This document discusses concepts related to forces and motion, including Newton's laws of motion. It defines key terms like force, mass, weight, friction. Examples are provided to demonstrate calculating unknown values like acceleration, force, coefficient of friction using the equations of motion. Diagrams and step-by-step working are included to illustrate problem solving. Concepts covered include calculating net force, resolving forces into components, and determining static and kinetic friction forces.

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2. – F ˆ F www.schoolDD.com 1
3
F ˈ F F
F F ˈ ˈ F F
F F F ˈ F F F
F
3.1 ,
F ʽ F F F F F
F F F ˈ F F 45 F F F
45 450 F ʽ F
(Force) F F F F F
F F F F
ˈ F ˈ F F ˈ (N)
(Mass) m F ˈ F F ˈ
(kg)
F F F
F F F
F ˈ
u = 0
F
v
ˈ
v1
F
v2
F v1 ˈ v2
F

3. – F ˆ F www.schoolDD.com 2
(Weight) w F F m
F F g F g F F F F
F ˈ (N)
F
F x - y
ˈ F
F θ x F x F F F
F 1 x F F cosθ F y F
2 F F sinθ F F F F F cosθ F F
F F sinθ .... F F α y F F α F
F F F F F.. F F θ = 30°, α = 60° ( F F cos30° =
F sin 60° F sin 30° = F cos60° ?) ʿ F F !
m
W = mg
“ F ”
F
- ˈ (= )
- F F F F F F
F F F g F
- ˈ F F F F F F
F 2
θ
F
y
F cosθ
x
F sinθ
1
2
F F x F cos θ
F F y F sin θ
F
- ˈ F F F F ˈ
F ( ) F F F
- F F F ˈ F
F F F

4. – F ˆ F www.schoolDD.com 3
F sin , cos , tan F
F sin , cos , tan F F F 30°, 60°, 37°, 53° 45°
F sin , cos , tan F F F F F F
F F F F F F ˆ F F F F
F F F ˈ F
F F sin , cos , tan F F
F F F F ʿ ʿ F F F F F
ˈ F F F
F F F ˈ 30°, 60°, 37°,
53° 45°
F F F sin , cos , tan 30°, 60°, 37°,
53° 45° F F F F F
θ
F F ˈ
θ
a
c 1
2
F sinθ F F 1՜2 F F
F sinθ = a/c
θ
b
c
3
4 F cosθ F F 3՜4 F F
F cosθ = b/c
30
60
1
2
√3
37
53
5
4
3
45
45
1
1
√2
F tanθ F F 5՜6 F F
F tanθ = a/b
θ
a
b
5
6

5. – F ˆ F www.schoolDD.com 4
F 1
F1 = 3 N F2 = 4 N m F
.
ˈ +
F F = F1 - F2
= 3 - 4
= - 1 ( F F F F )
∴ F 1 N F Ans
.
F2 ˈ x y F F1 F
x ΣFx = F1 + F2 cos60
= 3 + 4 (½)
= 5 N
y ΣFy = F2 sin60
= 4 ሺ
√ଷ
ଶ
ሻ
= 2√3 N
F F = ඥΣFxଶ ൅ Fyଶ
= ට5ଶ ൅ ሺ2ඥ3ሻ
ଶ
= 6.08 N
m
F2 F1
m
60° F1
F2
F1 F2 cos60
60°
F2 sin60
x
y
F2
α x
y
ΣFX
ΣFy
F F

6. – F ˆ F www.schoolDD.com 5
∴ F 6.08 N Ans
F F tan α =
Σ ୊౯
Σ ୊౮
=
ଶ√ଷ
ହ
= 0.7
∴ α = tan-1
0.7 Ans
.
F F = F1 + F2
= 3 + 4
= 7 N
∴ F 7 N Ans
3.2
F (Sir Issac Newton) ʽ F F
F F ˈ F F
F F 3 F
3 F
F 1 F F F F
m
F1
F2
F 1 F F
F F F ˈ F F
F 2 F F F ˈ F F F F
F F
F
F 3 Action = Reaction F F F
F
∑ F = 0
F
m
V , a = 0
m
V = 0 , a = 0

