integrate following expressions?

Question

It's a question from physics (system dynamics), but essentially its some basic math. (L is a constant, omega is angular speed, also constant), x is derived by t, ie time.

x'' = (7*L/12) * [omega]^2 * cos([omega]*t)

all starting conditions were 0 (x'_o= 0, x_o=0 - starting speed and position were 0)

The answer should be x = (7*L/12) * (1- cos([omega]*t)). I need the steps in between.


2nd question (C is const)

x'' = C * (r* [omega]^2 /4 )* sin ([omega] *t/2)

answer is

 x = C * (r* [omega]t /2 - r* sin ([omega] *t/2)

and I need the steps in between.

Dambarudhar · Accepted Answer

(1) Given, x"(t) = (7Lω^2/12) Cosωt ; Integrating once, we get 
 x' (t) = (7Lω^2/12) ∫Cosωt dt = (7Lω/12) ∫Cosωt (ωdt)  [θ = ωt , dθ = ωdt]
 = (7Lω/12) Sinωt + k ; when t = 0, x'(0) = 0  => k = 0
 Hence x'(t) = (7Lω/12) Sinωt ;  Integrating again, we get
 x(t) = (7Lω/12) ∫Sinωt dt = (7L/12) ∫Sinωt (ωdt) ;  [θ = ωt , dθ = ωdt]
 = (7L/12)*(- Cosωt) + k' ;  when t = 0, x(0) = 0  => 0 = - (7L/12) + k'  => k' = (7L/12) 
 Hence x(t) = (7L/12) {1 - Cosωt}
 (2) Given, x"(t) = (C r ω^2 / 4) Sin(ωt/2) ; Integrating once, we get
 x'(t) = (C r ω^2 / 4) ∫Sin(ωt/2) dt = (C r ω / 2) ∫Sin(ωt/2) (ωdt/2) ;  [θ = ωt/2 , dθ = ωdt/2]
 = (C r ω / 2)*{- Cos(ωt/2)} + k ; when t = 0, x'(0) = 0   => k = (C r ω / 2)
 Hence, x'(t) = (C r ω / 2)*{1 - Cos(ωt/2)} ; Integrating again, we get
 x(t) = (C r ω / 2)*∫{1 - Cos(ωt/2)} dt = (Cr)*∫{1 - Cos(ωt/2)} (ωdt/2) ;  [θ = ωt/2 , dθ = ωdt/2]
 x(t) = (Cr)*{(ωt/2) -  Sin(ωt/2)} + k" ;  when t = 0, x(0) = 0  => k" = 0
 Hence x(t) = C*{r*(ωt/2) - r*Sin(ωt/2)}

SS4 · Answer

x''(t) = 7ω²L/12 * cos(ωt)

x'(t) = 7ωL/12 * sin(ωt) + C

We are told x'(0) = 0 so x'(0) = 7ωL/12 * sin(0) +C = 0 --> C=0

x'(t) = 7ωL/12 * sin(ωt)

x(t) = -7L/12 cos(ωt) + C' (where C' is also a constant)

At t=0, x(t) = 0 and so -7L/12 + C' = 0 therefore C' = 7L/12

x(t) = -7L/12 cos(ωt) + 7L/12 = 7L/12 - 7L/12 cos(ωt) = 7L/12(1-cos(ωt))

shiner · Answer

first one is 0.5x^4- x^3 + 2.5x^2 -2x + C 2d one is -2.5cos(2?) + 0.5 sin(4?) + C to the guy above: in case you do not recognize then get yahoo solutions violation of -10 factors

Anon E. Moose アナンイムース · Answer

For a harmonic function:
x(t) = k∙cos(wt)
dx/dt = -kw∙sin(wt)
d²x/dt² = -kw²∙cos(wt) = -w² x(t)

The answer should be x = (7*L/12) * (-cos([omega]*t))

The answer should be x = C * r * (-sin([omega]*t/2))

integrate following expressions?

4 個解答