פתרון בגרות מכניקה — שאלון 36361

m = 164 kg ; g_M = 1.67 m/s² ; t_land = 12.45 s
v(0) = −20 m/s ; v(10) = 0 m/s ; v(12.45) ≈ −4.09 m/s
a₁ = +2.0 m/s² (0 ≤ t ≤ 10) ; a₂ = −1.67 m/s² (10 ≤ t ≤ 12.45)

0.00 s

0.00 s

105.01 m

−20.00 m/s

+2.00 m/s²

601.88 N

273.88 N

+328.00 N

32800.0 J

a = g_M = 1.67 m/s²

10 ≤ t ≤ 12.45 s ; a = 1.67 m/s²

a₂ = v(12.45) − v(10)12.45 − 10 = −4.09 − 02.45 = −1.67 m/s²

W = m·g_M = 164 · 1.67 = 273.88 N F = 601.88 N

ΣF = F − m·g_M = 601.88 − 273.88 = 328 N Fm·g_M = 601.88273.88 ≈ 2.20

a₁ = v(10) − v(0)10 − 0 = 0 − (−20)10 − 0 = 2010 = +2.0 m/s²

ΣF = m·a₁ ⟹ F − m·g_M = m·a₁ F = m·(a₁ + g_M)

F = 164 · (2.0 + 1.67) = 164 · 3.67 = 601.88 N

W_tot = ΔE_k = 0 − 12 · 164 · 20² = −32800 J W_tot = −(F − m·g_M)·d₁ = −(601.88 − 273.88) · 100 = −328 · 100 = −32800 J

Δt = 12.45 − 10 = 2.45 s

h = v(10)·Δt + 12·g_M·Δt² = 0 + 12 · 1.67 · (2.45)² h = 12 · 1.67 · 6.0025 = 10.0241752 = 5.0120875 m

m·g_M·h = 12·m·v_f² ⟹ v_f = √2·g_M·h = √16.74037225 = 4.0915 m/s v_f = g_M·Δt = 1.67 · 2.45 = 4.0915 m/s

d₁ = 12 · |v₀| · t₁ = 12 · 20 · 10 = 100 m

d₁ = |v₀|·t₁ − 12·a₁·t₁² = 20 · 10 − 12 · 2 · 10² = 200 − 100 = 100 m

H = d₁ + h = 100 + 5.0120875 = 105.0120875 m

W_g = m·g_M·H = 273.88 · 105.0120875 = 28760.71 J W_F = −F·d₁ = −601.88 · 100 = −60188 J W_tot = 28760.71 − 60188 = −31427.29 J ΔE_k = 12 · 164 · (4.0915)² − 12 · 164 · 20² = 1372.71 − 32800 = −31427.29 J

0 ≤ t ≤ 10 : y(t) = 105.0120875 − 20·t + t² 10 ≤ t ≤ 12.45 : y(t) = 5.0120875 − 0.835·(t − 10)²

y(10)H = 5.0120875105.0120875 ≈ 0.0477

m₁ = 1 kg ; m₂ = 4 kg ; α = 36.9° ; sinα = 0.6 ; cosα = 0.8
g = 10 m/s² ; a = 2 m/s² ; v₀ = 2.6 m/s ; T₁ = T₂ = T

0.00 s

0.20

36.9°

1.00 ×

0.00 s

0.00 m/s

2.00 m/s²

9.60 N

8.00 N

32.00 N

1.60 N

6.40 N

0.000 m

0.00 J

m₂·g·sinα = 4 · 10 · 0.6 = 24 N ; m₁·g·sinα = 1 · 10 · 0.6 = 6 N

m₁·g = 1 · 10 = 10 N

m₂·g = 4 · 10 = 40 N

T₁ = T₂ = T ; |a₁| = |a₂| = a

T − m₁·g·sinα − f₁ = m₁·a N₁ − m₁·g·cosα = 0 ⟹ N₁ = m₁·g·cosα f₁ = μ·N₁ = μ·m₁·g·cosα ⟹ T − m₁·g·sinα − μ·m₁·g·cosα = m₁·a

m₂·g·sinα − T − f₂ = m₂·a N₂ = m₂·g·cosα ; f₂ = μ·N₂ = μ·m₂·g·cosα m₂·g·sinα − T − μ·m₂·g·cosα = m₂·a

