Consider an ideal spring that has an unstretched length l0 = 3.9 m. Assume the spring has a constant k = 20 N/m. Suppose the spring is attached to a mass m = 6 kg that lies on a horizontal frictionless surface. The spring-mass system is compressed a distance of x0 = 1.7 m from equilibrium and then released with an initial speed v0 = 5 m/s toward the equilibrium position.
(1)What is the position of the block as a function of time. Express your answer in terms of t.
(2)How long will it take for the mass to first return to the equilibrium position?
(3)How long will it take for the spring to first become completely extended?
To find the position of the block as a function of time (x(t)), we need to consider the motion of the mass attached to the spring.
(1) The equation of motion for the mass-spring system is given by the equation:
m * d²x/dt² = -k * x
where m is the mass of the block, x is the position of the block, t is time, and k is the spring constant.
To solve this second-order differential equation, we assume a solution of the form x(t) = A * cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase angle.
First, we need to find the angular frequency ω. Using the relationship ω = √(k / m), we can substitute the given values:
ω = √(20 N/m / 6 kg) = √(10/3) rad/s
Next, we need to find the amplitude A and the phase angle φ. We use the initial conditions provided.
Given x0 = 1.7 m (initial displacement) and v0 = 5 m/s (initial velocity), we can use the following equations:
A = x0 = 1.7 m
φ = tan^(-1)(v0 / (ω * x0)) = tan^(-1)(5 m/s / (√(10/3) rad/s * 1.7 m))
Now we can express the position of the block as a function of time, x(t):
x(t) = A * cos(ωt + φ)
(2) To find the time it takes for the mass to first return to the equilibrium position, we need to calculate the period of the oscillation. The period T is given by:
T = 2π / ω
Using the value of ω calculated earlier, we can find the period T.
T = 2π / √(10/3) rad/s
(3) To find the time it takes for the spring to first become completely extended, we need to determine when the displacement reaches its maximum value, which is equal to the amplitude A.
We already found the amplitude A to be 1.7 m. Now, we need to calculate the time required for the displacement x(t) to reach this maximum value. Using the equation:
A = x(t) = A * cos(ωt + φ)
Simplifying, we have:
1 = cos(ωt + φ)
To find the value of t, we can solve for ωt + φ:
ωt + φ = cos^(-1)(1)
Since cos(0) = 1, the value inside the inverse cosine function must be 0:
ωt + φ = 0
Therefore, the time required for the spring to first become completely extended is t = 0.
Note: In the last question, it is important to note that the spring continues to oscillate, reaching maximum extension multiple times as time progresses. However, the first complete extension occurs at t = 0.