7. – F ˆ F www.schoolDD.com 6
F F
ˈ F F F
F 2 F F F ˈ F F F F F
F F F F ∑ ‫ܨ‬ F F F a F ∑ ‫ܨ‬ = ma
F 3 Action = Reaction F F F F F12 F
F21 F F
F F
F F F
F F ?
F F F F ˈ F ?
F F ?
F F F F a F g
F12 = F21
F21 F12
F
- F F 3 F F F F
F F ˂ F ˈ F
- 3 F ˈ F F F F F
F 3 F F F F ( F F ..?) F F
F F F F F F F F F
F
∑ F = ma
F F ˈ F
m
a
∑ F

8. – F ˆ F www.schoolDD.com 7
F 2
F 20 kg F F F 24 m/s
8
F.. F F F F F F F F F
F F ˈ F F”
F F F F F 2 ∑ F = ma
F ∑ F F F F
F ∑ F = F F = ma --- (1)
F F F m F F a ∴ F a F
F a F v = u + at ( )
24 = 0 + a (8)
a = 3 m/s2
F a = 3 (1) F
F = 20 (3)
∴ F = 60 N Ans
F = ?
m m
u = 0 ( F ) v = 24 m/s
t = 8 s
m = 20 kg
F
- F F F 2 F F F ˆ F
F ˈ F ʽ F F F F F
F F F F F F F
F F F F F ʽ F F F

9. – F ˆ F www.schoolDD.com 8
F 3
5 kg ˂ 100 m F
F ∑ F = ma
F F = m(g)
F = 5(10)
∴ F = 50 N Ans
F 4
200 N F F F 10 m/s2
F F 100 N
F F
F F ∑ F = ma
F F1 = ma1
∴ m =
୊భ
ୟభ
=
ଶ଴଴
10
= 20 kg
F a2 F F ∑ F = ma
F F2 = ma2
∴ a2 =
୊భ
୫
=
ଵ଴଴
20
= 5 m/s2
Ans
F
100 m
a = g
80
m = 5 kg
F = ?
F
- F
m
a1 = 10 m/s2
F1 = 200 N
F
m
a2 = ?
F2 = 100 N

10. – F ˆ F www.schoolDD.com 9
F F ˈ ∑ F = ma
F = ma
m F F F ‫ן‬ a
୊మ
୊భ
=
ୟమ
ୟభ
F F
ଵ଴଴
ଶ଴଴
=
ୟమ
ଵ଴
∴ a2 = 5 m/s2
Ans
F 5
m1 F F 20 m/s2
m2 F F 5 m/s2
F m1/m2
F ∑ F = ma
F F = m1a1 --- (1)
F = m2a2 --- (2)
(1) = (2) , m1a1 = m2a2
∴
୫భ
୫మ
=
ୟమ
ୟభ
=
ହ
ଶ଴
=
ଵ
ସ
Ans
F F ˈ ∑ F = ma
F = ma
F F F m ‫ן‬
ଵ
ୟ
୫భ
୫మ
=
ୟమ
ୟభ
F F
୫భ
୫మ
=
ହ
ଶ଴
∴
୫భ
୫మ
=
ଵ
ସ
Ans
m1
a1 = 20 m/s2
F
F
m2
a2 = 5 m/s2
F
୫భ
୫మ
= ?
F
- F F F F F ˈ F
F F F ʿ
F F F ˈ ( ) ʿ
F F F

11. – F ˆ F www.schoolDD.com 10
F 6
20 N F F F 10 F
50 m F F
∑ F = ma
F F = ma --- (1)
m F F F F a ‫׵‬ a F
a F s = ut + ½ at2
( )
50 = 0 + ½ a (102
)
a = 1 m/s2
F a = 1 (1)
F = ma
20 = m(1)
‫׵‬ m = 20 kg Ans
F 7
5 kg 6 N 8 N F
∑ F = ma --- (1)
a F m F F ∑ F ‫׵‬ ∑ F F
F ∑ F 6 N 8 N F
F F
F ”m
s = 50
t = 10F = 20
u = 0 ‫׶‬ m = ?
m
6 N
5 kg
8 N
F
∑ F = √6ଶ ൅ 8ଶ = √100 = 10 N
F
tanߙ =
଼
଺
=
ସ
ଷ
ߙ = 53°
8 N
10 N
6 N
ߙ

