> 7 0@bjbjUU 877)ljjjjjjj~:::8>:l:~>2;;";;;;;;$p `Mj;;;;;Cjj;;CCC;j;j;C;CCH9jj;;z)5~7:j@F0>CC~~jjjjLecture No. 3
Subject: Addition of Coplanar Forces
Objectives of Lecture:
To resolve a force into its rectangular components.
To express the forces using scalar notation.
To express the forces using Cartesian vector notation.
To determine the magnitude and orientation of resultant of the coplanar forces and to express the resultant force using scalar notation and Cartesian vector notation.
3.1 Resolution of a Force into its Rectangular Components
Each force may be resolved into its rectangular components, as shown in Fig.3.1:
F = Fx + Fy F2 = F2 x + F2 y
Fig.3.1
Scalar Notation of Forces
In scalar notation the components of a force can be represented either by positive scalars or by negative scalars, depending on the directional sense of the rectangular components of the given force.
For example in Fig.3.1,
The components of F can be represented by positive scalars ,Fx and Fy, since their sense of direction is along the positive x and y axes, respectively.
The components of F2 can be represented by a positive scalar,Fx ,and a negative scalar , Fy, since the sense of direction along the x axis is positive and the sense of direction along the y axis, is negative .
.
Cartesian Vector Notation of Forces
The components of a force may also be expressed in terms of Cartesian unit vectors: i, j, and k in the direction of the x, y, and z axes, respectively.
Cartesian unit vectors, i, j, and k have a dimensionless magnitude of unity, and their sense will be described analytically by a plus or minus sign depending on whether they are pointing along the positive or negative x or y or z axis
For example, the forces F and F2 , as shown in Fig.3.2, may be expressed as Cartesian vector, as follows:
F = Fx i + Fyj F2 = F2 x i +F2 y(j) = F2 x i  F2 y(j)
Fig3.2
Resultant of Coplanar Forces
The magnitude and direction of resultant of several coplanar forces may be determined using either the scalar notation or the Cartesian vector notation.
The following steps may be adopted:
Each force is first resolved into its rectangular components, i.e. into its x and y components.
Then the respective components are algebraically added.
The magnitude and direction of the resultant force is then determined by adding the resultants of the x and y components using the parallelogram law.
To understand the above procedure let us consider the example of addition of a system of coplanar forces, as shown in Fig.3.3(a):
Fig.3.3(a) Fig.3.3(b) Fig.3.3(c)
The forces F1, F2, and F3 are first resolved into their x and y components Fx and Fy ,as shown in Fig.3.3(b).
Then the respective components are added algebraically to determine the resultant components FRx and FRy, either using scalar notation or using Cartesian vector notation, as follows:
Using scalar notation
(!+) FRx = Fx FRx = F1x F2x + F3x
( !+ ) FRy = Fy FRy = F1y + F2y F3y
Using Cartesian vector notation
The coplanar forces F1, F2, and F3 may be expressed as Cartesian vector, as follows:
F1 = F1x i + F1yj
F2 = F2x i + F2yj
F3 = F3x i F3yj
The resultant vector is therefore
FR = F = F1 + F2 +F3
= F1x i + F1yj F2x i + F2yj + F3x i F3yj
= (F1x F2x + F3x )i + (F1y + F2y F3y )j
= (FRx)i + (FRy)j
Once the resultant components FRx and FRy are determined, they may be sketched along the x and y axes in their proper directions, and the magnitude of the resultant force, FR, and its direction, , can be determined from vector addition, as shown in Fig.3.3(c).
The value of FR may also be determined from the Pythagorean theorem, as follows:
FR = (F2Rx + F2Ry)0.5
Also, the value of , can be determined from trigonometry, as follows:
= tan1 FRy/ FRx 
Numerical Examples
Try the solved examples from Book (Examples 2 5 to 2 7, page 35 to 37)
Unsolved Examples
31 In each case resolve the force into x and y components. Report the results using Cartesian vector notation ( Fig. 3.1a,b,c,d).
Fig.3.1
32 Determine the magnitude of force F so that the resultant FR of the three forces is as small as possible (Fig3.2).
Fig.3.2
For a system of coplanar forces, as shown in (Fig.3.3),
Express F1, F2, and F3 as Cartesian vectors.
Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.
Fig.3.3
For a system of coplanar forces, as shown in (Fig.3.4),
a) Determine the magnitude and direction of force F1 so that the resultant force is directed vertically upward and has a magnitude of 800 N.
b) Determine the magnitude and direction measured counterclockwise from the positive x axis of the resultant force of the three forces acting on the ring A. Take F1 = 500 and = 20.
Fig.3.4
35 For a system of coplanar forces, as shown inFig.3.5
Express F1 and F2 as Cartesian vectors.
Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.
