a weight of mass m is attached to a spring and oscillates with simple harmonic motion. By Hooke's Law, the vertical displacement, y(t) satisfies the differential equation dy/dt=sqrt(k/m)*sqrt(A^2-y^2)
where A(Fixed) is the maximum displacement and k is a constant, solve this differential equation by separation of variables, assuming that y(0)=0.
To solve this differential equation by separation of variables, we need to isolate the variables on different sides of the equation and integrate both sides separately.
Given the differential equation:
dy/dt = sqrt(k/m) * sqrt(A^2 - y^2)
Step 1: Rearrange the equation to separate the variables.
1/sqrt(A^2 - y^2) dy = sqrt(k/m) dt
Step 2: Integrate both sides of the equation separately.
∫1/sqrt(A^2 - y^2) dy = ∫sqrt(k/m) dt
To integrate the left-hand side, we use the trigonometric substitution. Let y = Asinθ, and dy = Acosθ dθ.
∫1/sqrt(A^2 - y^2) dy = ∫1/sqrt(A^2 - A^2sin^2θ) Acosθ dθ
= ∫1/sqrt(A^2(1 - sin^2θ)) Acosθ dθ
= ∫1/sqrt(A^2cos^2θ) Acosθ dθ
= ∫1/Acosθ Acosθ dθ
= ∫dθ
= θ + C
Since we know that y(0) = 0, this implies that θ(0) = 0 as well. Therefore, the constant C in the above expression is 0.
∫1/sqrt(A^2 - y^2) dy = θ
Integrating the right-hand side is straightforward:
∫sqrt(k/m) dt = sqrt(k/m) t + C
Step 3: Substitute back y = Asinθ to find the solution.
θ = sqrt(k/m) t
Using the identity y = Asinθ:
Asinθ = Asin(sqrt(k/m) t)
Thus, the solution to the differential equation is:
y(t) = Asin(sqrt(k/m) t)
where A is the maximum displacement and k is a constant.