12. – F ˆ F www.schoolDD.com 11
F ∑ F = 10 (1)
F a =
∑ ୊
୫
=
ଵ଴
ହ
= 2 m/s2
F 53° 6 N Ans
3.3 (Frictional force)
f F F
F F F F F
F F F
F F F F
F ˈ F
f F F ߤ N
F ˈ 2
1. (Static friction) fs ˈ F
F F F F F F F ˈ
F F ( F )
2. F (Kinetic friction) fk ˈ F
ˈ F.. .. F F
F F F 0... fs (= ߤsN) fs
F F fk (= ߤkN) F F
F F F ˈ F ?
f = ࣆN
f
F
... .. ..
N
ߤ
mg
F
- ߤs > F ߤk
( F F F F .)
- F ˈ F F F
- (ߤsN) > F(ߤkN)
- N
- W = mg F F

13. – F ˆ F www.schoolDD.com 12
F
?
F 8
F 10 kg F F
F 70 N F F F 40 N
. F F
. F F F F
F F F
. fs = ? , ߤs = ?
fs
∑ Fx = max
F fs = max
70 fs = 10(0) ( F ‫׵‬ ax = 0)
‫׵‬ fs = 70 N Ans
ߤs fs = ߤsN --- (1)
∑ Fy = may
N mg = may
N 10(10) = 10(0) ( F ‫׵‬ ay = 0)
N = 100 N
F N = 100 (1)
fs = ߤsN
70 = ߤs(100)
‫׵‬ ߤs = 0.7 Ans
fk = ?
v
F = 40 N
N
mg
ߤk = ?fs = ?
v = 0 F
F = 70 N
N
mg
ߤs =?
m = 10 kg

14. – F ˆ F www.schoolDD.com 13
. fk = ? , ߤk = ?
F F fk
∑ Fx = max
F fk = max
40 fk = 10(0) ( F v ‫׵‬ ax = 0)
‫׵‬ fk = 40 N Ans
F ߤk fk = ߤkN --- (1)
∑ Fy = may
N mg = may
N 10(10) = 10(0) ( F ‫׵‬ ay = 0)
N = 100 N
F N = 100 (1)
fk = ߤkN
40 = ߤs(100)
‫׵‬ ߤk = 0.4 (< ߤs ) Ans
F
- F mg N F
F F F ˈ mg
F F F F N F F
F F F F
F mg N F F ( F ) F
- F ߤs F
mg
N
F F
N
mg
mg
N

15. – F ˆ F www.schoolDD.com 14
F 9
120 kg F F F F 20 m/s F
F 50 m F F
F F u F
u = 20 m/s
F F ( F ) F F
F 2 F F
F F F F
F F F 20 m/s F 1
∑ F = ma
F f = ma --- (1)
f F m F F a ‫׵‬ a F
a v2
= u2
+ 2as ( )
0 = 202
+ 2a (50)
a = -4 m/s2
(a ˈ F F u)
F a = -4 (1)
f = ma
f = 120 (-4)
f = -480 N ( F F )
‫׵‬ F 480 N Ans
F 10
500 N 37° 40 kg F
0.4 F F
F
F F
F ”
u = 20
v = 0 ‫׶‬
m
s = 50
t = 10
f = ?mm = 120
F F ”
500 N
N37°

16. – F ˆ F www.schoolDD.com 15
F F 500 N F F (x-y)
F
F a
∑ Fx = max
F 500 cos37 - f = ma --- (1)
a F m F F f ‫׵‬ f F
f = ߤN F ߤ F F N
N
∑ Fy = may
F N + 500 sin37 - mg = may
N + 500(3/5) 40(10) = 0 ( F ‫׵‬ ay = 0)
N = 100 N
F F f = ߤN = 0.4(100) = 40 N
F f = 40 (1)
500 cos37 - f = ma
500(4/5) - 40 = 40a
‫׵‬ a = 9 m/s2
Ans
a = ?500
37°
500 cos37
500 sin37
x
y
f
N
mg
m = 40
ߤ= 0.4
F F .
F ˈ . F F F F
F F F
F F ˈ .
F
- F ∑ F = ma F 2 ∑ F ( F)
F F m F F a F F F
F ∑ F F F F
F F
m
mg
N
f
T F
a
∑ F = ma
F F F = ma
F T f = ma

