2. An 8 lb weight attached to a spring exhibits simple harmonic motion. Determine the equation of motion if the spring constant is 4 lb/feet and the weight is released 4 inches above the equilibrium position with an upward velocity of 3 feet/sec.
To determine the equation of motion for the given scenario, we can use the following formula:
mx'' + kx = 0
Where:
m = mass of the weight
x = displacement from the equilibrium position
k = spring constant
x'' = second derivative of displacement with respect to time (acceleration)
Given:
Mass of the weight (m) = 8 lb
Spring constant (k) = 4 lb/feet
Weight released 4 inches above the equilibrium position (x_0 = 4 inches)
Upward velocity (v) = 3 feet/sec
To find the equation of motion, we need to determine the initial displacement (x_0) and the initial velocity (v) in terms of the standard units (feet).
1. Convert the initial displacement from inches to feet:
x_0 = 4 inches = 4/12 feet = 1/3 feet
2. Set up the initial conditions equation:
x(t = 0) = x_0
v(t = 0) = v
Given that the weight is released at the equilibrium position, the weight's initial displacement can be determined as:
x(t = 0) = x_0 = 0
Now we can write the equation of motion:
mx'' + kx = 0
Since we know the mass (m) and the spring constant (k), we can substitute them into the equation:
8x'' + 4x = 0
This is the differential equation that represents the equation of motion for the weight attached to the spring.
To determine the equation of motion for the given scenario, we can use the equation of motion for simple harmonic motion:
x(t) = A * cos(ωt + φ)
Where:
x(t) is the displacement from the equilibrium position at time t,
A is the amplitude of the motion,
ω is the angular frequency,
φ is the phase angle.
To find the values of A, ω, and φ, let's break down the information given in the question.
1. Amplitude (A):
The weight is released 4 inches above the equilibrium position. Since we're working in feet, we convert 4 inches to feet: 4 inches = 4/12 = 1/3 feet. So, the amplitude of the motion is 1/3 feet.
2. Angular frequency (ω):
The angular frequency is related to the spring constant (k) and the mass (m) of the weight by the equation:
ω = √(k / m)
Here, the spring constant (k) is given as 4 lb/feet, and the mass (m) of the weight is 8 lb. Plugging in the values, we get:
ω = √(4 lb/feet / 8 lb) = √(4/8) = √(1/2) = √2/2
So, the angular frequency is √2/2.
3. Phase angle (φ):
The phase angle depends on the initial conditions of the system. In this case, the weight is released with an upward velocity of 3 feet/sec. As the weight is released from the 4-inch position above the equilibrium, it will reach its maximum displacement and change direction (i.e., velocity becomes zero) at the equilibrium position.
At the maximum displacement, the weight is moving upwards. Hence, the phase angle is 0° or 2π radians.
Now that we have determined the values of A, ω, and φ, we can write the equation of motion:
x(t) = (1/3) * cos((√2/2)t)
So, the equation of motion for the given scenario is x(t) = (1/3) * cos((√2/2)t).