A weight attached to a spring is pulled 4 inches below the equilibrium position. Assuming that the frenquency 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 determine a trigonometric model that gives the position of the weight at time t seconds, we need to understand the motion of a mass-spring system.
In a mass-spring system, the weight attached to the spring experiences simple harmonic motion. This type of motion can be modeled using a sine or cosine function.
The general equation for the position of an object undergoing simple harmonic motion is given by:
x(t) = A * sin(ωt + φ)
Where:
- x(t) is the position of the object at time t,
- A is the amplitude (maximum displacement from the equilibrium),
- ω is the angular frequency (2π times the frequency),
- t is the time, and
- φ is the phase constant or initial phase.
In this case, the weight is pulled 4 inches below the equilibrium position. Since the amplitude represents the maximum displacement, we have A = 4 inches.
The frequency of the system is given as 8/π cycles per second. We can convert this to angular frequency ω by multiplying it by 2π:
ω = (8/π) * 2π = 16 radians per second
Now we can write the equation for the position of the weight at time t:
x(t) = 4 * sin(16t + φ)
Note that the phase constant φ is not given in the problem and represents the initial position of the weight at time t = 0. To determine the value of φ, we need additional information such as the initial position or velocity.
Thus, the trigonometric model that gives the position of the weight at time t seconds is:
x(t) = 4 * sin(16t + φ)