17. – F ˆ F www.schoolDD.com 16
F 11
F F ˈ F
F ߤs = 1.0 ߤk = 0.80 F θ F F
F F F F F F F
F F F ˆ F F F ˈ
F F ʽ F
θ F F F ˈ F
F
∑ F = ma
F F F = ma
F mg sinθ - fs = ma F a = 0 ( F )
‫׵‬ mg sinθ = fs --- (1)
fs fs = µs N F µs F F N
N
∑ F = ma
F N - mg cosθ = ma
N - mg cosθ = 0 ( F ‫׵‬ a = 0)
N = mg cosθ
“ ˈ F F F
F F F F ˂ 3
w = mg , f
,
N ˈ 2
mg sinθ mg cosθ ”
θ
N
mg sinθ
f
mg cosθ
W = mg
θ
‫׶‬ 90-θ ‫׵‬
θ
F 180 F

18. – F ˆ F www.schoolDD.com 17
F N F fs = µs N = 1.0 mg cosθ = mg cosθ
F fs (1)
mg sinθ = mg cosθ
ୱ୧୬θ
ୡ୭ୱθ
= 1.0
tanθ = 1.0
‫׵‬ θ = 45° Ans
F F F F ˈ F
F
∑ F = ma
F F F = ma
F mg sinθ fk = ma
mg sinθ - µk N = ma
mg sinθ - µk (mg cos θ) = ma (N = mg cos ߠ)
10 (
ଵ
√ଶ
) - 0.80 (10)(
ଵ
√ଶ
) = a ( F m F )
‫׵‬ a = √2 = 1.414 m/s2
Ans
F 12
90 N 10 kg F F 20 kg F
. F
. F
. F 90 N F 20 kg F
F ˈ F
. F a = ?
∑ F = ma
F
- F F F ˈ F F m F
F F F
F F F F F F F F F F F F F
F F
90 N
10 kg
20 kg
a = ?
F

19. – F ˆ F www.schoolDD.com 18
F F F = ma
m = 10 + 20 = 30 kg ( F ˈ F )
F 90 = 30 (a)
‫׵‬ a = 3 m/s2
Ans
. F R
F F F
10 kg
∑ F = ma
F F F = ma
90 - R = 10 (3)
‫׵‬ R = 60 N Ans
20 kg
∑ F = ma
F F F = ma
R = 20 (3)
‫׵‬ R = 60 N Ans ( F 10 kg)
F F F F F F
.
F a = ?
F R ˈ F - F
F F
F F 3 ”10kg
90 N R
a = 3 m/s2
20 kg
R
a = 3 m/s2
90 N
10 kg
20 kg
a = ?

20. – F ˆ F www.schoolDD.com 19
∑ F = ma
F F F = ma
m = 10 + 20 = 30 kg
F 90 = 30 (a)
‫׵‬ a = 3 m/s2
Ans ( F F F F F )
20 kg
∑ F = ma
F F F = ma
90 - R = 20 (3)
‫׵‬ R = 30 N Ans ( F F F F .)
10 kg
∑ F = ma
F F F = ma
R = 10 (3)
‫׵‬ R = 30 N Ans
20 kg
a = 3 m/s2
R 90 N
10kg
R
a = 3 m/s2
F
- ˈ F mg
N F F F F F
F F ˈ F F F F F F F F

