A vertical spring with a spring constant of 2.04 N/m has a 0.32-kg mass attached to it, and the mass moves in a medium with a damping constant of 0.025 kg/s. The mass is released from rest at a position 5.6 cm from the equilibrium position. How long will it take for the amplitude to decrease to 2.5 cm?
To solve this problem, we need to use the equation for the motion of a damped harmonic oscillator:
mx'' + bx' + kx = 0
where x is the displacement of the mass from equilibrium, m is the mass of the object, b is the damping constant, k is the spring constant, and primes denote derivatives with respect to time.
Given:
- m = 0.32 kg (mass of the object)
- b = 0.025 kg/s (damping constant)
- k = 2.04 N/m (spring constant)
- Initial amplitude = 5.6 cm = 0.056 m
- Final amplitude = 2.5 cm = 0.025 m
To find the time it takes for the amplitude to decrease to 2.5 cm (0.025 m), we need to find the decay constant before using it to determine the damping ratio.
1. Find the decay constant (γ):
We can calculate the decay constant using the formula:
γ = b / (2m)
Substituting the given values:
γ = 0.025 kg/s / (2 × 0.32 kg)
γ = 0.0391 s⁻¹
2. Find the damping ratio (ζ):
The damping ratio is defined as the ratio of the decay constant to the square root of the spring constant:
ζ = γ / √(k / m)
Substituting the given values:
ζ = 0.0391 s⁻¹ / √(2.04 N/m / 0.32 kg)
ζ = 0.0391 s⁻¹ / √(6.375 N/m)
ζ ≈ 0.0279
3. Find the angular frequency (ω):
The angular frequency can be calculated using the formula:
ω = √(k / m)
Substituting the given values:
ω = √(2.04 N/m / 0.32 kg)
ω = √(6.375 N/m)
ω = 2.525 rad/s
4. Find the time constant (T):
The time constant is the reciprocal of the angular frequency:
T = 1 / ω
Substituting the value of ω:
T = 1 / 2.525 rad/s
T ≈ 0.396 s
5. Find the time taken for the amplitude to decrease to 2.5 cm:
The time taken for the amplitude to decrease to a certain value is given by:
t = -ln(Af / Ai) / (ζω)
Substituting the values:
t = -ln(0.025 m / 0.056 m) / (0.0279 × 2.525 rad/s)
t ≈ -ln(0.4464) / (0.0704475)
t ≈ -(-0.8054) / (0.0704475)
t ≈ 11.446 s
Therefore, it will take approximately 11.446 seconds for the amplitude to decrease from 5.6 cm to 2.5 cm.
To determine how long it will take for the amplitude to decrease to 2.5 cm, we need to consider the damping effect on the system.
The equation that describes the motion of a damped harmonic oscillator is:
m * a + c * v + k * x = 0
where
m = mass of the object (0.32 kg)
a = acceleration of the object
c = damping constant (0.025 kg/s)
v = velocity of the object
k = spring constant (2.04 N/m)
x = displacement of the object from the equilibrium position
Since we're dealing with a damped harmonic oscillator, we can rewrite this equation as:
m * d^2x/dt^2 + c * dx/dt + k * x = 0
where
dx/dt = v
d^2x/dt^2 = a
Now, let's find the equation of motion for this system.
1. Write the equation of motion:
m * d^2x/dt^2 + c * dx/dt + k * x = 0
Substitute the given values:
0.32 * d^2x/dt^2 + 0.025 * dx/dt + 2.04 * x = 0
2. Simplify the equation:
0.32 * d^2x/dt^2 + 0.025 * dx/dt + 2.04 * x = 0
3. Solve the equation using any numerical solver or a differential equations solver.
The solution to this equation will give you the displacement x(t) as a function of time. From the solution, you can extract the amplitude as a function of time. Specifically, you're interested in finding the time at which the amplitude decreases to 2.5 cm (0.025 m).
Using a numerical solver, input the initial conditions (position, velocity) and solve for x(t). From the solution, you can extract the amplitude of motion as a function of time. Find the time at which the amplitude decreases to 0.025 m (2.5 cm).
Please note that the specific implementation of solving this equation may vary depending on the tools or software you have available.