A 4 ft spring measures 8 ft long after a mass weighing 8 pounds is attached to it. the medium through which the mass moves offers a damping force numerically equal to sqrt.2 times the instantaneous velocity. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 5 ft/s. Find the time at which the mass attains its extreme displacement from the equilibrium postion. What is the position of the mass at this instant?
To find the equation of motion for this system, we can start by applying Newton's second law, which states that the force acting on an object is equal to its mass multiplied by its acceleration.
Let's denote the position of the mass as x(t), where t represents time. The equation of motion can be written as:
m * x''(t) + c * x'(t) + k * x(t) = 0
where m is the mass of the object, x''(t) is the second derivative of x(t) with respect to time (acceleration), c is the damping coefficient, and k is the spring constant.
We are given that the mass is initially released from the equilibrium position with a downward velocity of 5 ft/s. This means that x(0) = 0 and x'(0) = -5 ft/s.
To solve the equation of motion, we need to find the values of c and k. We know that the damping force is numerically equal to sqrt(2) times the instantaneous velocity. Since the damping force is given by c * x'(t), we can rewrite it as:
c * x'(t) = sqrt(2) * x'(t)
Setting these two expressions equal to each other, we have:
c = sqrt(2)
Now, let's consider the spring's behavior before and after the mass was attached to it.
Before the mass was attached, the spring measured 4 ft long. The original rest position of the spring is therefore 4 ft. After the mass was attached, the spring measures 8 ft long, which is an extension of 4 ft. Hence, when the mass is at the equilibrium position (x = 0), the spring is compressed by 4 ft. This means that when x = 0, the force exerted by the spring, -k * x(t), is equal to the force exerted by the weight, mg. Therefore, we can write:
-k * 4 = m * g
where g is the acceleration due to gravity, approximately 32 ft/s^2.
Solving for k, we have:
k = (m * g) / 4
with m = 8 lb and g = 32 ft/s^2, we find:
k = (8 lb * 32 ft/s^2) / 4 ft = 64 lb/s^2
Now that we know the values of c and k, let's rewrite the equation of motion:
8 lb * x''(t) + sqrt(2) * x'(t) + 64 lb/s^2 * x(t) = 0
This is our equation of motion.
To find the time at which the mass attains its extreme displacement from the equilibrium position, we can solve the equation of motion using a differential equation solver or by assuming a solution of the form x(t) = A * e^(rt), where A is a constant and r is a complex number.
By solving the differential equation, we find that the solution is:
x(t) = e^(-sqrt(2)/8t) * (A * cos(sqrt(63)/8t) + B * sin(sqrt(63)/8t))
where A and B are constants determined by initial conditions.
Since the mass is initially released from the equilibrium position with a downward velocity of 5 ft/s, we have x(0) = 0 (initial displacement = 0) and x'(0) = -5 ft/s (initial velocity = -5 ft/s).
Using these initial conditions, we can find the values of A and B:
x(0) = A * cos(0) + B * sin(0) = 0
x'(0) = -sqrt(2)/8A * sin(0) + sqrt(63)/8B * cos(0) = -5
From the first equation, we find that A = 0.
Using this result, the second equation simplifies to:
sqrt(63)/8B = -5
Solving for B, we find:
B = -8/((5)(sqrt(63)))
Now that we have the values of A and B, we can rewrite the equation of motion:
x(t) = e^(-sqrt(2)/8t) * (B * sin(sqrt(63)/8t))
The time at which the mass attains its extreme displacement from the equilibrium position can be found by setting x(t) = +/- 8 ft (since the spring is extended by 4 ft from its equilibrium position). Solving for t, we find:
sin(sqrt(63)/8t) = +/- 1
This occurs when sqrt(63)/8t = (2n + 1) * pi/2, where n is an integer. Solving for t, we find:
t = [sqrt(63)(2n + 1) * pi] / (8 * sqrt(2))
Finally, to find the position of the mass at this instant, we substitute the value of t back into the equation of motion:
x(t) = e^(-sqrt(2)/8t) * (B * sin(sqrt(63)/8t))
where B = -8/((5)(sqrt(63))).