21. – F ˆ F www.schoolDD.com 20
F 13
F 120 N 5 kg 10 kg 15 kg F F
F T1 T2 F µ = 0.10
F a
∑ F = ma
F F F = ma
F F - f1 - f2 - f3 = ma
F - µN1 - µN2 - µN3 = (m1+ m2 + m3)a
120 - µ(N1 + N2 + N3 ) = (15 + 10 + 5)a
120 - 0.1(15(10) + 10(10) + 5(10)) = 30 a ( F N = mg)
‫׵‬ a = 3 m/s2
Ans
F T1
15 kg
∑ F = ma
F ”
T1 F F F 10 kg 15 kg 15 kg F m1
15 kg 10 kg 5 kg
T1 T2 F
m1 m2 m3
T1 T2 F
a = ?
f1 f2
N2
m1g m2g m3g
N1 N3
f3µ = 0.10
m1
m1g
f1
T1 = ?
N1
a = 3 m/s2

22. – F ˆ F www.schoolDD.com 21
F F F = ma
F T1 - f1 = m1a
T1 - µN1 = m1a
T1 - µ m1g = m1a
T1 = m1(a + µg)
T1 = 15(3 + 0.1(10))
‫׵‬ T1 = 60 N Ans
F T2
5 kg
∑ F = ma
F F F = ma
F F T2 f3 = m3a
F T2 µN3 = m3a
F T2 µ(m3g) = m3a
120 T2 0.1(5x10) = 5(3)
‫׵‬ T2 = 100 N Ans
F
- F F F F
F F F F F F
F F F F F F F F
F F F F
m
mg
T
T
T
m
mg
m1 m2
FT
m1 m2
FT T
m3
m3g
N3
f3
T2 = ? F = 120 N
a = 3 m/s2

23. – F ˆ F www.schoolDD.com 22
F 14
80 N 2 kg 3 kg F F F F
F
F a
F ˈ F T F ˈ F
∑ F = ma
F F F = ma
F F m1g m2g = (m1 + m2)a
80 2(10) 3(10) = (2 + 3)a
‫׵‬ a = 6 m/s2
Ans
F T
3 kg
3 kg
2 kg
F = 80 N
T a
a = 6 m/s2m2
T = ?
m2g
a = ?
m2
m1
F = 80 N
m1g
m2g

24. – F ˆ F www.schoolDD.com 23
∑ F = ma
F F F = ma
F T m2g = m2a
T 3(10) = 3(6)
‫׵‬ T = 48 N Ans
F 15
2 kg 3 kg F F F F F
F F
F a
3 kg F F 2 kg F
F a F F F
∑ F = ma
F F F = ma
F m2g m1g = (m1 + m2)a
3(10) 2(10) = (2 + 3)a
‫׵‬ a = 2 m/s2
Ans
F
”
T T
2 3
a = ?a = ? m1 m2
m2gm1g

25. – F ˆ F www.schoolDD.com 24
F
2 kg
∑ F = ma
F F F = ma
F T m1g = m1a
T 2(10) = 2(2)
‫׵‬ T = 24 N Ans
F 16
F F F F F F
F a
F m1
∑ F = ma
F F
F a
F F F F
T = ?
a = 2 m/s2
m1
m1g
10 kg
37°
T2
T2
T1
T1
10 kg
15 kg
25 kg
F
10 kg
37°
m2
m3
m1
m2g m2g cos37
m2g sin37
N
m1g
m3g
a
a
a

26. – F ˆ F www.schoolDD.com 25
F F F = ma
F m1g m2g sin37 m3g = (m1 + m2 + m3)a
25(10) 10 (10)(3/5) 15(10) = (25 + 10 + 15)a
‫׵‬ a = 0.8 m/s2
Ans
F a ˈ F F a ˈ F
F T1
m1
∑ F = ma
F F F = ma
F m1g T1 = m1a
25(10) T1 = 25(0.8)
‫׵‬ T1 = 230 N Ans
F T2
m3
∑ F = ma
F F F = ma
F T2 m3g = m3a
T2 15(10) = 15(0.8)
‫׵‬ T2 = 162 N Ans
T1 = ?
a = 0.8 m/s2
m1
m1g
T2 = ?
a = 0.8 m/s2
m3
m3g

