Moment (Torque)

The moment, or torque, of a force about a point is a measure of the rotational effect produced by the force. It is calculated as the product of the force magnitude and the perpendicular distance from the point of rotation to the line of action of the force. This concept is fundamental in statics and mechanics for determining the tendency of a force to cause a body to rotate about an axis or point.

Force Resolution into Components

Force resolution into horizontal and vertical components involves decomposing a force vector into perpendicular directions. This is achieved using trigonometric functions such as sine and cosine. The practice of resolving forces simplifies the analysis of complex problems by allowing the consideration of forces along mutually perpendicular axes, which is essential in statics and dynamics.

Trigonometric Decomposition

Trigonometric decomposition refers to the use of trigonometric functions to break down a vector into its components. When a force is applied at an angle, the cosine function is used to determine the horizontal component while the sine function is used for the vertical component. This process is crucial in calculating moments when the force is not acting perpendicular to the lever arm.

Lever Arm

The lever arm is the perpendicular distance from the axis or point about which the moment is calculated to the line of action of the force. Understanding and determining the lever arm is important because the moment of the force depends directly on this distance. Accurately computing the lever arm, especially when forces are applied at an angle, is critical for correct moment analysis in an engineering context.

Transcript

00:01 In this exercise, we have a force of 20 pounds that's applied on a rod that has the length of 9 inches, as shown here in the picture.

00:10 The red arrow represents the force that is applied at an angle, alpha, of 25 degrees from the horizontal.

00:22 And the rod makes an angle of 65 degrees with the horizontal, as shown here in the picture.

00:30 And our goal is to find what is the moment m that the force causes on the rod.

00:41 First, remember that the moment is equal to the vector force that joins the center of rotation to the point where the force is applied.

00:56 So in this case, this vector that i just drew here is r.

01:02 It's the vector that connects these two points and then we have to take r vector f so this is the vector product between r and f so in order to find the momentum m we have to find what are the vectors r and f so notice that r is equal to the length of the rod actually minus the length of the rod times the cosine of 65 degrees in the x -dd and i'm setting a coordinate system such that the x direction points to the right and the y direction points upward minus plus l times sine of 65 j so and using that l is equal to nine inches you have to this is minus nine inches times the cosine of 65 i plus 9 inches times the sign of 65 j so r is equal to minus 3 .8 i plus 8 .16 j this is the value of r i i'm going to highlight it and now we can find the force vector okay notice that the force vector as a negative y component that is equal to, actually let's start with the x component, so i have the force of 20 pounds times the cosine of 25 i minus 20 pounds times the sign of 25 pounds times the sign of 25 j.

03:08 So the force is equal to 18 .13 i minus 8 .45j pounds.

03:22 And they went up here of the vector r, i forgot to write, it's inches...

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