2. 5
5.1 W ˈ F F ˈ Nm
(J)
F
W = Fs
s
F F
¬ ˄ ?
F ?
¬ F F ?
¬ F ˈ F F? ?
¬ F F F F ˈ F
F? ?
- F F F F F
F
θ F cosθ W = F cosθ s
s
- F F F F ˈ F
F F
F
fs W = - fs s
s ˀ mg s W= -Fs
F
.=
- F F F s (W = Fs ) s
F ˆ F www.schoolDD.com 1
3. - F F F F
F1= 0 F2
W = ሺ۴ ା ۴ ሻ x
x
- F F ( +,-) F
FW=F Fs
F 1
3 F F1= 20 , F2 = 5 F3= 6 F F
5
. F1 , F2 F3
.
F2
F1
F3
60°
5.0
.
F1 F F1 F F ˈ F F1cos60
F 5.0 m
W1 = F1 cos60 s1
= 20(1/2) 5
W1 = 50 N-m (J) Ans
F2 , W2 = F2 s2
= 5(0) ( F2 ˈ )
F
W2 = 0 J Ans
F3 , W3 = F3 s3
= -6(5) ( F3 ˈ )
W3 = -30 J Ans
F ˆ F www.schoolDD.com 2
4. .
F1+ F2+ F3 , W = W1+ W2+ W3
= 50 + 0 + (-30)
W = 20 J Ans
W =F Fs
= (F1 cos60 - F3) s
= (20(1/2) 6) 5
W = 20 J Ans
F 2
F ˄ 10 2.5 F
˄ ˈ F F F
F
a=0 ˄ F
mg F . ˄
2.5 m
˄
˄ F a=0
∑ F = ma F F = mg = 10(10) = 100 N
= 2.5 m
W = Fs
= -100(2.5) ( F ˈ )
W = -250 J Ans
¬ F F ˄ F F F ˄ F
F F ˈ 10 m F F ?
¬ F F F ˆˉ F ˈ
F ?
F ˆ F www.schoolDD.com 3
5. F 3
F F F ˂ F
4 F F F 10 F F
2
0.5 /
a = 0.5 F F F
T 10
mg
W = Fs F s F F F
∑ F = ma
F mg T = ma
4(10) T = 4(0.5)
T = 38
W = Fs
= -38(10)
W = -380 J Ans
F 4
25 10
2 F . 0.1 ˈ F
F F F
F sin37
F = 25 N F = 25
37°
10 kg 37°
f F cos37
mg
f=0
∑F = ma
F F cos37 = ma
25(4/5) = 10 a
a = 2 m/s2
F ˆ F www.schoolDD.com 4
6. s s = ut + ½ at2
= 0 + ½ (2)(22)
F
s=4 m
W = F s = F cos37 s
= 25(4/5)(4)
W = 80 J Ans
ˀ f = µN = µ(mg - F sin37)
= 0.1(10(10) 25(3/5))
f = 8.5 N
∑ F = ma
F F cos37 - f = ma
(20-8.5) = 10 a
a = 1.15 m/s2
s s = ut + ½ at2
= 0 + ½ (1.15)(22)
s = 2.3 m
W = F F s = (F cos37 f) s
= (20-8.5)(2.3)
W = 26.45 J Ans
F 5
F F 100 F F F 5 F F F 8
F 10 F F 20
F = 100 N
5.0 m
N
4.0 m
θ 2F cosθ
θ f 10.0 m
5.0 m 4.0 m
mg
F = 100 N
W = F F s = (2F cosθ f) s
= (2(100)(3/5)-20)(10)
W = 1000 J Ans
F ˆ F www.schoolDD.com 5
7. F 6
F F F F F 40 37
3 F F F ˈ 60
F F F
N
3.0 m 3.0 m
f
37° 37°
F F
mg sin37 mg cos37
mg
F F F F
F = mg sin37 + f
= 40(10)(3/5) + 60
F = 300 N
W = Fs ( s = 5 m)
= 300(5)
W = 1500 J Ans
F 7
40 30 F 4 F
F ˈ 2 F
t=2 t=2
a
N
F = 40 F = 40
30° 30°
mg sin30 mg cos30
mg
W=Fs , FF s
= ma
∑F
F - mg sin30 = ma
F ˆ F www.schoolDD.com 6
8. 40 4(10)(1/2) = 4a
a = 5 m/s2
s s = ut + ½ at2
= 0 + ½ (5)(22)
s = 10 m
W = Fs ( s = 10 m)
= 40(10)
W = 400 J Ans
F 8
F F 5 50 FF F F
FF
3.0
2.0
5.0
F F F ˈ 2 F F F F (3 ) F F
(2 )
F F 3
F F1= mg (3/5) F F F F F
F ( F F )
F1 = 50(10)(3/5) = 300 N
s1 = 3/2 = 1.5 m ( ˈ cm. F F )
F
W1 = F1 s1
= 300(1.5)
W1 = 450 J
F F F 2
F2 = mg (2/5)
F2 = 50(10)(2/5) = 200 N
s2 = 3.0 m ( cm. F F F F 3 m)
W2 = F2 s2
= 200(3.0)
F ˆ F www.schoolDD.com 7
9. W2 = 600 J
W = W1 + W2
= 450 + 600
W = 1050 J Ans
F 9
F 5x5 3 2 F
( F F =1 / F )
F
cm.