N₁ = 1 · 10 · 0.8 = 8 N ; N₂ = 4 · 10 · 0.8 = 32 N m₁·g·sinα = 6 N ; m₂·g·sinα = 24 N

g·sinα·(m₂ − m₁) − μ·g·cosα·(m₁ + m₂) = (m₁ + m₂)·a

10 · 0.6 · (4 − 1) − μ · 10 · 0.8 · (1 + 4) = (1 + 4) · 2 18 − 40μ = 10 ⟹ 40μ = 8 μ = 840 = 0.2

T = m₁·a + m₁·g·sinα + μ·m₁·g·cosα = 2 + 6 + 1.6 = 9.6 N T = m₂·g·sinα − μ·m₂·g·cosα − m₂·a = 24 − 6.4 − 8 = 9.6 N

f₁ = 0.2 · 8 = 1.6 N ; f₂ = 0.2 · 32 = 6.4 N ; f₁ + f₂ = 8 N ΣF = 24 − 6 − 8 = 10 N ; a = 105 = 2 m/s²

f₁ = 1.6 N ; f₂ = 6.4 N ; f₁ + f₂ = 8 N

m₁·g·sinα − T − μ·m₁·g·cosα = m₁·a T − m₂·g·sinα − μ·m₂·g·cosα = m₂·a

g·sinα·(m₁ − m₂) − μ·g·cosα·(m₁ + m₂) = (m₁ + m₂)·a 10 · 0.6 · (1 − 4) − 0.2 · 10 · 0.8 · (1 + 4) = 5a ⟹ −18 − 8 = 5a a = −265 = −5.2 m/s²

T = m₁·g·sinα − μ·m₁·g·cosα − m₁·a = 6 − 1.6 + 5.2 = 9.6 N T − m₂·g·sinα − μ·m₂·g·cosα = 9.6 − 24 − 6.4 = −20.8 N = 4 · (−5.2) ✓

t = v₀|a| = 2.65.2 = 0.5 s d = v₀²2·|a| = 6.7610.4 = 0.65 m

E_k = 12 · 5 · 2.6² = 16.9 J ; h = 0.65 · 0.6 = 0.39 m ΔE_p = (m₂ − m₁)·g·h = 3 · 10 · 0.39 = 11.7 J W_f = 8 · 0.65 = 5.2 J ; 11.7 + 5.2 = 16.9 J ✓ |F| = 16.90.65 = 26 N ; |a| = 265 = 5.2 m/s² ✓

μ·(N₁ + N₂) = 0.2 · 40 = 8 N ; g·sinα·(m₂ − m₁) = 18 N

# | 1 | 2 | 3 | 4 | 5
R (m) | 16 | 20 | 24 | 28 | 32
v_max (m/s) | 11.31 | 12.65 | 13.86 | 14.97 | 16
v_max² (m²/s²) | 128 | 160 | 192 | 224 | 256

1 · R = 16 m

11.31 m/s

0.80

1.00×

16.00 m

11.31 m/s

128.00 m²/s²

8.00 m/s²

0.707 rad/s

8.89 s

8.00 N/kg

m·g ; N ; f_s

ΣF_y = N − m·g = 0 ⟹ N = m·g

a_c = v²R ΣF_r = f_s = m·a_c = m · v²R

f_s = m·ω²·R = m · 4π²·RT²

f_s ≤ f_s,max = μ_s·N

f_s,max = m · v_max²R

μ_s·m·g = m · v_max²R

v_max² = μ_s·g·R

12816 = 16020 = 19224 = 22428 = 25632 = 8 m/s²

(16 , 128) ; (20 , 160) ; (24 , 192) ; (28 , 224) ; (32 , 256)

v_max² = 8·R

v₁² = 8 · 18 = 144 m²/s² ; v₂² = 8 · 36 = 288 m²/s²

v₂² − v₁²R₂ − R₁ = 288 − 14436 − 18 = 14418 = 8

m²/s²m = m/s² 8 m/s²

256 − 12832 − 16 = 12816 = 8 m/s²

a_c,max = μ_s·g = 8 m/s²

v_max² = (μ_s·g)·R ⟹ μ_s·g = 8 m/s²

μ_s = 8g = 810 = 0.8

μ_s = v_max²g·R = 12810 · 16 = 128160 = 0.8 μ_s = 25610 · 32 = 256320 = 0.8

T = 2π·Rv_max ; v_max = √μ_s·g·R = √8·R T = 2π·R√8·R = 2π√R8

T₁ = 2π · 1611.31 = 100.5311.31 = 8.89 s T₅ = 2π · 3216 = 201.0616 = 12.57 s T₁ = 8.89 ; T₂ = 9.93 ; T₃ = 10.88 ; T₄ = 11.75 ; T₅ = 12.57 s