Fig.3.5
Try the other unsolved examples from Book
(Examples 2 41 to 2 58, page 39 to 42)
Lecture No. 3
Subject: Addition of Coplanar Forces
3.0 Objectives of Lecture:
To resolve a force into its rectangular components.
To express the forces using scalar notation.
To express the forces using Cartesian vector notation.
To determine the magnitude and orientation of resultant of the coplanar forces and to express the resultant force using scalar notation and Cartesian vector notation.
3.1 Resolution of a Force into its Rectangular Components
Each force may be resolved into its rectangular components, as shown in Fig.3.1:
F = Fx + Fy F2 = F2 x + F2 y
Fig.3.1
Scalar Notation of Forces
In scalar notation the components of a force can be represented either by positive scalars or by negative scalars, depending on the directional sense of the rectangular components of the given force.
For example in Fig.3.1,
The components of F can be represented by positive scalars ,Fx and Fy, since their sense of direction is along the positive x and y axes, respectively.
The components of F2 can be represented by a positive scalar,Fx ,and a negative scalar , Fy, since the sense of direction along the x axis is positive and the sense of direction along the y axis, is negative .
Cartesian Vector Notation of Forces
The components of a force may also be expressed in terms of Cartesian unit vectors: i, j, and k in the direction of the x, y, and z axes, respectively.
Cartesian unit vectors, i, j, and k have a dimensionless magnitude of unity, and their sense will be described analytically by a plus or minus sign depending on whether they are pointing along the positive or negative x or y or z axis
For example, the forces F and F2 , as shown in Fig.3.2, may be expressed as Cartesian vector, as follows:
F = Fx i + Fyj F2 = F2 x i +F2 y(j) = F2 x i  F2 y(j)
Fig3.2
Resultant of Coplanar Forces
The magnitude and direction of resultant of several coplanar forces may be determined using either the scalar notation or the Cartesian vector notation.
The following steps may be adopted:
Each force is first resolved into its rectangular components, i.e. into its x and y components.
Then the respective components are algebraically added.
The magnitude and direction of the resultant force is then determined by adding the resultants of the x and y components using the parallelogram law.
To understand the above procedure let us consider the example of addition of a system of coplanar forces, as shown in Fig.3.3(a):
Fig.3.3(a)
The forces F1, F2, and F3 are first resolved into their x and y components Fx and Fy ,as shown in Fig.3.3(b).
Fig.3.3(b)
Then the respective components are added algebraically to determine the resultant components FRx and FRy, either using scalar notation or using Cartesian vector notation, as follows:
Using scalar notation
(!+) FRx = Fx FRx = F1x F2x + F3x
( !+ ) FRy = Fy FRy = F1y + F2y F3y
Using Cartesian vector notation
The coplanar forces F1, F2, and F3 may be expressed as Cartesian vector, as follows:
F1 = F1x i + F1yj
F2 = F2x i + F2yj
F3 = F3x i F3yj
The resultant vector is therefore
FR = F = F1 + F2 +F3
= F1x i + F1yj F2x i + F2yj + F3x i F3yj
= (F1x F2x + F3x )i + (F1y + F2y F3y )j
= (FRx)i + (FRy)j
Once the resultant components FRx and FRy are determined, they may be sketched along the x and y axes in their proper directions, and the magnitude of the resultant force, FR, and its direction, , can be determined from vector addition, as shown in Fig.3.3(c).
Fig.3.3(c)
The value of FR may also be determined from the Pythagorean theorem, as follows:
FR = (F2Rx + F2Ry)0.5
Also, the value of , can be determined from trigonometry, as follows:
= tan1 FRy/ FRx 
Numerical Examples
Try the solved examples from Book
(Examples 2 5 to 2 7, page 35 to 37)
Unsolved Examples
31 In each case resolve the force into x and y components. Report the results using Cartesian vector notation ( Fig. 3.1a,b,c,d).
Fig.3.1
32 Determine the magnitude of force F so that the resultant FR of the three forces is as small as possible (Fig3.2).
Fig.3.2
For a system of coplanar forces, as shown in (Fig.3.3),
Express F1, F2, and F3 as Cartesian vectors.
b) Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.
Fig.3.3
For a system of coplanar forces, as shown in (Fig.3.4),
a) Determine the magnitude and direction of forceF1 so that the resultant force is directed vertically upward and has a magnitude of 800 N.
b) Determine the magnitude and direction measured counterclockwise from the positive x axis of the resultant force of the three forces acting on the ring A. Take F1 = 500 and = 20.
Fig.3.4
35 For a system of coplanar forces, as shown inFig.3.5
a) Express F1 and F2 as Cartesian vectors.
Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.
Fig.3.5
Try the other unsolved examples from Book
(Examples 2 41 to 2 58, page 39 to 42)
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