2.0 m
3.0 m
2.0 m
5.0x5.0 m mg
cm. cm.
F = mg
F = 1x103(5x5x2)(10) = 5x105 N
s = 2.0 m
W =Fs
= 5x105(2.0)
W = 1x106 J Ans
F 10
F F 0.4 F 10 F F
0.1 F F
u = 10 v=0
t = 0.1
m
F m
s
W = F s -------1
F F = ma ( a v = u + at a = (v-u)/t
F ˆ F www.schoolDD.com 8
10. F = m (v-u)/t
F = 0.4(0-10)/(0.1) = - 40 N
s s = (u + v)/2 x t
s = (10 + 0)/2 x 0.1
s = 0.5 m
F F, s 1, W =Fs
= - 40 (0.5)
W = - 20 J Ans
F 11
F F F s ˈ
F (N)
10
F 0 --> 4 m F
F F = 10 N
s (m)
2 4 6 8
-10
F . F F s . ˈ F ˈ
W = W0◊4 + W4◊6 + W6◊8
= (10 x 40) + (1/2 x 2 x 10) + (1/2 x 2 x -10)
W = 40 J Ans
,, ,,
5.2 P F F ˈ F F ˈ / F (W)
܅
P= ܜ P=
=
ୱ
= Fv ( v )
୲ ୲
¬ F F F F F
F
F ˆ F www.schoolDD.com 9
11. F 12
F 1 10 20 F
m
F s = 10
a=0 t = 20
m
mg
P = W/t
ୱ
= ୲ ( F v a=0 F = mg )
୫ ୱ
= ୲
య
= ଵ୶ଵ ሺଵሻ ሺଵሻ
ଶ
P = 5x10 W Ans
3
F 13
F F 37 ° F
F F 15 20
F (N)
F 10
37°
s s (m)
5 10 15
P = W/t -------1
F W = (F cos37) s
= Fs cos 37 ( = . F x cos 37 )
= ½ x (15 + 10) x10 x (4/5)
W = 100 J
F W 1, P = 100/20
P = 5 W Ans
F ˆ F www.schoolDD.com 10
12. F 14
F 100 30 ° F 0.01 m/s F .
F F F 1/√3 F
N v = 0.01
F
f
mg sin30
mg cos30
mg
30°
P = W/t = F v -------1
Fv F F ∑ F = ma
F - mg sin30 - f = 0 ( v a = 0, f = µN = µmg cos 30 )
F 100(10)(1/2) (1/√3 )(100)(10)( √3 /2) = 0
F = 1000 N
F F 1, P = 1000 (0.01)
P = 10 W Ans
F ˆ F www.schoolDD.com 11
13. 5.3
5.3.1 F Ek ˈ F F ˈ (J)
v m F v
2
Ek = mv F F 1/2 mv2
m F F ˈ kg m2/s2 = kg m/s2 m
=Nm=J F
5.3.2. F Ep F F ˈ
F F ˈ (J) F ˈ
1. F F F
m
m F F F ˈ
Ep = mgh h F F F F
h F F mgh
F F F F ˈ kg m/s2 m =
Nm=J F
F F
2. F F
m F F
Ep = kx2
F ˈ x
F F F F
k F 2
1/2 kx
F F F ˈ N/m m2 =
x Nm=J F
5.4. F
F F F
∑ ۳ = ∑ ۳ F
∑ ۳ - W1→2 = ∑ ۳ F F
F ˆ F www.schoolDD.com 12
14. ∑ Eଵ = F 1 ( Ep1 + Ek1 )
∑ Eଶ = F 2 ( Ep2 + Ek2 )
W1→2 = F 1 2
F ˈ F F
1. F F ( F F )
u=0
1 m
= ∑ Eଶ
∑ Eଵ
Ep1 + Ek1 = Ep2 + Ek2
h mgh + 0 = 0 + ½ mv2
v mgh = ½ mv2
2 m
v = ∑ Eଶ
∑ Eଵ
2 m Ep1 + Ek1 = Ep2 + Ek2
h 0 + ½ mu2 = mgh + ½ mv2
u ½ mu2 = mgh + ½ mv2
1 m
2.