ω = v_maxR = √8R ω₁ = √816 = 0.707 rad/s ⟹ T₁ = 2π0.707 = 8.89 s ω₅ = √832 = 0.500 rad/s ⟹ T₅ = 2π0.500 = 12.57 s

a_c = v_max²R = μ_s·g = 8 m/s² a_c = 4π²·RT² = const ⟹ T² ∝ R

t (ms) | 0 – 10 | 10 – 20 | 20 – 40 | 40 – 100
F (N) | 0 | 0 ... 40 | 40 | 40 ... 0
m = 2 kg ; v₀ = 0.8 m/s ; 1 ms = 0.001 s

0.0 ms

0.8 m/s

2.0 kg

40 N

0.0 ms

0.00 N

0.00 m/s²

0.80 m/s

1.60 kg·m/s

0.00 N·s

−0.30 m/s

−2.20 kg·m/s

J = Σ F·Δt

10 ms = 0.010 s ; 20 ms = 0.020 s ; 40 ms = 0.040 s ; 100 ms = 0.100 s 40 − 00.020 − 0.010 = 400.010 = 4000 N/s 0 − 400.100 − 0.040 = −400.060 = −666.7 N/s

J₁ = 12 · 0.010 · 40 = 0.2 N·s J₂ = 0.020 · 40 = 0.8 N·s J₃ = 12 · 0.060 · 40 = 1.2 N·s J = 0.2 + 0.8 + 1.2 = 2.2 N·s

J = 0.090 + 0.0202 · 40 = 0.1102 · 40 = 2.2 N·s

J = −2.2 N·s

p₀ = m·v₀ = 2 · 0.8 = 1.6 kg·m/s

J = 2.2 N·s

Jp₀ = 2.21.6 = 1.375 = 118

J = Δp = p_f − p₀ ; p_f = 1.6 − 2.2 = −0.6 kg·m/s

J = Δp = m·v_f − m·v₀

−2.2 = 2·v_f − 2 · 0.8 −2.2 = 2·v_f − 1.6 ⟹ 2·v_f = −0.6 v_f = −0.62 = −0.3 m/s

Δv = Jm = 2.22 = 1.1 m/s v_f = v₀ − Δv = 0.8 − 1.1 = −0.3 m/s

0.30.8 = 0.375 = 38

E_k,0 = 12 · 2 · 0.8² = 0.64 J E_k,f = 12 · 2 · 0.3² = 0.09 J ΔE_k = 0.09 − 0.64 = −0.55 J

a = ΣFm = F_wallm

a_max = −402 = −20 m/s²

−20 − 00.020 − 0.010 = −2000 m/s³ 0 − (−20)0.100 − 0.040 = 333.3 m/s³

Δv = −( 12 · 0.010 · 20 + 0.020 · 20 + 12 · 0.060 · 20 ) Δv = −(0.1 + 0.4 + 0.6) = −1.1 m/s

Δt = 100 − 10 = 90 ms = 0.090 s

J = F·Δt ⟹ F = JΔt = 2.20.090 = 24.44 N

a = ΔvΔt = 1.10.090 = 12.22 m/s² F = m·a = 2 · 12.22 = 24.44 N

0 ; 24.44 ; 40 N

m₁ = 60 g = 0.06 kg ; ℓ₁ = 20 cm = 0.20 m ; T₁ = 0.5 s ; g = 10 m/s²
ω = 2·πT ; T = 2·π·√mk ; T₂ = 1.2 · T₁

