Consider an undamped spring-mass system with mass 3 kg and a spring which is stretched 1 meter by 5 Newtons. Suppose an oscillating force 3 cos(ωt) is acting on the system. What value of ω causes resonance? For this value of ω, find a formula for x(t), the distance between the weight and equilibrium t seconds after the weight starts at x = 0 at rest
To determine the value of ω that causes resonance in an undamped spring-mass system, we need to consider the natural frequency of the system which is determined by the mass and the spring constant.
First, let's find the spring constant, denoted as k. The spring constant represents how stiff the spring is and is given by Hooke's Law as the force divided by the displacement:
k = F / x
In this case, the force applied is 5 Newtons and the displacement is 1 meter:
k = 5 N / 1 m = 5 N/m
The natural frequency of the system, denoted as ω_0, is given by the formula:
ω_0 = √(k / m)
where m is the mass of the system. In this case, the mass is 3 kg:
ω_0 = √(5 N/m / 3 kg) ≈ 1.08 rad/s
Resonance occurs when the frequency of the external force matches the system's natural frequency. In this case, the external force is described by 3 cos(ωt), so we need to find the value of ω for resonance.
Since the external force is a cosine function, the frequency of the external force is given by the coefficient of t, ω.
For resonance to occur, ω must be equal to ω_0, the natural frequency of the system. Therefore, ω = ω_0 ≈ 1.08 rad/s.
To find the equation for x(t), we need to solve the differential equation that represents the motion of the spring-mass system. In this case, since the system is undamped, the equation is:
m * x''(t) + k * x(t) = F_ext(t)
where m is the mass, k is the spring constant, x(t) is the displacement of the weight from equilibrium position, and F_ext(t) is the external force. In this case, F_ext(t) = 3 cos(ωt).
Substituting the values, we get:
3 kg * x''(t) + 5 N/m * x(t) = 3 cos(ωt)
To solve this equation, we can guess a solution of the form x(t) = A cos(ωt + φ), where A represents the amplitude and φ represents the phase angle.
Differentiating this equation twice with respect to t, we get:
x''(t) = -Aω^2 cos(ωt + φ)
Plugging this back into the equation, we get:
-3Aω^2 cos(ωt + φ) + 5 N/m * A cos(ωt + φ) = 3 cos(ωt)
Simplifying, we have:
(-3Aω^2 + 5 N/m * A) cos(ωt + φ) = 3 cos(ωt)
Comparing the coefficients of the cosine terms, we have:
-3Aω^2 + 5 N/m * A = 3
Solving this equation for A, we obtain:
A = 3 / (5 N/m - 3ω^2)
Thus, the equation for x(t) is:
x(t) = [3 / (5 N/m - 3ω^2)] * cos(ωt + φ)
Note that the phase angle φ cannot be determined solely from the given information and needs additional conditions, such as initial conditions or boundary conditions, to determine its value.