A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the frequency of the system is 8/pi cycles per second, determine a trigonometric model that gives the position of the weight at time t seconds.
To find a trigonometric model that gives the position of the weight at time t seconds, we can use the formula for simple harmonic motion.
In simple harmonic motion, the position of an object can be described as a sinusoidal function of time. The general formula for the position of an object undergoing simple harmonic motion is given by:
x(t) = A * cos(ωt + φ) + x₀
Where:
- x(t) represents the position of the object at time t seconds
- A represents the amplitude of the motion (maximum displacement from the equilibrium position)
- ω represents the angular frequency of the motion
- φ represents the phase constant (initial phase)
- x₀ represents the equilibrium position (position at rest)
In this specific scenario, the weight is pulled down 4 inches below the equilibrium position, so the amplitude A is 4 inches. The frequency of the system is given as 8/π cycles per second.
To convert the frequency to angular frequency, we know that ω = 2πf, where f is the frequency. Substituting the given frequency, we have ω = 2π * (8/π) = 16π radians per second.
Since the weight is pulled down, the initial phase φ is 0.
The equilibrium position is the position at rest, which is the original position of the weight. In this case, since we pulled the weight down 4 inches, the equilibrium position x₀ is -4 inches.
Putting all the values together, the trigonometric model for the position of the weight at time t seconds is:
x(t) = 4 * cos(16πt) - 4
This model will give you the position of the weight at any given time t seconds.