0.20 m

T = 0.500 s

0.0264 kg

T₂ = 0.600 s · 20.0%

0.000 s

0.000 s

0.200 m

20.00 cm

0.000 m/s

−31.58 m/s²

12.566 rad/s

0.500 s

9.475 N/m

0.0600 kg

0.0000 J

0.1895 J

0.0279 m

m·g = k·x₀ ΣF = m·g − k·(x₀ + y) = −k·y ω = √km ; T = 2·π·√mk

ω₁ = 2·πT₁ = 2·π0.5 = 4·π = 12.566 rad/s

A = ℓ₁ = 20 cm = 0.20 m

y(t) = A·cos(ω·t + φ₀) y(0) = A·cos φ₀ = +A ⟹ cos φ₀ = 1 ⟹ φ₀ = 0 v(0) = −A·ω·sin φ₀ = 0 ✓

t = T₁2 = 0.25 s ⟹ y = 0.20·cos(π) = −0.20 m t = T₁ = 0.5 s ⟹ y = 0.20·cos(2·π) = +0.20 m

v(t) = −0.20 · 4·π · sin(4·π·t) = −2.513·sin(4·π·t) m/s a(t) = −0.20 · 16·π² · cos(4·π·t) = −31.58·cos(4·π·t) m/s²

y = ℓ₁2 = 0.202 = 0.10 m

|v| = ω·√A² − y² |v| = 4·π·√0.20² − 0.10² = 12.566·√0.03 = 12.566 · 0.17321 |v| = 2.1766 m/s ≈ 2.18 m/s

k = m₁·ω₁² = 0.06 · 157.914 = 9.4748 N/m 12·m·v² + 12·k·y² = 12·k·A² E = 12 · 9.4748 · 0.20² = 0.18950 J E_p = 12 · 9.4748 · 0.10² = 0.04737 J E_k = 0.18950 − 0.04737 = 0.14212 J v = √2·E_km₁ = √4.7374 = 2.1766 m/s ✓

0.10 = 0.20·cos(4·π·t) ⟹ cos(4·π·t) = 0.5 ⟹ 4·π·t = π3 t = 112 = 0.0833 s = T₁6 v = −2.513·sinπ3 = −2.513 · 0.86603 = −2.1766 m/s

T₂ = 1.2 · T₁ = 1.2 · 0.5 = 0.6 s

T₁ = 2·π·√m₁k ; T₂ = 2·π·√m₁ + m₂k T₂T₁ = √m₁ + m₂m₁

(T₂T₁)² = m₁ + m₂m₁ = 1 + m₂m₁ (1.2)² = 1.44 = 1 + m₂m₁ ⟹ m₂m₁ = 0.44 m₂ = 0.44 · 0.06 = 0.0264 kg = 26.4 g

k = 4·π²·m₁T₁² = 39.478 · 0.060.25 = 2.36870.25 = 9.4748 N/m k = 4·π²·(m₁ + m₂)T₂² = 39.478 · 0.08640.36 = 3.41090.36 = 9.4748 N/m

m₁·g = k·x₁ ; (m₁ + m₂)·g = k·x₂ Δ = x₂ − x₁ = m₂·gk

Δ = 0.0264 · 109.4748 = 0.2649.4748 = 0.02786 m Δ = 0.0279 m ≈ 2.79 cm

x₁ = m₁·gk = 0.69.4748 = 0.06333 m = 6.333 cm x₂ = (m₁ + m₂)·gk = 0.8649.4748 = 0.09119 m = 9.119 cm Δ = 9.119 − 6.333 = 2.786 cm ✓

Δ = g·(T₂² − T₁²)4·π² = 10 · (0.36 − 0.25)39.478 = 1.139.478 = 0.02786 m ✓

m·a = −k·y ⟹ a = −km·y = −ω²·y a = −(4·π)²·y = −157.9·y

a_max = ω²·A = 157.9 · 0.20 = 31.6 m/s² ; a(y = 0) = 0

ω₂ = 2·π0.6 = 10.472 rad/s ; −ω₂² = −109.7 1/s²

R_M = 1.74·10⁶ m ; M_M = 7.35·10²² kg ; M_E = 5.98·10²⁴ kg
r_M = 3.84·10⁸ m ; G = 6.67·10⁻¹¹ N·m²/kg² ; m = 164 kg
h = 200 km = 2.00·10⁵ m ; r_S = R_M + h = 1.94·10⁶ m

