2 - Couple, Resultants, and 3D Force Systems

Question 1

What is a couple?

Answer 1

A couple is the moment produced by two equal, opposite, and noncollinear forces.

E.g. the two parallel vector forces F and -F

Question 2

What are equivalent couples?

Answer 2

Equivalent couples refer to when the moment of two couples are equal when the two force couples lie in two parallel planes or the same plane and they have equal moments of the same sense.

Question 3

What is the effect of a force acting on a body?

Answer 3

The effect of a force acting on a body is the tendency to push or pull the body in the direction of the force and to rotate the body about any fixed axis which does not intersect the line of the force.

Question 4

What is a force-couple system?

Answer 4

A force-couple system is when the tendency of a force to push or pull and rotate a body in a direction or around a fixed axis is replaced by an equal parallel force and a couple to compensate for the change in the moment of the force.

Question 5

What are resultants?

Answer 5

The resultant of a system of forces is the simplest force combination that can replace the original forces without altering the external effect on the rigid body to which the forces are applied.

Question 6

What is an equilibrium of a body?

Answer 6

Equilibrium of a body is the condition in which the resultant of all forces acting on the body is zero.

When the resultant of all forces on a body is not zero, the acceleration of the body is obtained by equating the force resultant to the product of the mass and acceleration of the body.

Question 7

What is the moment-arm rule?

Answer 7

The moment arm or lever arm is the perpendicular distance between the line of action of the force and the center of moments.

Question 8

What is the formula for a moment?

Answer 8

M = F*d

or

M = F(a+d) - Fa

Where:

d,a = perpendicular distances
F = Force

Question 9

What is the moment of a couple?

Answer 9

M = F*d

Question 10

What do couples do?

Answer 10

Consider the action of two equal and opposite forces F and -F a distance d apart. These two forces cannot be combined into a single force because their sum in every direction is zero. Their only effect is to produce a tendency of rotation. The combined moment of the two forces about an axis normal to their plane and passing through any point such as O in their plane is the couple M.

Question 11

What is the expression for the moment of a couple using vector algebra?

Answer 11

M = r_A x F + r_B x (-F)
= (r_A - r_B) x F
= r x F

Question 12

How can we represent couple vectors as clockwise or counterclockwise?

Answer 12

Since the couple vector M is always perpendicular to the plane of the forces which constitute the couple, in the two-dimensional analysis we can represent the senes of a couple vector as clockwise or counterclockwise.

Question 13

What are equivalent couples?

Answer 13

Changing the values of F and d does not change a given couple as long as the product Fd remains the same. Likewise, a couple is not affected if the forces act in a different but parallel plane. Therefore, you can produce numerous configurations of equivalent couples which still produce the same moment M.

Question 14

What is the expression of a resultant?

Answer 14

R = F1 + F2 + F3 + … = Sigma F

Rx = Sigma Fx

Ry = Sigma Fy

R = ((Sigma Fx)^2 + (Sigma Fy)^2)^0.5

theta = tan^-1 (Ry/Rx)

Question 15

What is the algebraic method of finding a resultant?

Answer 15

1. Move all force to a chosen point using the force-couple method.
2. Add all forces at the chosen point to form the resultant force R, and add all couples to form the resultant couple Mo.
3. Find the line of action of R by requiring R to have a moment of Mo about point O. Using Mo = R*d

Question 16

What is the principle of moments?

Answer 16

The Principle of Moments states that when a body is balanced, the total clockwise moment about a point equals the total anticlockwise moment about the same point.

In other words, The moment of a resultant force about any point equals the sum of the moments of the original forces of the system about the same point.

Question 17

What are the expressions of rectangular components of force?

Answer 17

Fx = Fcos(thetax)
Fy = Fcos(thetay)
Fz = F cos(thetaz)

F = (Fx^2 + Fy^2 + Fz^2)^0.5

F = Fxi + Fyj + Fzk

F = F( icos(thetax) + jcos(thetay) + kcos(thetaz))
= F ( li + mj + nk )
n_F = li + mj + nk

Question 18

How do you find the scalar components of a force given two points on a line of action?

Answer 18

Given two points A and B, you find vector force F:

F = Fn_F = F * ( AB / |AB| )

Where:

n_F is the unit vector of F

Question 19

How do you find the scalar components of a force given two angles on a line of action?

Answer 19

Assuming theta and phi are known. theta being on xy plane and phi being on z plane.

Break down the components first into xy and z planes.

Fxy = Fcos(phi)
Fz = Fsin(phi)

Then into x and y:

Fx = Fxy cos(theta) = F cos(phi)cos(theta)
Fy = Fxy sin(theta)  = F cos(phi)sin(theta)

Question 20

How do you find the rectangular components for F using dot product?

Answer 20

Given two vectors P and Q:

P*Q = |P| |Q| cos(alpha)

Question 21

What is the magnitude of the projection of F in a direction n?

Answer 21

Fn = F*n

Which is a scalar quantity. To find the vector quantity, multiply its scalar value by unit vector n:

Fn = (F*n)n

Question 22

What is a wrench?

Answer 22

A wrench is a force and couple system in which the force and couple are parallel.

The perpendicular part of the resultant moment can be replaced by sliding the resultant force along the line perpendicular to the plane of the resultant force and resultant couple.

What remains after this operation is a wrench that has a moment equal to the component of the resultant couple parallel to the force.