Coplanar Forces
RP
1.In the square OABC of side 2m,O,A,and B are the points (0,0),
(2,0) and (2,2) respectively. Forces of magnitude 5,8,3 and 5 2
newtons act along OA,BA,OC and AC respectively.Show that this
system of forces is equivalent to a couple and find its moment.
If a fifth force of magnitude 6 2 newtons acts along OB,find
the equation of the line of action of the resultant of the enlarged system.
(J/156/9.2/Q20)
Ans: 6 Nm clockwise ; y = x +1
2.(a)In this question the unit of distance is the metre and the
unit of force is the newton.The force P = 2i acts through the
point with position vector 5j.The force Q = 6i acts through
the point with position vector j.The resultant of P and Q acts
through the point with position vector nj.Find the value of n
and the magnitude of the resultant of P and Q.
(b) ABC is an isosceles triangle,right-angled at A with
AB=1m.Forces of 8N,4N and 6N act along BA,BC and CA respectively .Find the single force that could replace this system
and find where its line of action cuts AB.
[(a)/UM/129/7G/Q1,
(b)/UM/128/7F/Q10]
Ans: (a) n = 2 , 8i (b) 11.3 N at 73.6 0 to CA ; 89 cm fr A on BA produced .
3.ABCD is a rectangle with AB=70 cm and AD=20 cm.Forces of 5 N,
2 N,3 N and 6 N act along AB,BC,CD and AD respectively.Find the
single force that could replace this system and find where its
line of action cuts AB.
The force along AB is now replaced by a force through A which
reduces the system to a couple.Find the magnitude and direction
of this force and the moment of the couple.
(UM/128/7F/Q14)/M1.
Ans: 8.25 N , 75 0 58' to AB , 25 cm fr A . 8.54 N , 69 0 27' to AB ,
2 Nm in the sense ABCD .
4.Forces of 5N and 3N act along the sides AB, AC
respectively of an equilateral triangle ABC of side 12 m.
Find the magnitude and direction of their resultant . The line of
action of the resultant intersects BC at D. By taking moments
about D , or otherwise, find the length of BD.
(K/127/1)
0
Ans: 7N at 38.2 to AC ; 4.5 m .
5.A rectangle is defined by the four points A(0,0), B(5,0),
C(5,3) and D(0,3), distances being measured in metres.Forces
of magnitude 6N, 8N, 4N and 2N act along AB, BC, CD and
DA respectively in directions indicated by the letters.
Calculate:
(a) the magnitude of the resultant of this system of forces;
(b) the angle between the line of action of the resultant and the
x-axis.
The line of action of this resultant cuts the x-axis at (a,0).
(c) Find a value for a by consideration of moments about A.
Hence determine the equation of the line of action of this
resultant.
(K/127/2)
1
Ans:(a) 2 10 N at tan 3 to x-axis (b) 71.6
2
© a = 8 m , y = 3x - 26
3
0
6.The square ABCD has each side of length 6 m. Forces of
magnitude 1, 2, 8, 5, 5 2 and 2 2 N
act along AB, BC, CD, DA, AC and DB respectively, in the
directions indicated by the order of the letters. Prove that these
forces are equivalent to a couple. Calculate the magnitude and
sense of this couple.
(K/127/3)
Ans: 48 Nm in sense ABCD.
7.OABC is a square of side 1 m. Forces of magnitude 2, 3, 4
and 5 2 newtons respectively in directions indicated by the
order of the letters and a couple of moment 7 Nm acts in the
plane of the square in the sense OCBA. Find the magnitude and
direction of the resultant of the system and the equation of its
line of action referred to OA and OC as axes. What is the
magnitude and direction of the least force introduced at A if the
resultant of the original system and this new force is to pass
through O?
(K/127/4)
1
Ans: 65N at tan 8 to OA, line of action 3/8 m from O in
direction OA & 3 m from O in direction CO so y = 8x + 3 ; 3 N
in direction BA .
8.A rectangle ABCD has AB = 3cm and BC = 4cm . Forces, all
measured in newtons and of magnitudes 2, 4, 6, 8 and k , act
along AB, BC, CD, DA and AC respectively, the direction of
each force being shown by the order of the letters. The resultant
of the five forces is parallel to BD.
5
Find k and show that the resultant has magnitude
6
newtons.Find the distance from A of the line of action of the
resultant.
(K/127/5)
35
Ans: k =
; 43.2 cm
6
9. A rigid rectangular lamina ABCD, with AB = 4a and BC =
3a, is subject to forces of magnitudes 10P, P, 2P, 3P acting
along CA, AD, DC, CB respectively in the directions indicated
by the order of the letters .
(i) Find the magnitude of the resultant of the four forces.
(ii) Find the tangent of the acute angle between the line
of
action of the resultant and the edge AB of the lamina.
(iii) Find the distance from A of the point where the line of
action of the resultant meets AB.
(iv) Indicate clearly on a diagram the line of action and the
direction of the resultant.
(v) Find the magnitude and sense of the couple G which, if
added to the system , would cause the resultant force to act
through E, the midpoint of CD.
(vi) In the case when G is not applied, find forces S along
AB, T along AD and U along BC which, when added to the
system, would produce equilibrium.
(K/127/6)
Ans: (i) 10 P (ii) 4/3 (iii) 9a/4 (iv)
1
1
(v) 30a in sense ABCD (vi) 6P N , 3 P N , 4 P N
2
2