200 km

164 kg

1940.0 km

1589.7 m/s

2.52704 ·10⁶ m²/s²

7667.9 s

2.130 h

1.3026 m/s²

1.6193 m/s²

213.63 N

0.4436 N

2.077 ·10⁻³

2.0722 ·10⁸ J

−2.0722 ·10⁸ J

4.764 ·10⁷ J

−2.5485 ·10⁸ J

1589.7 m/s

0.0 m/s

0.0000 ·10⁸ J

0.0 s

F_M = G·M_M·mr_S² F_E = G·M_E·md²

r_Mr_S = 3.84·10⁸1.94·10⁶ = 197.9 d ≈ r_M = 3.84·10⁸ m

F_EF_M = G·M_E·mr_M²G·M_M·mr_S² = M_EM_M·r_S²r_M²

M_EM_M = 5.98·10²⁴7.35·10²² = 81.36 r_Sr_M = 1.94·10⁶3.84·10⁸ = 5.0521·10⁻³ r_S²r_M² = 2.5524·10⁻⁵ F_EF_M = 81.36 · 2.5524·10⁻⁵ = 2.077·10⁻³ ≈ 2.08·10⁻³

F_MF_E = 12.077·10⁻³ = 482

d = r_M − r_S = 3.84·10⁸ − 1.94·10⁶ = 3.8206·10⁸ m F_EF_M = 81.36 · (1.94·10⁶)²(3.8206·10⁸)² = 81.36 · 2.5783·10⁻⁵ = 2.098·10⁻³

r_S = R_M + h = 1.74·10⁶ + 2.00·10⁵ = 1.94·10⁶ m

G·M_M·mr_S² = m·v_A²r_S v_A² = G·M_Mr_S

v_A² = 6.67·10⁻¹¹ · 7.35·10²²1.94·10⁶ = 4.90245·10¹²1.94·10⁶ = 2.52704·10⁶ m²/s² v_A = √2.52704·10⁶ = 1589.7 m/s ≈ 1.59·10³ m/s

g_M = G·M_MR_M² = 4.90245·10¹²3.0276·10¹² = 1.6193 m/s² a_c = g_M·R_M²r_S² = 1.6193 · 1.74²1.94² = 1.3026 m/s² v_A = √a_c·r_S = √1.3026 · 1.94·10⁶ = 1589.7 m/s

T_S = 2·π·r_Sv_A = 2·π·1.94·10⁶1589.7 = 7.67·10³ s

G·M·mr² = m·ω²·r = m·4·π²T²·r T²r³ = 4·π²G·M

T_M²r_M³ = 4·π²G·M_E ; T_S²r_S³ = 4·π²G·M_M T_S² · r_M³r_S³ · T_M² = M_EM_M = 81.36

T_S = 2·π·r_Sv_A = 2·π·1.94·10⁶1589.7 = 7.67·10³ s

W = ΔE_k = E_k,f − E_k,i

E_k,i = 12·m·v_A² ; E_k,f = 0 W = 0 − 12·m·v_A² = −12·m·v_A²

W = −12 · 164 · 2.52704·10⁶ = −82 · 2.52704·10⁶ W = −2.072·10⁸ J

W_g = U_i − U_f = G·M_M·m·1R_M − G·M_M·m·1r_S G·M_M·m = 4.90245·10¹² · 164 = 8.04002·10¹⁴ W_g = 8.04002·10¹⁴ · 5.9249·10⁻⁸ = +4.764·10⁷ J

E_i = 12·m·v_A² − G·M_M·mr_S = 2.07217·10⁸ − 4.14434·10⁸ = −2.07217·10⁸ J E_f = 0 − G·M_M·mR_M = −8.04002·10¹⁴1.74·10⁶ = −4.62070·10⁸ J W_eng = E_f − E_i = −4.62070·10⁸ + 2.07217·10⁸ = −2.54853·10⁸ J W = W_g + W_eng = +4.7636·10⁷ − 2.54853·10⁸ = −2.0722·10⁸ J

v_A = 1.59·10³ m/s ⟹ v_f = 0

m₁·Δv₁ + m₂·Δv₂ = 0 Δv₁ = −m₂m₁·Δv₂

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