Ayunan sederhana yang digantungkan pada langit-langit lift yang diam memiliki periode ayunan T.

Summary

The video script covers various physics problems related to harmonic motion, including pendulum periods, energy conservation, and oscillations. It begins by exploring the relationship between the period of a simple pendulum in a moving elevator, where the periods change depending on whether the elevator is stationary, moving up, or moving down. The script also discusses problems involving spring vibrations, calculating kinetic energy and potential energy, as well as concepts like amplitude and phase in harmonic motion. Several examples and formulas are worked through, providing a comprehensive look at key principles in mechanical oscillations.

Takeaways

• 😀 The first scenario involves a simple pendulum and its period of oscillation when the lift is at rest, moving upward with constant acceleration, and moving downward with deceleration. The relationship between the periods of oscillation in these three conditions is explained.
• 😀 When the lift is at rest, the period of oscillation, T1, is determined by the formula T1 = 2π√(l/g), where l is the length of the pendulum and g is the acceleration due to gravity.
• 😀 In the case of the lift moving upwards with acceleration, the period T2 is affected by the additional acceleration. The formula becomes T2 = 2π√(l/(g + a)).
• 😀 If the lift moves downward with deceleration, the period T3 is influenced by the negative acceleration. The formula becomes T3 = 2π√(l/(g - a)).
• 😀 Comparing the three periods T1, T2, and T3, it is concluded that T1 is greater than T2, and T2 is greater than T3.
• 😀 The second part discusses a harmonic oscillator on a spring. The maximum speed of the particle is given as 16 m/s, with an amplitude of 50 cm. The speed when the kinetic energy equals the potential energy is calculated.
• 😀 The angular frequency of the spring's vibration is determined using the equation vmax = ω * A, where ω is the angular frequency and A is the amplitude. The angular frequency is calculated to be ω = 32 rad/s.
• 😀 The equation for the speed of the particle during oscillation is derived from the relationship between the total mechanical energy and the velocity at any point in the oscillation. The final speed is found to be 8√2 m/s.
• 😀 The third part explores a particle experiencing simple harmonic motion. When the particle is at its equilibrium position, the displacement is zero, and both the potential energy and the force are also zero, resulting in a minimum force and energy.
• 😀 In a case where the kinetic energy equals the potential energy, the phase angle of the particle is calculated. The phase difference at this point is determined to be 1/8.

Q & A

What is the relationship between the periods T1, T2, and T3 of a simple pendulum in an elevator?

The relationship between the periods T1, T2, and T3 is derived based on the elevator's movement. T1 is the period when the elevator is at rest. T2 is the period when the elevator is moving upwards with constant acceleration, and T3 is the period when the elevator is moving downwards with constant deceleration. The periods are related by the equation T1 > T2 = T3.

How does the acceleration of the elevator affect the period of the pendulum?

When the elevator is moving upwards, the period decreases because the effective gravitational acceleration increases. When the elevator is moving downwards, the period increases due to the reduction in effective gravitational acceleration. The net effect of these changes depends on the elevator's acceleration relative to gravity.

What is the equation for the period of a simple pendulum in an elevator at rest?

When the elevator is at rest, the period of the pendulum (T1) is given by the formula T1 = 2π√(l/g), where l is the length of the pendulum and g is the gravitational acceleration.

What changes occur in the formula for the period when the elevator is moving upwards?

When the elevator is moving upwards, the period T2 is given by T2 = 2π√(l / (g + a)), where a is the constant upward acceleration of the elevator. The acceleration due to gravity increases, reducing the period.

What is the effect of downward acceleration on the period of the pendulum?

When the elevator is moving downwards with constant deceleration, the period T3 is given by T3 = 2π√(l / (g - a)), where a is the constant downward acceleration. The period increases due to the reduced effective gravitational acceleration.

Why does the downward acceleration cause the period to increase?

The downward acceleration reduces the effective gravitational acceleration, which makes the pendulum swing slower, increasing the period of oscillation.

How does the relationship between the periods simplify when comparing T1, T2, and T3?

When comparing T1, T2, and T3, it simplifies to the relationship T1 > T2 = T3. This means the period when the elevator is at rest is greater than the periods when the elevator is moving upwards or downwards.

What is the formula used to calculate the maximum speed of a mass on a spring in simple harmonic motion?

The maximum speed (V_max) of a mass in simple harmonic motion is given by the formula V_max = ω * A, where ω is the angular frequency and A is the amplitude of the oscillation.

How is the energy of a mass on a spring in simple harmonic motion related to its speed and amplitude?

In simple harmonic motion, the total mechanical energy is constant and is divided into kinetic and potential energy. The maximum kinetic energy occurs when the speed is maximum, and the maximum potential energy occurs when the displacement is maximum. The speed and amplitude are related through the formula V_max = ω * A, where the amplitude A influences the maximum speed.

What is the formula for the speed of a mass when its kinetic energy equals its potential energy in simple harmonic motion?

When the kinetic energy equals the potential energy, the speed of the mass is given by V = ω * √(A² - y²), where ω is the angular frequency, A is the amplitude, and y is the displacement. This condition occurs when the system is at a point where the energies are balanced.