Final answer:
The equation of motion for a spring with a mass attached to it is x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant. With a force constant of 12 N/m and a mass of 1 kg, the angular frequency is 2√3. The equation of motion for the spring as a function of time is x(t) = A cos(2√3 t).
Step-by-step explanation:
The equation of motion for a mass attached to a spring undergoing simple harmonic motion is given by the equation:
x(t) = A cos(ωt + φ)
Where:
- x(t) is the displacement of the mass from its equilibrium position at time t
- A is the amplitude of the oscillation
- ω is the angular frequency and is given by ω = sqrt(k/m), where k is the force constant of the spring and m is the mass attached to the spring
- φ is the phase constant
In this case, the force constant is 12 N/m (as stated in the question) and the mass is 1 kg. Therefore, the angular frequency is ω = sqrt(12/1) = sqrt(12) = 2√3. The displacement from equilibrium at time t is given by:
x(t) = A cos(2√3 t + φ)
To find the equation of motion as a function of time, we need to determine the phase constant φ. At t = 0, the mass is stretched 1 m past equilibrium. Plugging in these values into the equation, we have:
1 = A cos(0 + φ)
Since cos(0 + φ) = cos(φ), we have:
1 = A cos(φ)
From this equation, we can determine that φ = 0 (since cos(0) = 1). Therefore, the equation of motion for the spring as a function of time is:
x(t) = A cos(2√3 t)