the formula of damping harmonic motion and worked example
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Damped harmonic motion occurs when a mass on a spring experiences a resistive force proportional to its velocity. This resistance causes the oscillations to gradually decrease in amplitude over time. The motion is described by a second-order differential equation that combines both the spring force and the damping force. The solution to this equation gives us the position of the mass as a function of time, which follows an exponentially decaying sinusoidal pattern, as shown in the graph.
The motion of a damped harmonic oscillator is described by a second-order differential equation. This equation combines Newton's second law with the spring force and the damping force. The spring force is proportional to displacement, following Hooke's law, while the damping force is proportional to velocity. When we combine these forces, we get the equation: m times d-squared-x over dt-squared, plus c times dx over dt, plus k times x equals zero. Here, m represents the mass, c is the damping coefficient, k is the spring constant, and x is the displacement from equilibrium.
The solution to the damped harmonic oscillator equation depends on the amount of damping in the system. For the most common case, called underdamping, the solution takes the form of an exponentially decaying cosine function. The position as a function of time is given by: x of t equals A times e to the negative gamma t times cosine of omega-prime t plus phi. Here, gamma is the damping rate, equal to c over 2m. Omega-zero is the natural frequency of the undamped oscillator, equal to the square root of k over m. And omega-prime is the damped frequency, which is slightly lower than the natural frequency. The constants A and phi are determined by the initial conditions of the system. As we increase the damping coefficient, we can observe how the oscillations decay more rapidly.
Let's work through a specific example of damped harmonic motion. We have a 1 kilogram mass on a spring with spring constant k equals 10 newtons per meter. The system has a damping coefficient c equals 2 newton-seconds per meter. The mass is initially displaced 0.1 meters from equilibrium and released from rest. Our task is to find the equation describing the position as a function of time. First, we calculate the key parameters. The damping rate gamma equals c divided by 2m, which is 2 divided by 2 times 1, giving us 1 per second. The natural frequency omega-zero equals the square root of k over m, which is the square root of 10 radians per second. The damped frequency omega-prime equals the square root of omega-zero squared minus gamma squared, which gives us 3 radians per second. Next, we apply the initial conditions to find the constants A and phi. Since the mass starts at position 0.1 meters, we have x of 0 equals 0.1. And since it's released from rest, the initial velocity is zero. Solving these equations, we get A equals 0.1 and B equals 1/30. Therefore, our final solution is: x of t equals e to the negative t times the quantity 0.1 times cosine of 3t plus 1/30 times sine of 3t. The graph shows how the position oscillates while gradually decreasing in amplitude due to damping.
To summarize what we've learned about damped harmonic motion: First, it's a physical system that combines a restoring force from a spring with a resistive force proportional to velocity. Second, the motion is governed by a second-order differential equation: m times d-squared-x over dt-squared, plus c times dx over dt, plus k times x equals zero. Third, for the common underdamped case, the solution is an exponentially decaying oscillation described by x of t equals A times e to the negative gamma t times cosine of omega-prime t plus phi. Fourth, depending on the damping coefficient, the system can be underdamped with oscillations, critically damped with the fastest return to equilibrium without oscillation, or overdamped with a slow return to equilibrium. Finally, damped harmonic motion has numerous applications in engineering and physics, including shock absorbers in vehicles, pendulum clocks, and electronic RLC circuits.