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.
The trigonometric model is y(t) = 4cos(8/πt).
To determine a trigonometric model for the position of the weight at time t seconds, we'll start by setting up the equation for simple harmonic motion.
The general equation for simple harmonic motion is given by:
x(t) = A * cos(ωt + φ)
Where:
- x(t) represents the position of the weight at time t seconds.
- A is the amplitude of the motion.
- ω is the angular frequency of the motion.
- φ is the phase constant.
In this case, we are given that the weight is pulled down 4 inches below the equilibrium position. This means the amplitude (A) is 4 inches.
We are also given that the frequency of the system is 8/π cycles per second. The frequency (f) is related to the angular frequency (ω) by the equation:
ω = 2πf
Substituting the given frequency into the equation, we have:
ω = 2π * (8/π) = 16 radians per second
Now, we can substitute the values into the general equation for simple harmonic motion:
x(t) = 4 * cos(16t + φ)
Lastly, we need to determine the value of the phase constant (φ). To find φ, we will use the initial condition given that the weight is pulled down 4 inches below the equilibrium position. At t = 0 seconds, x(t) = -4 inches.
Substituting the values into the equation, we have:
-4 = 4 * cos(0 + φ)
Dividing both sides by 4, we get:
-1 = cos(φ)
To determine the value of φ, we take the inverse cosine (cos⁻¹) of -1:
φ = cos⁻¹(-1) = π
So, the trigonometric model that gives the position of the weight at time t seconds is:
x(t) = 4 * cos(16t + π)
To determine a trigonometric model that gives the position of the weight at time t seconds, we will start by identifying the general form of the equation. In a simple harmonic motion, the position as a function of time can be expressed as:
x(t) = A * cos(ωt + φ) + x₀
Where:
- x(t) is the position of the weight at time t seconds.
- A is the amplitude of the motion.
- ω is the angular frequency, given by ω = 2πf, where f is the frequency in cycles per second.
- φ is the phase constant.
- x₀ is the equilibrium position.
Let's determine these values step by step:
1. Amplitude (A):
The amplitude of the motion is the maximum distance the weight will move from the equilibrium position. In this case, the weight is pulled down 4 inches below the equilibrium position. Therefore, the amplitude is 4 inches.
A = 4 inches
2. Angular frequency (ω):
The angular frequency, ω, is given by the formula ω = 2πf, where f is the frequency in cycles per second.
Given that the frequency of the system is 8/π cycles per second, we can substitute this value to find ω.
ω = 2π * (8/π)
= 16 radians per second
3. Phase constant (φ):
The phase constant, φ, represents the initial phase of the motion at time t = 0 seconds. In this case, we are not given any specific information about the initial phase, so we can assume it to be zero.
φ = 0
4. Equilibrium position (x₀):
The equilibrium position is the mean position of the weight or the point where the weight would rest without any disturbance. Since the weight is pulled down 4 inches below this position, the equilibrium position will be 4 inches above the pulled-down position.
x₀ = 4 inches
Now that we have determined all the values, we can substitute them into the general equation for simple harmonic motion to obtain the trigonometric model:
x(t) = 4 * cos(16t)
This equation gives the position of the weight at time t seconds.