Moment of a Force - Mechanics Quiz

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Questions and Answers

What is the formula used to calculate the moment about the y axis for a force applied at an angle?

$M_y = F(d_1 + d)$

$M_y = F(d cos θ)$ (correct)

$M_y = F(d sin θ)$

$M_y = F(d × θ)$

In scalar analysis, how is the moment arm defined?

The angle of application of the force

The length of the force line

The total distance from the force to the point of rotation

The perpendicular distance from the line of action of the force to the axis (correct)

What is the generalized expression for the moment about any axis a?

$M_a = F(d sin θ)$

$M_a = F(d + θ)$

$M_a = Fd_a$ (correct)

$M_a = r × F$

Which expression correctly uses the cross product to find the moment about a point O?

$M_o = r × F$ Signup and view all the answers

How is the component of the moment along the y axis determined using vector analysis?

$M_y = j • (r × F)$ Signup and view all the answers

Which statement accurately describes the scalar triple product in this context?

It finds the projection of the moment onto the a axis. Signup and view all the answers

What does the notation $u_a· (r × F)$ represent?

The projection of the moment onto axis a Signup and view all the answers

What is the result of the determinant form of the moment calculated using Cartesian vectors?

All components are included in a single expression. Signup and view all the answers

What does a positive scalar value for $M_a$ indicate about its direction along the axis?

It acts in the same direction as $u_a$. Signup and view all the answers

Which equation correctly expresses the moment $M_a$ when using vector analysis?

$M_a = u_a · (r × F)$ Signup and view all the answers

What is the relationship between the perpendicular distance $d_a$ and the moment $M_a$?

$M_a = Fd_a$ Signup and view all the answers

If the resultant moment $M_y$ is calculated as $-230 lb.ft$, what does this signify?

The moment acts in the negative y direction. Signup and view all the answers

When determining $M_a$ as a Cartesian vector, which of the following expressions is accurate?

$M_a = M_a u_a$ Signup and view all the answers

What does the right-hand rule help to determine in the context of moments?

The sense of twist about the axis. Signup and view all the answers

Which statement is true about a force that is parallel to a coordinate axis?

It produces no moment about that axis. Signup and view all the answers

What is the effect on the moment $M_z$ when the calculation results in $-80 lb.ft$?

It acts in the -z direction. Signup and view all the answers

What is the formula used to determine the moment $M_{AB}$ produced by the force F?

$M_{AB} = u_{AB} · (r × F)$ Signup and view all the answers

Which of the following correctly describes the vector $u_{AB}$?

$u_{AB} = 0.8944i + 0.4472j$ Signup and view all the answers

What does a positive moment result indicate about the direction of $M_{AB}$?

It aligns with the direction of $u_{AB}$ Signup and view all the answers

What would be the consequence of defining the axis AB using a unit vector directed from B toward A?

The sign of $M_{AB}$ would change to negative. Signup and view all the answers

In the moment calculation $M_{OA} = u_{OA} (r × F)$, what does r represent?

The position vector from the axis to the line of action of the force Signup and view all the answers

What is the magnitude of the moment $M_{AB}$ calculated in Example 4.8?

80.50 N·m Signup and view all the answers

Which component of the moment expression $M_{AB}$ contributes the negative sign when using $-u_{AB}$?

The unit vector $u_{AB}$ Signup and view all the answers

What is the significance of the determinant used in calculating $M_{AB}$?

It allows for the calculation of the cross product. Signup and view all the answers

Study Notes

Moment of a Force about a Specified Axis

Determining the moment of a force about a specific axis is important in mechanics
Scalar analysis involves finding the perpendicular distance (moment arm) from the axis to the line of action of the force
Moment = Force × moment arm (Ma = Fda)
Vector analysis uses the dot product to find the component of the moment along the specified axis
Moment along axis a = unit vector along a • (position vector × force vector)(Ma = ua • (r × F))

Scalar Analysis

The moment arm is the perpendicular distance from the axis to the line of action of the force
The moment about the axis is calculated by multiplying the force by the moment arm
The direction of the moment is determined by the right-hand rule

Vector Analysis

The moment of the force about a point on the axis is calculated using the cross product (Mo = r × F)
The component of the moment along the specified axis is found by taking the dot product of the unit vector along the axis with the moment about the point (Ma = ua • (r × F))
The variables used in the equation:
- ua represents the unit vector along the axis
- r represents the position vector from a point on the axis to a point on the line of action of the force
- F represents the force vector
The resulting scalar indicates the direction of the moment along the axis

Important Points

Determining the moment of a force about a specific axis requires finding the perpendicular distance from the force line of action to the axis
Vector analysis involves the cross product of the position vector and force vector, followed by a dot product with a unit vector specifying the axis direction (Ma= ua • (r × F))
A negative scalar result indicates the moment is in the opposite direction of the unit vector
The moments along each axis can be found, and Cartesian components can be expressed as a vector. (Ma = Maua)

Examples

Examples provided show how to apply scalar and vector analyses to determine the resultant moment of forces about specific axes
They illustrate the calculation of moments about axes (x,y,z)
Using examples, the direction of the resultant moments is either positive or negative accordingly.

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Description

Test your understanding of the moment of a force about a specified axis. This quiz covers both scalar and vector analysis techniques used in mechanics. You will learn how to calculate moments using the moment arm and cross product.