u=0
1 m ∑ Eଵ = ∑ Eଶ
Ep1 + Ek1 = Ep2 + Ek2
h v
mgh + 0 = 0 + ½ mv2
2 m mgh = ½ mv2
θ
F ˆ F www.schoolDD.com 13
15. 3. ˀ
u=0 s
1 m ∑ EଵW1◊2 = ∑ Eଶ
h f Ep1 + Ek1 f s = Ep2 + Ek2
v
mgh + 0 f s = 0 + ½ mv2
2 m mgh f s = ½ mv2
θ
4. ˈ
v2
2
∑ Eଵ = ∑ Eଶ
r
Ep1 + Ek1 = Ep2 + Ek2
+
0 + ½ mv12 = mgh + ½ mv22
½ mv12 = mg(2r) + ½ m v22
1
v1
∑ Eଵ = ∑ Eଶ
u=0
Ep1 + Ek1 = Ep2 + Ek2
1 mgh + 0 = 0 + ½ mv2
h
2 mgh = ½ mv2
v
5.
m u=0
1
∑ Eଵ = ∑ Eଶ
h Ep1 + Ek1 = Ep2 + Ek2
m v=0 mg(h+x) + 0 = ½ kx2 + 0
m x mg(h+x) = ½ kx2
2 k
F ˆ F www.schoolDD.com 14
16. 6.
m u=0
1
∑ Eଵ W1◊2 = ∑ Eଶ
h
Ep1 + Ek1 f x = Ep2 + Ek2
m mg(h+x) + 0 f x = 0+0
f m x mg(h+x) = fx
2
v=0
7. ˆ F F
1 2
∑ Eଵ W1◊2 = ∑ Eଶ
u f
v=0 Ep1 + Ek1 f x = Ep2 + Ek2
m x 0 + ½ mu2 f x = 0 + 0
½ mu2 = f x
F 15
F 1 ˂ 20
. F F
. F
u=0
m
h = 20
v
m
. Ep = ? F
Ep = mgh
= 1(10)(20)
Ep = 200 J Ans
F ˆ F www.schoolDD.com 15
17. . Ek = ?
Ek = ½ mv2 -----1
Fm v v2 = u2 + 2as
= 0 + 2(10)(20)
v = 20 m/s
F v 1, Ek = ½ (1)(202)
Ek = 200 J Ans
F F F F ˈ F F
F
F 16
5 F 100 N/m F 7 F F
F
k =100
0.07 - 0.05 = 0.02 m
0.05 Ep = ½ kx2
= ½ (100)(0.022)
Ep = 0.02 J Ans
0.07
F 17
0.1 F 100 N/m F 20 F
F F F F F F
. . A, B, C, D E
. F EF
B
O
D
u = 0 0.2 1.0
0.5 vF = 0
k = 100 A C E F
10.0
F ˆ F www.schoolDD.com 16
18. . v A , vB , vC , vD , vE = ?
F F F F F
2
F ∑ E = Epo = ½ kx
= ½ (100)(0.22)
∑ E = 2 J
O --> A ∑ E = ∑ E F
2
2 = ½ mvA
2 = ½ (0.1)( vA2)
vA = 6.3 m/s Ans
O --> B ∑ E = ∑ E
2 = mgh + ½ mvB2
2 = 0.1(10)(1) + ½ (0.1)( vB2)
vB = 4.5 m/s Ans
O --> C ∑ E = ∑ Eେ
2 = ½ mvC2
2 = ½ (0.1)( vC2)
vC = 6.3 m/s Ans
O --> D ∑ E = ∑ Eୈ
2 = mgh + ½ mvD2
2 = 0.1(10)(0.5) + ½ (0.1)( vD2)
vD = 5.5 m/s Ans
O --> E ∑ E = ∑ E
2 = ½ mvE2
2 = ½ (0.1)( vE2)
vE = 6.3 m/s Ans
. W E◊ F = ?
O --> F ∑ E - WE◊ F = ∑ E
2 - W E◊ F = 0
WE◊F = 2 J Ans
F F O F F E --> F F
F F
F ˆ F www.schoolDD.com 17
19. F 18
m F F F F .
0.4
u=0
m
A
v=0
2.0
m m
B f C
s=?
A --> C ∑ E - WB ◊ C = ∑ E େ
mgh - f s = 0
mgh - µmg(s) = 0
s = h/µ = 2/0.4
s = 5 m Ans
F 19
F F F F F 1 .
