A 2.0 kg block is attached to the end of a spring with a spring constant of 350 N/m and forced to oscillate by an applied force F = (15 N)sin(ωt),where ω = 35 rad/s. The damping constant is b = 15 kg/s. At t = 0,the block is at rest with the spring at its rest length.(a) Use numerical integration to plot the displacement of the block for the first 1.0 s. Use the motion near the end of the 1.0 s interval to estimate the amplitude, period, and angular frequency. Repeat the calculation for (b) ω = √k/m and (c) ω = 20 rad/s.
Good
To solve this problem, we'll need to use numerical integration to calculate the displacement of the block over time. We'll start by calculating the acceleration of the block at each time step.
The equation of motion for a damped harmonic oscillator is given by:
m * d^2x/dt^2 + b * dx/dt + k * x = F
where:
m = mass of the block (2.0 kg)
b = damping constant (15 kg/s)
k = spring constant (350 N/m)
x = displacement of the block from equilibrium position
F = applied force (15 N * sin(ωt))
To perform numerical integration, we'll use the Euler method, which involves approximating the derivative as the difference between two values over a small time interval. We'll divide the time interval into small steps and calculate the displacement at each step.
(a) For ω = 35 rad/s:
The amplitude, period, and angular frequency can be estimated by observing the motion near the end of the 1.0 s interval.
1. Set up the parameters:
- Set the time step, dt, to a small value (e.g., 0.01 s) to ensure accuracy.
- Set the initial conditions: x = 0 (block at rest), v = 0 (initial velocity).
2. Perform numerical integration:
- Loop over time from t = 0 to t = 1.0 s.
- At each time step, calculate the acceleration using the equation of motion above (F = (15 N)sin(ωt)).
- Update the velocity v using v = v + (a * dt).
- Update the displacement x using x = x + (v * dt).
3. Plot the displacement:
- Store the values of displacement x at each time step.
- Plot the graph of x vs. t.
(b) For ω = √k/m:
Repeat the same numerical integration steps as in part (a), but change the value of ω to the square root of k/m.
(c) For ω = 20 rad/s:
Again, repeat the same numerical integration steps, but change the value of ω to 20 rad/s.
By observing the motion near the end of the 1.0 s interval for each case, you can estimate the amplitude (maximum displacement), period (time for one complete oscillation), and angular frequency (2π divided by the period).