Contoh Soal Undamped Free Vibration (Seri Getaran Mekanis part2)

Summary

This video explains the principles of free vibration in a spring-mass system using a toy car as an example. The mass is modeled as 2 kg, and the spring constant is 205 N/m. It demonstrates how to calculate the natural frequency, amplitude, and phase shift of the oscillation, with initial conditions provided. The mathematical steps are detailed, using the equation of motion for simple harmonic motion. The key takeaway is understanding the intrinsic nature of oscillations, with the natural frequency dependent only on the mass and spring constant, regardless of initial conditions.

Takeaways

• 😀 The script discusses a simplified example of a vibration problem in physics, using a toy car with a spring to demonstrate the principles of free vibration.
• 😀 The system is modeled as a mass-spring system, where the mass is 2 kg and the spring constant is 205 N/m.
• 😀 The problem involves determining the natural frequency (Omega n) and amplitude of oscillation for the toy car in free vibration after an initial displacement.
• 😀 Initial conditions include a displacement of 2 mm upwards and an initial velocity of 1 mm/s.
• 😀 The natural frequency (Omega n) can be calculated using the formula: Omega n = sqrt(k/m), where k is the spring constant and m is the mass of the object.
• 😀 Omega n is an intrinsic property of the system, independent of initial conditions such as initial displacement or velocity.
• 😀 The formula for natural frequency is applied and yields Omega n = 10 rad/s.
• 😀 The equation of motion for the system is M * x'' + K * x = 0, which results in a solution of the form x(t) = A * cos(Omega n * t + phi), where A is the amplitude and phi is the phase.
• 😀 To find the amplitude (A) and phase (phi), the initial conditions are used to solve a system of equations involving displacement and velocity at t = 0.
• 😀 From the initial conditions, the amplitude A is calculated to be 2 mm, and the phase angle phi is found to be approximately 87.5 degrees.
• 😀 The final equation describing the motion of the system is x(t) = 2.00 * cos(10t + 87.5°), where the amplitude is 2 mm and the natural frequency is 10 rad/s.

Q & A

What is the primary purpose of the example in the script?

The primary purpose of the example is to demonstrate how to solve a basic vibration problem in physics using a toy car system as a model for a mass-spring system.

What are the given parameters for the system in the example?

The given parameters for the system are: mass (m) = 2 kg, spring constant (k) = 205 N/m, initial displacement (x0) = 2 mm, and initial velocity (v0) = 1 mm/s.

How is the toy car system modeled in the problem?

The toy car system is modeled as a mass-spring system with the mass representing the car and the spring constant representing the car's suspension, simplifying the problem for easier calculations.

What is the equation for the natural frequency (ωn) in this system?

The natural frequency (ωn) is calculated using the formula ωn = √(k/m), where k is the spring constant and m is the mass.

What is the value of the natural frequency (ωn) for this system?

The natural frequency (ωn) is calculated as √(205 N/m ÷ 2 kg), which equals approximately 10 rad/s.

How does the initial condition affect the natural frequency of the system?

The initial condition does not affect the natural frequency of the system, as ωn is an intrinsic property of the mass and spring constant, independent of the initial displacement or velocity.

What is the general form of the equation of motion for the system?

The general form of the equation of motion for the system is: m * x''(t) + k * x(t) = 0, where x(t) is the displacement as a function of time.

What is the solution to the equation of motion for a simple harmonic oscillator?

The solution to the equation of motion is given by x(t) = A * cos(ωn * t + φ), where A is the amplitude and φ is the phase angle.

How are the amplitude (A) and phase angle (φ) determined in this case?

The amplitude (A) and phase angle (φ) are determined using the initial conditions: x(0) = 2 mm and v(0) = 1 mm/s. These values are used to solve for A and φ.

What are the final values for the amplitude (A) and phase angle (φ) in this system?

The final values for the amplitude and phase angle are: A ≈ 2.00 mm and φ ≈ 87.5°.

What is the final form of the displacement equation for this system?

The final form of the displacement equation is x(t) = 2.00 * cos(10 * t + 87.5°), where 10 rad/s is the natural frequency and 87.5° is the phase angle.