Components of a Force
Forces acting at some angle from the coordinate axes can be
resolved into mutually perpendicular forces called components.
The component of a force parallel to the x-axis is called the xcomponent, parallel to y-axis the y-component, and so on.
Components of a Force in XY Plane
Given the slope of the line of
action of the force as v/h (see
figure to the right)
,
Components of a Force in 3D Space
Given the direction cosines of
the force:
Given the coordinates of
any two points along the
line of action of the force (in
reference to the figure
shown, one of the points is
the origin):
Let d = distance OB
Vector Notation of a Force (Also called Rectangular
Representation of a Force)
Where λ is a unit vector. There are two cases in determining λ; by
direction cosines and by the coordinates of any two points on the
line of action of the force.
Given the direction cosines:
Given any two points P1(x1, y1) and P2(x2, y2) on the line of action
of the force:
Where i, j, and k are unit vectors in the direction of x, y and z
respectively.
,
,
Note:
Also note the following:
Fx  F cos  x  F
dy
dx
d
, Fy  F cos  y  F
, FZ  F cos  Z  F Z
d
d
d
Thus,
In simplest term
Problem 001
Determine the x and y
components of the forces shown
below in Fig P-001
Solution 001
Rectangular Representation
From the above vector notation of forces, Fx is the coefficient
of i and Fy is the coefficient of j.
Problem 002
Compute the x and y
components of each of the four
forces shown in Fig. P-002.
Solution 002
Rectangular Representation
The coefficients of i and j from the vector notations are the
respective x and y components of each force.
Problem 003
Which of the following correctly
defines the 500 N force that passes
from A(4, 0, 3) to B(0, 6, 0)?
A. 256i - 384j + 192k N
B. -256i + 384j - 192k N
C. -384i + 192j - 256k N
D. 384i - 192j + 256k N
Solution 003
From the figure
Unit vector from A to B:
Rectangular representation of F:
Answer: B
Problem 004
Referring to Fig. 1.4,
determine the angle between
vector A and the y-axis.
A. 65.7°
B. 73.1°
C. 67.5°
D. 71.3°
Solution 004
Answer: D
Problem 005
Find the components in the x, y,
u and v directions of the force P
= 10 kN shown in Fig. P-005
Solution 005
Problem 006
The force P of magnitude 50 kN is acting at 215° from the x-axis.
Find the components of P in u 157° from x, and v negative 69°
from x.
Solution 006
From the figure:
Check:
(okay!)
By Sine Law:


answer
answer
Problem 007
A block is resting on an incline of slope 5:12
as shown in Fig. P-007. It is subjected to a
force F = 500 N on a slope of 3:4. Determine
the components of F parallel and
perpendicular to the incline.
Solution 007
answer
answer
Problem 008
A force P = 800 N is shown in Fig. P-008.
a.
b.
c.
d.
Find the y-component of P with respect to x and y axis.
Find the y'-component of P with respect to x' and y' axis.
Find the y-component of P with respect to x' and y axis.
Find the y'-component of P with respect to x and y' axis.
Solution 008
Part (a)
y-component of P with respect to x
and y axis
answer
Part (b)
y'-component of P with respect to x'
and y' axis
Part (c)
y-component of P with respect to x'
and y axis
The figure formed is in the shape
of equilateral triangle. Thus,
answer
Part (d)
y'-component of P with respect to x and y'
axis
By Sine Law


answer
Problem 009
The body on the 30° incline in Fig. P-009 is acted upon by a force
P inclined at 20° with the horizontal. If P is resolved into
components parallel and perpendicular to incline and the value of
the parallel component is 1800 kg, compute the value of the
perpendicular component and that of P.
Solution 009
Click here to show or hide
the solution
Perpendicular component
answer
Value of P
answer
Problem 010
The triangular block shown
in Fig. P-010 is subjected to the
loads P = 1600 lb and F = 600
lb. If AB = 8 in. and BC = 6 in.,
resolve each load into
components normal and
tangential to AC.
Solution 010
Problem 011
The magnitude of vertical
force F shown in Fig. P016 is 8000 N. Resolve F
into components parallel
to the bars AB and AC.
Solution 011
By Sine Law:
answer

answer
Problem 012
If the force F shown in Fig.
P-017 is resolved into
components parallel to the
bars AB and BC, the
magnitude of the
component parallel to bar
BC is 4 kN. What are
the magnitudes of F and its
component parallel to AB?
Solution 012
By Sine law
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
answer
answer