M1 F O R C E S| 1

A Level Mechanics (9709/ Paper 42)

Prac ice ( rces) up down
202m20 Find 1006s
1 1 22
FSind1006120205
Left Right i FLink 81.113
FlosD afging
2010140 10081720
i Flosa 2021120 FCOSL
40.522 g
20640
i land 4 22 100620
X 5816
F 95
1005in20

Three coplanar forces of magnitudes 20 N, 100 N and F N act at a point. The directions of these forces
are shown in the diagram.

Given that the three forces are in equilibrium, find F and !. [6]
Ans: F = 95(.0), α = 58.6

2 1 21
2421472464 246127USD 361084
24 N
Let Right ftp.fpsind ploso 1043624
12
242in 4362in Pino420
"
! PN
20 N
602in PSin0 20 "
tanD
Ploso 605in 20
60 3 20 4266
D tan
PLOSO 16 36 N
382in 59
Coplanar forces of magnitudes 24 N, P N, 20 N and 36 N act at a point in the directions shown in the
diagram. The system is in equilibrium.

Given that sin ! = 3, find the values of P and ".

i P 18.7 [6]
5
Ans: P =18.7 θ = 59 .0
 Up down
2 22 2

LeffRight afp botfusino 16km504

202in
14 1610550 206860 FLOSO Fukind Fphind
168 504
FnCosD 2010560 14 1610550 166550 205in60 60
Frod 2 30.422
14.284 7206160
1621750
i
tano 2 4205in 60
44 Fr 3306N
0Coplanar
64.84
forces of magnitudes 60 N, 20 N, 16 N and 14 N act at a point in the directions shown in the
diagram.

Find the magnitude and direction of the resultant force. [6]

Ans: R = 33.6 N, 64.8° above the 14 N force

up down
1 22 1

Left Right 165in55 02 825 25

96525 166555 22 1621055 Q PSin25 1625455
1 22 peg 5344021425168in
D 2 PGin25 V
655 0 143
34.40

Coplanar forces of magnitudes P N, Q N, 16 N and 22 N act at a point in the directions shown in the
diagram. The forces are in equilibrium.

Find the values of P and Q. [5]

Ans: P = 34.4, Q = 1.43

5 9709/41/M/J/16/Q4 UP down
50 482inxtpsin.tn
Left Right 142in90 9
Cos 1461904 48684 intrin
50 488
page 486,2 141012
Ploso 486K 148in
agging
4845 1425 Bind PSind 50 483in
14108α
go
42.16 PSind 23.12
Coplanar forces of magnitudes 50N, 48N, 14N and PN act at a point in the directions shown in the
7
diagram. The system is in equilibrium. Given that tan ∝ = , find the values of P and 𝜃. [6]
24
Ans: P= 48.1, θ = 28.7
Hand
EI P 48
1
D 28.7
 9709/42/O/N/10/Q3 90 2
Left Right
62in
F 66k452m
6 52in
565190 4
bind F
5Cosα
52in 90 4 51082
F 68in 5108N
6692452in 62in 5698
11652 Gina to
A particle P is in equilibrium on a smooth horizontal table under the action of four horizontal forces of
tana11 magnitudes 6N, 5N, F N and F N acting in the directions shown. Find the values of ∝ and F. [6]
α tank 84.8 F 5.52 Ans: 84.8°, 5.52
2 2
FN up down
Left Right 4N
affinx faint 421230 6
46530 71052 3 Abinzod 30" !" 3 N Fbind 4
i tank 46520
P
Flora F 4.62
2 0

tail 3
α 83.4 i F 403 6N

Coplanar forces, of magnitudes F N, 3 N, 6 N and 4 N, act at a point P, as shown in the diagram.

(a) Given that ! = 60, and that the resultant of the four forces is in the direction of the 3 N force,
find F. [3]

(b) Given instead that the four forces are in equilibrium, find the values of F and !. [5]
Ans: F = 4.62, (b) F = 4.03 and α = 83.4
2 21
QSin60 PSinb0
660 8660 150 0 n 60!
R PLinbO
P6S60 50 5 kg
XN
1006in60 7
100 82 X
8
860 2
60!

Q Left Right
100 5053 X
A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal
force of magnitude X N, as shown in the diagram. The two strings are both inclined at 60! to the
vertical.
i (a) Given that X = 100, find the tension in the lower string. [4]
(b) Find the least value of X for which the block remains in equilibrium in the position shown. [4]
Ans: 675N , 200N
see
 M1 F O R C E S| 4

9709/42/F/M/21/Q3 2 RLin3O PLimb

2782 BR.BE
P
R1os30 P 6560
RE 2 Q 60! 31
P ER 4
2R
30!

83.2 R
13 v
R 2
253
A particle Q of mass 0.2 kg is held in equilibrium by two light inextensible strings PQ and QR. P is
a fixed point on a vertical wall and R is a fixed point on a horizontal floor. The angles which strings
PQ and QR make with the horizontal are 60! and 30! respectively (see diagram).

Find the tensions in the two strings. [5]

Ans: TP = 3.46 N and TR = 2 N

1 9709/42/O/N/20/Q3
45!
445 9010160 Things120 60 10
60!
141 8ings 205in601
7645 20660 9202in60
14.1 Things m 273
T TN 20 N

m kg

Iom
A block of mass m kg is held in equilibrium below a horizontal ceiling by two strings, as shown in the
diagram. One of the strings is inclined at 45! to the horizontal and the tension in this string is T N.
The other string is inclined at 60! to the horizontal and the tension in this string is 20 N.

