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Understanding Physics Topics is essential for solving complex problems in many fields, including engineering and medicine.
Are Parallel Forces Acting in the Same Direction?
Forces acting in the same direction are called like parallel forces. \(\vec{F}_1\), \(\vec{F}_2\) and \(\vec{F}_3\) are examples of like parallel forces [Fig.]. When two forces act in opposite directions, they
constitute a pair of unlike parallel forces. \(\vec{F}_4\) and \(\vec{F}_5\) are two unlike parallel forces [Fig.]. The straight line along which a force acts is called the line of action of that force. For example, AB is the line of action of forces \(\vec{F}_1\) and \(\vec{F}_2\), CD is the line of action of force
Resultant of Two Like Parallel Forces
Let \(\vec{F}_1\) and \(\vec{F}_2\) be a pair of like parallel forces acting at points A and B [Fig.]. Their resultant has to be calculated. O is a point on the plane of the forces. Line OX is drawn perpendicular to the lines of action of \(\vec{F}_1\) and \(\vec{F}_2\). OX is taken as the x-axis. OY is the y-axis on the same plane.
Since \(\vec{F}_1\) and \(\vec{F}_2\) do not have any components along x-axis, the resultant \(\vec{R}\) (say) will also have no component along x -axis and its line of action will be perpendicular to the line OX, i.e., parallel to the lines of action of \(\vec{F}_1\) and \(\vec{F}_2\). Hence, component of \(\vec{R}\) along y-axis is,
R = F1 + F2 …….. (1)
Hence, the magnitude of the resultant is the sum of the magnitudes of the forces and its direction will be the same as the direction of the parallel forces.
To find the line of action of R, let us assùme that \(\vec{R}\) acts at point C. \(\vec{R}\) is the resultant of \(\vec{F}_1\) and \(\vec{F}_2\) – it implies that the effect of forces \(\vec{F}_1\) and \(\vec{F}_2\) together, should be the same as that of \(\vec{R}\) acting alone.
Hence, the algebraic sum of moments due to \(\vec{F}_1\) and \(\vec{F}_2\) about the point O should be equal to the moment of \(\vec{R}\) about O,
or, F1 × OA + F2 × OB = R × OC
or, F1(OC – AC) + F2(OC + CB) = (F1 + F2) × OC
or, F1 × AC = F2 × BC
or, \(\frac{F_1}{F_2}\) = \(\frac{B C}{A C}\) ………. (2)
Hence, the line of action of the resultant divides the distance AB in two parts in the inverse ratio of the magnitudes of the forces.
Suppose the line A’B’ is obtained by joining any two points on the lines of action of F1 and F2. Obviously, the line of action of the resultant divides it in two parts in the inverse ratio of the magnitude of the forces \(\vec{F}_1\) and \(\vec{F}_2\).
Position of the point C can also be determined in terms of points A and B. In Fig., if OA = x1, OB = x2 and OC = x, then, taking moments about the point O of the forces \(\vec{F}_1\), \(\vec{F}_2\) and resultant \(\vec{R}\) we get,
F1x1 + F2x2 = R × x = (F1 + F2)x
∴ x = \(\frac{F_1 x_1+F_2 x_2}{F_1+F_2}\) …… (3)
Resultant of Two Unlike Parallel Forces
Let \(\vec{F}_1\) and \(\vec{F}_2\) be two unlike parallel forces acting at points A and B as shown in Fig. Let F1 > F2. Suppose \(\vec{R}\) is the resultant of these two forces. Then, R = F1 – F2
Clearly, the resultant acts along the direction of the greater force.
Taking moments about point O and from the definition of resultant we get,
F1 × OA – F2 × OB = R × OC
or, F1(OC + CA) – F2(OC + CB) = (F1 – F2) × OC
or, \(\frac{F_1}{F_2}\) = \(\frac{B C}{A C}\)
and x = \(\frac{F_1 x_1-F_2 x_2}{F_1-F_2}\) ……. (4)
Hence, the line of action of the resultant divides externally the distance of separation of the two forces in the inverse ratio of their magnitudes.
Resultant of Three or More Parallel Forces
When a body is acted upon by the parallel forces \(\vec{F}_1\), \(\vec{F}_2\), \(\vec{F}_3\) ,……. simultaneously, the resultant force \(\vec{R}\) can be obtained by framing an equation similar to equation (1) [Fig.].
R = F1 + F2 + F3 + ……. = \(\sum_i F_i\) ……. (5)
Any one of the forces may be taken as positive. Correspondingly, forces in the same direction as this force are positive and those in the opposite direction are negative. If the forces are coplanar, the distance of the line of action of the resultant from any point O on that plane is,
x = \(\frac{F_1 x_1+F_2 x_2+F_3 x_3 \cdots}{R}\) or, x = \(\frac{\sum_i F_i x_i}{\sum_i F_i}\) …….. (6)
where x1, x2, x3, … are the perpendicular distances of the point O from the lines of action of the forces \(\vec{F}_1\), \(\vec{F}_2\), \(\vec{F}_3\),……. etc.
The moment of the resultant about the point O can be positive or negative, The direction of rotation caused by the resultant about the point O is determined by this positive or negative sign.