f
8, 0N
f = F. cos cos 15o
5, 0N 0 , 7m
f = 7, 42N 42N
∆E t = f.d f. d = (7, (7, 42N 42N )).(4, (4, 06m 06m) = 30, 30, 1J
W = F.d. cos θ = (8, (8, 0N ) N ).(0, (0, 70m 70m).(cos (cos 0o ) = 5, 6J F. cos cos 15o − f = 0 ⇒ N + F. + F. sin sin 15o − mg = mg = 0,
∆E t = F at .d = (5, (5, 0N ) N ).(0, (0, 70m 70m) = 3, 5J at .d =
m
F
f
N f = 7, 42N 42N
27 27kg kg N = mg −F. sin15o = (3 ( 3, 75kg 75kg)(9 )(9,, 8m/s2 )−(7, (7, 68N 68N )). sin15o = 33, 33 , 0N
9, 2m 32o 0, 20
f = µ c .N ⇒ µc =
F at at
35 35,, 0N
F. cos cos 32o = µc .N P = N + N + F. F. sin(32o ) F =
f (7, (7, 42N 42N )) = = 0, 225 N (33, (33, 0N ) N )
4, 00 00kg kg 0, 600
µc .N ⇒ F = 55, 55 , 5N cos cos 32o
3, 00 00m m 40 40,, 0J
W = F.d. cos θ = (55, (55, 5N ) N ).(9, (9, 2m). cos cos 32o = 433J 433J W = F.d. cos θ = (35, (35, 0N )(3 N )(3,, 00m 00m). cos0o = 105J 105J o F at at = µ c .(mg − F. sin32 )
∆E t = F at .d = (27. (27.9, 8 − 55, 55, 5. sin32o ).0, 2.9, 2 = 433J 433 J at .d =
∆E t = F at .d = µ µ c .N.d = .N.d = µ µ c .m.g.d = .m.g.d = (0, (0, 600)(4, 600)(4, 00kg 00kg)(9 )(9,, 80m/s 80m/s2 )(3, )(3, 00m 00m) at .d =
3, 57 57kg kg 40, 40, 0J
4, 06 06m m o
7, 68 68N N 15
(70, (70, 6 −
40, 40, 0)J 0)J = = 30, 30, 6J 70, 70, 6J (105 − 70, 70, 6)J 6)J = 34, 34, 4J
W = F.d. cos θ = (7, (7, 68N 68N )).(4, (4, 06m 06m). cos cos 15o = 31, 31 , 1J
520Kg 520 Kg 500m 500 m
300m 300 m
0.25 U U
U
U = m.g.h
U = (520)(9, 8)(300) = 1, 53.106 J
K i = ∆U + ∆E ter = m.g.(h + µ.d),
K i =
1 mv 2 2
m
1 2 v2 h v = g(h + µ.d) =⇒ d = i − = 1, 2m 2 2µg µ
5, 29N 20m/s 0., 65N
h ∆E ter = f .h
f k = µ k .N = µ k .m.g. sin( π2 − θ) = µ k .m.g. cos θ ∆E at = f k .d = µ k .m.g.d. cos θ d d = 500 x cos θ cos θ = d x = 400m ∆E ter = µk .m.g.d. cos θ = µk .m.g.d. xd = µk .m.g.x = (0.25)(520)(9.8)(400) = 5, 1.105 J
K f + U f + ∆E ter = K i + U i
0 + m.g.h + f.h =
K f = K i + U i − U f − ∆E ter
h h = h =
6
K f = 0 + 1, 53.10 − 0 − 5, 1.10
2 m.vo 2(m.g+f )
=
2 m.vo 2(m.g+f )
(0,54)(20)2 2(0,54.9,8+0,265)
K f = 63m/s
m =
5,29 9,8
= 0, 54Kg
= 19, 4m
5
∆E ter = 2.f.h
K f = 0 + 1, 02.106 J
1 mv 2 2
1 .m.vo2 2
K f + ∆E ter = K i
v =
1 1 m.v 2 + 2.f.h = m.vo2 2 2 1 1 m.vo2 m.v 2 = m.vo2 − 2.f. 2 2 2(m.g + f ) v 2 = v o2 (1 − v 2 = (20)2 (1 −
h
vo 1, 1m e µk = 0, 60
6, 0m/s d
2.f ) (m.g + f )
2.0, 265 )= 5, 29 + 0, 265
361, 836 = 19m/s
x = 0, 10m o
4.0Kg
30 128J
0, 30 t = 0 1kg ˆ ˆ v = (18m/s)i + (24m/s) j t = 0 t = 6s
∆U
∆y = v o .t − y
1 .g.t2 2
∆y = −32m ∆U = m.g.∆y = −318J = 3, 2.10
2
−
J
N = m.g. cos θ
25cm
15g
f k = µ k .m.g. cos θ
K i = K f + ∆U + ∆E ter K i = 0 + m.g.d. sin θ + µk .m.g.d. cos θ dx d
d =
x
Du = −(dx)gx =⇒ =
m h
K i 128 = = 4, 3m. m.g(sin θ + µk . cos θ) 4.9, 8.(sin 30o + 0, 3. cos 30o ) 0
h ∆U =
1 1 du = − gx.dx = − .g.h2 = − .m.g.h 2 2
25J 7, 5cm
E =
1 .k.x2 2
m = 15g
h = 25cm
∆U = −0, 018J
k = 89N/cm.