0.1 F
. F F
. F
A
u=0 vD = 0
D
0.50
hmax
f
B 1.0 C
. hmax = ?
A --> D ∑ E - WB ◊ C = ∑ E ୈ
mgh - f s = mghmax
F ˆ F www.schoolDD.com 18
20. mgh - µmg(s) = mghmax
hmax = h - µ(s) = 0.5 0.1(1)
hmax = 0.4 m Ans
. F
F F
∑ E = WB◊ C
mgh = fs
mgh = µmg(s)
s = h/µ = 0.5/0.1
s = 5.0 m
- F F C F s = 5.0 m Ans
¬ - F F B ?
F 20
F 1.0 F 1 / F F
k = 100 / F F
v=1
k =100
m
x
m
F F ˈ F F
F = ∑ Eଶ
∑ Eଵ
½ mv2 = ½ kx2
1.0(12) = 100(x2)
x2 = 1/100
x = 0.1 m Ans
F ˆ F www.schoolDD.com 19
21. F 21
F 0.2 60 F
60° l = 2
u=0
h
v
F F F F ˈ F
F ∑ Eଵ= ∑ Eଶ
mgh = ½ mv2
v2 = 2gh = 2g(l l cos60)
v2 = 2(10)(2-2(1/2))
v = √20 m/s Ans
F 22
1 F F F F
F 0.10 F F F F
F F
k
m
x1 = 0.1 m
x2 = ?
m
F F F
F F = mg
kx1 = mg
k = 1(10)/0.1 = 100 N/m
F F F F ˈ F F
F ∑ Eଵ = ∑ Eଶ
mgh = ½ kx2
1(10) x2 = ½ (100)( x22)
x2 = 0.2 m Ans
F ˆ F www.schoolDD.com 20
22. F 23
10 1000 F 1
F ( F F )
m u=0
F F F F
h =1000 ∑ Eଵ W1◊2 = ∑ Eଶ
mg(h+x) f x = 0
m
mg(h+x) = f x
f x =1.0
m 10(10)(1000+1) = f (1)
v=0 f = 100100 N Ans
F 24
F 1 ˀ µk = 0.5 37 F F F k = 100
N/m F 1.0 F F F
1.0
u=0
m x=?
v=0
m
37°
F F F F ˈ F F F F
F W1◊2 =
∑ Eଵ ∑ Eଶ
mgh f s = ½ kx2
mg (1+x) sin37 - µmg cos37(1+x) = ½ kx2
1(10)(1+x)(3/5) 0.5(1)(10)(4/5)(1+x) = ½ (100) x2
6(1+x) 4(1+x) = 50 x2
25 x2 x 1 = 0
ିሺିଵሻേඥሺିଵሻమ ିସሺଶହሻሺିଵሻ
x = ଶሺଶହሻ
x = 0.22 m Ans
F ˆ F www.schoolDD.com 21
23. F 25
F m F F F h F F ˈ F
r F ˈ F
A u=0
m m
N=0
h 2r v
B
h m
r
+ 2r mg
A --> B
F F F F F A ˈ F F B
F ∑ E = ∑ E
mg (h-2r) = ½ mv2
2g (h-2r) = v2 ----1
F B ˈ F F N=0
mg F ˈ F F
୫୴మ
Fc = ୰
୫୴మ
mg = ୰
v2 = r g
F v2 = r g 1, 2g (h-2r) = r g
ହ
h = ଶr Ans
5.6
F F F F
F F F
F = F
F ˆ F www.schoolDD.com 22
24. F F
F = F
Eff ˈ
F F F F 100 F ˈ %
܅F
Eff = ܅F
x 100
F
W F = Fs
F W F = mgh
s s = h
h m
F Eff = 100 % , W F = W F
m F s = mgh
F = mg
mg FF
W F = Fs
W F = mgh
s s = 2h
F Eff = 100 % , W F = W F
F s = mgh
m F
h F = mg/2
F
m
mg
F ˆ F www.schoolDD.com 23
25. F
s
h m
mg W F = Fs
W F = mgh
F Eff = 100 % , W F = W F
F s = mgh
୦
s m F = mg ୱ
F s>h
F h
m
mg
R
W F = Fs
r F W = mgh
F
s s = 2 ߨR h = 2 ߨr
h m F Eff = 100 % , W F = W F
Fs = mgh
m ୰
F = mg ୖ
mg F R>r
F
r W F = F (2 ߨr)
W F = mgh
m F
F Eff = 100 % , W F = W F
mg F (2 ߨr) = mgh
h ୦
F = mg ሺ2 గrሻ
F r>h
F ˆ F www.schoolDD.com 24