27. – F ˆ F www.schoolDD.com 26
3.4
F F F F F F
1 2 F F F F T
3 F F F F F N
F F F F F F ?
F F F F ?
F 17
F 1 kg F F F F F F F
. F
. F
. F
. F F 0.5 m/s2
. F F 0.5 m/s2
F
- F F F F F F
1
m1
T1
T1
T2 = 2T1
1
m2
T1
T2 = T1/2
1
m1
T2 = T1/2
mm = 1 kg
1
mg
T
m
2
m2g
m g
m1g
T
T T
T
m1
m2
3
mg
mm
N

28. – F ˆ F www.schoolDD.com 27
F F F F
. F
F F
∑ F = ma
F F F = ma
F T mg = ma
T 1(10) = 1(0)
‫׵‬ T = 10 N ( F ) Ans
. F
F F
∑ F = ma
F F F = ma
F T mg = ma
T 1(10) = 1(0)
‫׵‬ T = 10 N ( F ) Ans
. F
F F
m
mg
a = 0
T
m
mg
v , a = 0
T
m
mg
v , a = 0
T

29. – F ˆ F www.schoolDD.com 28
∑ F = ma
F F F = ma
F mg T = ma
1(10) T = 1(0)
‫׵‬ T = 10 N ( F ) Ans
. F F 0.5 m/s2
F F
∑ F = ma
F F F = ma
F T mg = ma
T 1(10) = 1(0.5)
‫׵‬ T = 10.5 N ( F ) Ans
. F F 0.5 m/s2
F F
∑ F = ma
F F F = ma
F mg T = ma
1(10) T = 1(0.5)
‫׵‬ T = 9.5 N ( F F ) Ans
m
mg
a = 0.5 m/s2
T
m
mg
a = 0.5 m/s2
T

30. – F ˆ F www.schoolDD.com 29
F 18
5 kg 15 kg F F F F
F F F F F F F
F F
F 15 kg F 5 kg
F a
∑ F = ma
F F F = ma
F m1g m2g = (m1 + m2) a
15(10) 5(10) = (15 + 5)a
‫׵‬ a = 5 m/s2
T
m2
∑ F = ma
F F F = ma
F T m2g = m2a
T 5(10) = 5(5)
‫׵‬ T = 75 N Ans
15 5
m2g
m g
m1g
T
T T
T
m1
m2
a a
a
m2
m2g
a = 5 m/s2
T = ?

31. – F ˆ F www.schoolDD.com 30
F 19
F 100 kg F F F F
F F F F F F
F F F F F 5 m/s2
F
F F F F F F F F F
F F F a = 5 m/s2
( )
∑ F = ma
F F F = ma
F mg N = ma
100(10) N = 100(5)
N = 500 N
‫׵‬ F F = 50 ˈ F Ans
F F F !!”
F 20
F F 20 F F F F ˈ F F
. F F
. F F F 1 m/s2
. F F F 1 m/s2
. F F 1 m/s
. F F F 1 m/s2
. F F F 1 m/s2
. F F 1 m/s
. F
F
F F F F F F F
N F
F F F 3
”
N
a = 5 m/s2
mg

32. – F ˆ F www.schoolDD.com 31
. F F a = 0
∑ F = ma
N - mg = 0
N = mg
N = 100 (10) = 1000 N
‫׵‬ F F = 100 Ans
. F F F a = 1 m/s2
∑ F = ma
N - mg = ma
N = m (a + g)
N = 100 (1 + 10)
N = 1100 N
‫׵‬ F F = 110 Ans
. F F F a = -1 m/s2
∑ F = ma
N - mg = ma
N = m (a + g)
N = 100 (-1 + 10)
N = 900 N
‫׵‬ F F = 90 Ans
. F F v = 1 m/s ‫׵‬ a = 0
∑ F = ma
N - mg = 0
N = mg
N = 100 (10)
N = 1000 N
‫׵‬ F F = 100 Ans
. F F F a = 1 m/s2
∑ F = ma
mg - N = ma
N = m (g - a)
N = 100 (10 - 1)
F ”
F F F ˈ F
F
F 2 F ”
F F F a ˈ - F
F
F F ”
F F ‫׵‬ F ˈ
F F”
N
a = 0
mg
N
a = 1 m/s2
mg
N
a = -1 m/s2
mg
N
v
a = 0mg
N
a = 1 m/s2
mg