m = 2kg h = 40cm
k = 1960n/m
K i + U i = K f + U f 0 = −m.g(h + x) +
1 .k.x2 2
U (x)
U (x)
x
M.g = 19, 6N, h = 0, 4m e k = 1960n/m
U (x)
F dx F (2) =⇒ x ≤ −3, F (x) = 0
U (x) =
−3 ≤ −2 ≤
−π ×
x ≤ −2, F (x) = −1
θ =
x ≤ −1, F (x) = 2(x + 1, 5)
−1 ≤
θ × 3π
π
θ
−3
x ≤ −1, F (x) = 1
1, 6m
1 ≤ x ≤ 2, F (x) = 2(x − 1, 5) 2 ≤ x ≤ 3, F (x) = −1 x ≥ 3, F (x) = 0 U (x) x ≥ 3 =⇒ U (x) = U (0) = 0
x≤3
0 ≤ x ≤ 1 =⇒ U (x) =
x
dx = −x
0
x = 1 −→ U (1) = 1
1 ≤ x ≤ 2 =⇒ U (x) = −1 −
x2 − 3x + 1 x = 1 −→ U (1) = 1 − 3 + 1 = −1 x = 2 −→ U (2) = 4 − 6 + 1 = −1 2 ≤ x ≤ 3 =⇒ U (x) = −1 +
x
(2x − 3)dx = −1(x2 − 3x)
1
x
1
=
x
dx = −1 + x − 2 = x − 3
2
x = 2 −→ U (2) = −1 x = 3 −→ U (3) = 0
θ = − π2 F (θ) = −
dU (L) dU (Rθ) 1 dU (θ) =− =− dL d(Rθ) R dθ 1 .1, 6. cos θ 1, 6 − cos θ = 0, 1a
F (θ) = −
U (θ)
−→
dF = 0 −→ sin θ = 0 dθ a = −
cos(− π3 ) = −5m/s2 0, 1
75kg
7, 0m 12m 7, 0m
U (x) = 3x2 − x3 U (x)
[x] = metros
−2 ≤
[U ] = J oules
x≤2
W = m.g. cos θ W = (75kg)(9, 80m/s2 )(7, 0m)(cos 180o ) = −5100J W = 75.9, 8.7. cos0 = 5100J 90o W = 0
3x2 − x3 = 0
x = 0 ou x = 3 U (x) = 6x − 3x2
x = 0 ou x = 2
W t = 0
U (0) = 0 ou U (2) = 4
U (x)
x F (x) =
dU (x) dx
10kg
2
= 6x − 3x
0, 250 2, 00m 4, 00m
100g U (θ ) = 1, 6sin θ
θ
θ = 180
U (θ)
−m.g.µ.2πr
W =
W = −(0, 250)(10)(9, 80)(2π)(2) = −308J W = −616J
1, 5kg 0, 25 0, 6kg 1, 2N 3, 0m 3, 0m
−(1, 2N )(3, 0m)
= −3, 6J −3, 6J
−7, 2J
W 1 = W 2 = −(3, 675).(8, 0) = −29, 4J e W t = W 1 + W 2 = −59J W = −(3, 675).d d = 11, 3m (diagonal).W = −42J F.d = −(3, 675).(8, 0) = −29, 4J
−59J
