CE 201 - Statics

Lecture 10

FORCE SYSTEM RESULTANTS

So far, we know that for a particle to be in equilibrium, the resultant of the force system acting on the particle is to be equal to zero. That is:

F = 0In a later stage, we are going to see that F = 0 is required for a particle to be in equilibrium, but it is not sufficient to make that particle in equilibrium since we know that forces could create momentum which will tend to rotate the particle.

In the coming lectures, moments about a point or axis will be discussed.

CROSS PRODUCT

If we have two vectors A and B, then the CROSS PRODUCT of the two vectors is:

C = A B

(C is equal to A cross B)

The magnitude of C is:

C = AB sin The direction of C is perpendicular to

the plane containing A and B.

Then

C = A B = ( AB sin ) uC

AB

C

uC

AB sin

Laws of Operation

(1) A B B A BUT A B = - B A

AB

C = A B

AB

-C = A B

Laws of Operation

(2) Multiplying by a Scalar

a (A B) = (a A) B = A (aB) = (A B) a

(3) A (B + D) = (A B) + (A D)

Cartesian Vector Formulation

Use C = AB sin () to find the cross product of two vectors.

ij = k; ik = -j ; ii = 0

jk = i; ji = -k; jj = 0

ki = j; kj = -i; kk = 0

ij = ij sin 90 = (1)(1)(1) = 1

ii = ii sin (0) = 0

If A and B are two vectors, then:A B = (Ax i + Ay j + Az k) (Bx i + By j + Bz k)

= AxBx (i i) +AxBy (i j) + AxBz (i k)

+ AyBx (j i) +AyBy (j j) + AyBz (j k)

+ AzBx (k i) +AzBy (k j) + AzBz (k k)

= (AyBz – AzBy) i – (AxBz – AzBx) j + (AxBy – AyBx) k

This equation can be written as:

i j k

A B = Ax Ay Az

Bx By Bz

This is nothing but a determinant whose first raw consists of unit vectors i, j, and k and its second and third raws represent the x, y, and z components of the two vectors A and B.

MOMENT OF A FORCE – SCALAR FORMATION

A moment of a force about a point or axis is an indication or measure of the tendency of the force to rotate the particle about that point or axis.

Fx is perpendicular to the handle of the wench and located a distance (dy) from point (O). Fx tends to turn the pipe about the z-axis. The turning effect will be larger if Fx or dy gets larger. This turning effect is sometimes referred to as TORQUE, but most frequently it is called MOMENT (M0)z

y

x

z

O

Fxdy

If Fz is applied to the wrench at dy from O. Fz will tend to rotate the pipe about the x-axis and (MO)x is produced.

The force Fz and distance dy lie in the (x-z) plane, which is perpendicular to the x-axis ( the moment axis)

y

x

z

O

Fz

dy

If Fy is applied to the wrench, no moment will be produced at point (O) since the line of action of Fy passes through point (O).

y

x

z

O Fy

dy

Generally, the moment of force F about point (O) which lies in a plane perpendicular to the moment axis, has a magnitude and direction. Therefore, the moment is a vector quantity.

Magnitude

MO = F d

Where d is called the moment arm, which is perpendicular distance from (O) to the line of action of force F.

F d

O

Direction

F

dO

F

dO

MO

MO

Resultant Moment of a System of Coplannar Forces

F1, F2, and F3 lie in the (x-y) plane. Their distances from point (O) are d1, d2, and d3, respectively.

+ MO = F d

y

x

z

F1

F2

F3d1

d2d3

O

Assignment 3

3-52, 54, 57, 60, 64, 65