33. – F ˆ F www.schoolDD.com 32
N = 900 N
‫׵‬ F F = 90 Ans
. F F F a = -1 m/s2
∑ F = ma
mg - N = ma
N = m (g - a)
N = 100 (10 - (-1))
N = 1100 N
‫׵‬ F F = 110 Ans
. F F v = 1 m/s ‫׵‬ a = 0
∑ F = ma
mg - N = 0
N = mg
N = 100 (10)
N = 1000 N
‫׵‬ F F = 100 Ans
. F a = g
∑ F = ma
mg - N = mg
N = 0 N
‫׵‬ F F = 0 Ans
F 21
50 kg F F F F 0.5 m/s2
F F F F 40 kg F
F !! ! F F F
F F
F F a = g”
N
a = -1 m/s2
mg
N
v
a = 0mg
N
a = g
mg

34. – F ˆ F www.schoolDD.com 33
( ) F
∑ F = ma
F F F = ma
F mg N T = ma
50(10) 400 T = 50(0.5)
‫׵‬ T = 75 N Ans
F 22
F 40 kg F 30
F F ˈ
( F ) F
F F F F F N F F F
F F F F N
T = ?
mg
N = 400 N
a = 0.5
30°
30°
N
mg
mg sin30
a
N = ?
mgay
30°
ax = a cos30
ay = a sin30a

35. – F ˆ F www.schoolDD.com 34
F mg sin30
mg
∑ F = ma
F F F = ma
F mg sin30 = ma
a = g sin30
= 10(1/2)
a = 5 m/s2
‫׵‬ ay = a sin30 = (5x1/2) = 2.5 m/s2
∑ F = ma
F F F = ma
F mg N = may
40(10) N = 40(2.5)
N = 300 N
‫׵‬ F F = 30 Ans
F 23
70 kg F F
. F F 1 m/s2
. F F 1 m/s2
. F F a = 1 m/s2
∑ F = ma
F
( ) F
T mg”
mg
a = 1 m/s2
T

36. – F ˆ F www.schoolDD.com 35
F F F = ma
F T mg = ma
T 70(10) = 70(1)
‫׵‬ T = 770 N Ans
. F F a = 1 m/s2
∑ F = ma
F F F = ma
F mg T = ma
70(10) T = 70(1)
‫׵‬ T = 630 N Ans
F F F F ˈ F ?”
F 24
F F 8 kg F F F 10 kg F
F F F F F F
F F F F F F F
F F ‫׵‬ F F a1 = 0
F
∑ F = ma
mg
a = 1 m/s2
T
T
m1g
a1 = 0

37. – F ˆ F www.schoolDD.com 36
F T = m1g a = 0
T = 10(10)
‫׵‬ T = 100 N
∑ F = ma
F F F = ma
F T m2g = m2a2
100 8(10) = 8 a2
a2 = 2.5
‫׵‬ F F F F 2.5 m/s2
Ans
F 25
F 2 F F F ˈ 0.25
F F
F F F
F F ˈ F F F
F F 3 F
”
F F N1 F
F F F F N2
F F F F F
a2 = ?
T
m2g
6 kg
4kg
F = 100 N
f2
f1
N1
N2
T
F = 100 N
m2g
a
f1
a
N1
T
m1g

38. – F ˆ F www.schoolDD.com 37
F m1 = 4
∑ F = ma
F F F = ma
F T f1 = m1a
T ߤN1 = m1a
T 0.25(4x10) = 4a
T = 4a + 10 ---- (1)
F F m2 = 6
∑ F = ma
F F F = ma
F F T f1 f2 = m2a
F T (1) f = µN
F (4a + 10) ߤN1 ߤN2 = m2a
N1 = m1g , N2 = N1 + m2g F
F (4a + 10) ߤm1g ߤ(N1 + m2g) = m2a
100 (4a + 10) 0.25(4x10) 0.25(4x10 + 6x10) = 6a
‫׵‬ a = 5.5 m/s2
Ans
F a (1)
T = 4a + 10
T = 4(5.5) + 10
‫׵‬ T = 32 N Ans
F F F
∑ F = ma
F F F = ma
F - f2 - f1 - f1 = (m1 + m2)a
100 - 0.25(10x10) - 0.25(4x10) - 0.25(4x10) = 10a
‫׵‬ a = 5.5 m/s2
Ans

