91Ó°ÊÓ

Prove that a couple, together with a force in the same plane, is equivalent to a single force. Describe completely the possible resultants of a force of \(10 \mathrm{~N}\) acting in the same plane as a couple of magnitude \(20 \mathrm{Nm}\).

A system of coplanar forces consists of forces of \(4,3,2,5\) and 6 newton acting along the sides \(\mathrm{AB}, \mathrm{BC}, \mathrm{CD}, \mathrm{DE}\) and \(\mathrm{EF}\) respectively of a regular hexagon \(\mathrm{ABCDEF}\) of side \(2 \mathrm{~m}\), the forces acting in the directions indicated by the order of the letters. Find, in magnitude and direction, the force \(P\) newtons, acting at \(\mathrm{F}\), which will reduce the system to a couple and find the magnitude and sense of this couple. If the force \(P\) newtons is replaced by a force of 7 newton along \(A F\), show that the system now reduces to a single force and find the magnitude of this resultant and the point of intersection of its line of action with \(\mathrm{AB}\), produced if necessary.

A system of forces acting in the plane of perpendicular axes \(O x\) and \(O y\) consists of: a force \(10 P\) along \(\mathrm{Ox}\), a force \(-9 P\) along \(\mathrm{O} y\), a force \(13 P\) along \(\mathrm{OA}\), where \(\mathrm{A}\) is the point \((12 a, 5 a)\). a force \(20 P\) along \(A B\), where \(B\) is the point \((8 a, 8 a)\). Find the magnitude, direction and equation of the line of action of the resultant of this system. A clockwise coplanar couple of magnitude \(240 P a\) is added to the system. Find the magnitude, direction and equation of the line of action of the resultant of the new system.

Find the magnitude and direction of the resultant of each of the following two systems of forces which act in the plane of a rectangle \(A B C D\). in which \(A B=4 a\), and \(\mathrm{BC}=3 a\) (a) Forces \(4 P\) along \(A B, P\) along \(A D\) and \(10 P\) along \(D B\). (b) Forces which have total moment \(-P a,+15 P a\) and \(-5 P a\) about A. B and D respectively ( \(+\) indicates the sense \(\mathrm{ABC}\) ). Find also, in each case, the point of intersection of the line of action of the resultant with the line \(\mathrm{AB}\) (produced if necessary).

Six forces act round the sides of a hexagon. Find the equation of the line of action of their resultant. (a) Their magnitudes are \(P, 2 P, 4 P, 3 P, P, 2 P\) along \(\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{BC}}, \overrightarrow{\mathrm{CD}}, \overrightarrow{\mathrm{ED}}, \overrightarrow{\mathrm{FE}}, \overrightarrow{\mathrm{AF}}\) respectively. (b) The hexagon is regular. (c) The co-ordinates of vertex \(\mathbf{B}\) are \((1,1)\).