Find T and m. [5]

Ans: T = 14.1, m = 2.73

11 9709/43/O/N/20/Q3
A string is attached to a block of mass 4 kg which rests in limiting equilibrium on a rough horizontal
table. The string makes an angle of 24 above the horizontal and the tension in the string is 30 N.

(a) Draw a diagram showing all the forces acting on the block. [1]

(b) Find the coefficient of friction between the block and the table. [5]

up down
Ans: μ = 0.986
R
124 R 3061724 40
F up R 40
3061724

Up 3010824 27.8
 27.8 986
M1 F O R C E S| 5

TSinb0 R 2562
12
Tsin 60 256120 8
2 2

18h
TN
Left Right of 14.804.5212 23.99
7660 25217204M
60
2510120 R
r 60!
2.5 kg
1.52k 23.49 19
0
i tanto
2525720TUR R 23,4 2

TfginoI
20!
tk
8 8.55 30 23.49 R256
A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at 20! to the horizontal by a
force of magnitude T N making an angle of 60! with a line of greatest slope of the plane (see diagram).
The coefficient of friction between the particle and the plane is 0.3.

Find the greatest and least possible values of T . [8]

Ans: P = 3.29, θ = 55.3

1 2 2
30 N 15 N

" !
" !
PN

33 N

Coplanar forces of magnitudes 30 N, 15 N, 33 N and P N act at a point in the directions shown in the
diagram, where tan ! = 43 . The system is in equilibrium.

! "2 ! "2
14.4 28.8
(a) Show that + = 1. [4]
30 − P P + 30
(b) Verify that P = 6 satisfies this equation and find the value of ". [2]
2 Ans: θ = 36.9
2 522 2
up down
1 2
A 2621260 731230 2
Left Right 60!
XN

20560 7 76130Thinbon MabinG

B 2
7 6560
711 0530 3,301 R
0.2 kg

8
2 1 42N
A smooth ring R of mass 0.2 kg is threaded on a light string ARB. The ends of the string are attached
to fixed points A and B with A vertically above B. The string is taut and angle ABR = 90!. The angle
T between the part AR of the string and the vertical is 60!. The ring is held in equilibrium by a force of
magnitude X N, acting on the ring in a direction perpendicular to AR (see diagram).

Calculate the tension in the string and the value of X . [5]

Ans: X = 2, tension in string = 0.536 N
i x 2 T 0.536
 M1 F O R C E S| 6
15 9709/42/M/J/21/Q4

41206525
A particle of mass 12 kg is stationary on a rough plane inclined at an angle of 25! to the horizontal. A
pulling force of magnitude P N acts at an angle of 8! above a line of greatest slope of the plane. This
force is used to keep the particle in equilibrium. The coefficient of friction between the particle and
the plane is 0.3.
ok up Down
eft Right 725 L
P2in8Ans:1206525
Find the greatest possible value of P. [6]

UP 12051725 1658
paying
York P = 80.8

Down
16 9709/42/M/J/23/Q5
UP
eft Right PN
GRABS R
0.6 kg
P3in35 1610135
65mn35 POS35 UR 3.441 19656 1.0480
35C 3,610535
441 80.819 P 1.407
PHBbdtgfg.gg
35!
0.465740 4914
A particle of mass 0.6 kg is placed on a rough plane which is inclined at an angle of 35! to the
horizontal. The particle is kept in equilibrium by a horizontal force of magnitude P N acting in a
vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the
particle and plane is 0.4.

Find the least possible value of P. [6]

Ans: P=1.41

17 9709/41/M/J/17/Q3

30!
40!

25 N
AN BN

Two light inextensible strings are attached to a particle of weight 25 N. The strings pass over two
smooth fixed pulleys and have particles of weights A N and B N hanging vertically at their ends. The
sloping parts of the strings make angles of 30! and 40! respectively with the vertical (see diagram).
The system is in equilibrium. Find the values of A and B. [6]

Ans: A = 17.1, B = 13.3

 M1 F O R C E S| 7
1 2 2
TN

20!
B

30!
FN

A block B, of mass 2 kg, lies on a rough inclined plane sloping at 30! to the horizontal. A light rope,
inclined at an angle of 20! above a line of greatest slope, is attached to B. The tension in the rope
is T N. There is a friction force of F N acting on B (see diagram). The coefficient of friction between
B and the plane is ".

(a) It is given that F = 5 and that the acceleration of B up the plane is 1.2 m s−2.
(i) Find the value of T . [3]

(ii) Find the value of ". [3]

(b) It is given instead that " = 0.8 and T = 15.Determine whether B will move

up the plane. [3]

Ans: T =18.5, μ = 0.455, the block does not move

1 Three horizontal forces of magnitudes 150N, 100N and PN have directions as shown in the diagram.
The resultant of the three forces is shown by the broken line in the diagram. This resultant has
magnitude 120N and makes an angle 𝟕𝟓° with the 150N force. Find the values of P and 𝜽.

Ans: θ = 80.7 or P = 117

2 22

A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre C. The wire is fixed in
a vertical plane and the ring is kept at rest by a light string connected to A, the highest point of the
circle. The string makes an angle of 25! to the vertical (see diagram).

Find the tension in the string and the magnitude of the normal reaction of the wire on the ring. [6]
Ans: T = 72.5 N, R = 40 N
The diagram shows a particle of mass 0.6kg on a plane inclined at 25 degree to the horizontal. The particle
is acted on by a force of magnitude P N directed UP the plane to a line of greatest slope. The coefficient of
friction between the particle and the plane is 0.36. Given that particle is in equilibrium, Find the set of
possible values of P.