39. – F ˆ F www.schoolDD.com 38
F 26
F F F F F F F F
F F F 0.50 ( F )
F
∑ F = ma
F F F = ma
F N = ma --- (1)
F
F f = mg
ߤN = mg
N = mg/ߤ
F N (1)
N = ma --- (1)
mg/ߤ = ma
a = g/ ߤ F F
a = 10/ 0.5
‫׵‬ a = 20 m/s2
Ans
F F F F F F F F
F
F ˇ F F
F
a
a = ?
N
f
mg

40. – F ˆ F www.schoolDD.com 39
F 27
F F F F F F
F F 10 m/s2
F
∑ F = ma
F F F = ma
F Tsinθ = ma --- (1)
F
F Tcosθ = mg --- (2)
(1)/ (2) , Tsinθ/ Tcosθ = ma/mg
tanθ = a/g
tanθ = 10/10
tanθ = 1
‫׵‬ θ = 45° Ans
F
”
a = 10 m/s2
T
Tsinθ
mg
Tcosθ
θ
a = 10 m/s2
θ
F
- F F F F F ! F F
F F F ˈ F F F
F F F F ʿ F F
F F ˆ F F

41. – F ˆ F www.schoolDD.com 40
3.6 F
F F F F
F F F F F F F ˈ
F F
F
F F
F F F
........
m1 m2 ˈ
R ˈ F
G ˈ F F F F 6.673 x 10-11
Nm2
/kg2
FG ˈ F
F F F F
F F F F F
FG =
ୋ୫భ୫మ
ୖమ
mg =
ୋ୫୫౛
ୖ౛
మ
me =
୥ୖ౛
మ
ୋ
FG =
۵‫ܕ‬૚‫ܕ‬૛
‫܀‬૛
m
mg
FGRe
me
FG FG
R
m1 m2
F
FG FG
m1
m2
R

42. – F ˆ F www.schoolDD.com 41
F Re = 6.38 x 106
m
‫׵‬ me = 9.8(6.38 x 106
)2
/(6.67 x 10-11
)
me = 5.98 x 1024
kg
F F F F F F
FG =
ୋ୫భ୫మ
ୖమ
mg =
ୋ୫୫౛
ୖమ
‫׵‬ g =
ୋ୫౛
ୖమ
ˈ F F F g R F
F 28
F 1.0 2.0 F F
1.0 F G = 6.6 x 10-11
Nm2
/kg2
m1 = 1.0x10-12
x10-3
= 1.0x10-15
kg
m2 = 2.0x10-9
x10-3
= 2.0x10-12
kg
R = 1.0 x10-6
m
FG =
ୋ୫భ୫మ
ୖమ
F F FG =
଺.଺୶ଵ଴షభభ୶ଵ.଴୶ଵ଴షభఱ୶ଶ.଴୶ଵ଴షభమ
ሺଵ.଴୶ଵ଴షలሻమ
‫׵‬ FG = 1.32 x 10-25
N Ans
FG FG
R
m1 m2
m
mg
FG
R
me

43. – F ˆ F www.schoolDD.com 42
F 29
F F 900 F F F F 3 F
F F F
F
FG =
ୋ୫భ୫మ
ୖమ
G , m1 , m2 F
F F FG ‫ן‬
ଵ
ୖమ
୊ృమ
୊ృభ
=
ୖభ
మ
ୖమ
మ
F F
୊ృమ
ଽ଴଴
=
ୖభ
మ
ሺଷୖభሻమ
‫׵‬ FG2 = 100 N Ans
m
mg1
FG1R1
me
FG1 = mg1 = 900 N FG2 = mg2 = ?
m
mg2
FG2
R2 = 3R1
me