1. Дано:

ΔL = 0,01 м

g = 10 Н/кг

Т - ?

Решение:

Т = 2π*√(m/k)

В состоянии покоя силы, действующие на груз:

Fупр = Fтяж

kΔL = mg

ΔL = mg/k = (m/k)*g

m/k = ΔL/g => √(m/k) = √(ΔL/g)

T = 2π*√(ΔL/g) = 2*3,14*√(0,01/10) = 0,1985... = 0,2 c

Ответ: 0,2 с.

2. Дано:

t = T/6

φ = 0

Eп(t)/Eк(t) - ?

Решение:

Ек = mυ(t)²/2

Eп(t) = Е - Ек(t), Е = Eк(max) = mυ(max)²/2 =>

=> Eп(t) = mυ(max)²/2 - mυ(t)²/2 = (m/2)*(υ(max)² - υ(t)²)

υ(t) = x' = [A*(sin(ωt) + φ)]' = Aω*cos(ωt), где Аω = υ(max)

υ(t)² = A²ω²*cos²(ωt)

υ(max)² = А²ω²

T = 2π/ω => ω = 2π/Τ

(υ(max)² - υ(t)²) = А²*(2π/Τ)² - A²*(2π/Τ)²*cos²[(2π/Τ)*(Τ/6)] = A²*4π²/Τ² - (Α²*4π²/Τ²)*cos²(π/3) = (4Α²π²/Τ²)*(1 - cos²(π/3)) = (4Α²π²/Τ²)*sin²(π/3)

υ(t)² = A²ω²*cos²(ωt) = (4Α²π²/Τ²)*cos²(π/3)

Eп(t)/Eк(t) = (m/2)*(υ(max)² - υ(t)²) / (m/2)*υ(t)² = υ(max)² - υ(t)² / υ(t)² = [ (4Α²π²/Τ²)*sin²(π/3) ] / (4Α²π²/Τ²)*cos²(π/3) = sin²(π/3) / cos²(π/3) = tg²(π/3) = 3

Ответ: 3.