Physics Mechanics. Lecture 32 Oscillations II

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1 Physics Mechanics Lecture 32 Oscillations II

2 Gravitational Potential Energy A plot of the gravitational potential energy U g looks like this:

3 Energy Conservation Total mechanical energy of an object of mass m a distance r from the center of the Earth is: This confirms what we already know as an object approaches the Earth, it moves faster and faster.

4 Satellite Orbits and Energies The tangential velocity v needed for a circular orbit depends on the gravitational potential energy U g of the satellite at the radius of the orbit. The needed tangential velocity v is independent of the mass m of the satellite (provided m<<m).

5 Connections between Uniform Circular Motion and Simple Harmonic Motion The position as a function of time: The angular frequency:

6 The Period of a Mass on a Spring Since the force on a mass on a spring is proportional to the displacement, and also to the acceleration, we find that. Substituting the time dependencies of a and x gives:

7 Kinematics of Simple Harmonic Motion

8 Harmonic Motion v and a The position as a function of time is: Then the velocity is: and the acceleration is:

9 Vertical Oscillations

10 Vertical Oscillations

11 Example: A Vertical Oscillation A 200 g block hangs from a spring with spring constant 10 N/m. The block is pulled down to a point where the spring is 30 cm longer than it s unstretched length, then released. Where is the block and what is its velocity 3.0 s later?

12 Energy Conservation in Oscillatory Motion In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring: Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:

13 Energy Conservation in Oscillatory Motion As a function of time, So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa.

14 Energy Conservation in Oscillatory Motion This diagram shows how the energy transforms from potential to kinetic and back, while the total energy remains the same.

15 Example: Using Conservation of Energy A 500 g block on a spring is pulled a distance of 20 cm and released. The subsequent oscillations are measured to have a period of 0.80 s. At what position (or positions) is the speed of the block 1.0 m/s?

16 Energy Conservation in Oscillatory Motion anywhere max center

17 The Simple Pendulum A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass). The angle θ that it makes with the vertical varies with time approximately as a sine or cosine.

18 The Simple Pendulum Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ, whereas the restoring force for a spring is proportional to the displacement (which is θ in this case). This is not a Hooke s Law force.

19 The Small Angle Approximation However, for small angles, sin θ and θ are approximately equal.

20 The Simple Pendulum Substituting θ for sin θ allows us to treat the pendulum in a mathematically identical way to the mass on a spring. Therefore, we find that the period of a pendulum depends only on g and the length of the string (and is independent of mass): Note that T does not depend on m, the mass of the pendulum bob.

21 The Physical Pendulum A physical pendulum is an extended solid mass that oscillates around its center of mass. It cannot be modeled as a point mass suspended by a massless string because there is energy in the rotation of the object as its center of mass moves. Examples:

22 The Physical Pendulum In this case, it can be shown that the period depends on the moment of inertia I about the pivot, where l is the distance from the pivot to the CM: Substituting the moment of inertia of a point mass a distance L from the axis of rotation gives, as expected,

23 Example: A Comfortable Pace The pace of a comfortable walk can be estimated if we model each leg as a swinging physical pendulum. Is this estimate valid? Note that pace period is proportional to L ½.

24 Damped Oscillations In most physical situations, there are nonconservative forces of some sort, which will tend to decrease the amplitude of the oscillation. In many cases (e.g. viscous flow) the damping force is proportional to the speed: This causes the amplitude to decrease exponentially with time:

25 Damped Oscillations This exponential decrease of an underdamped oscillation is shown in the figure:

26 Damped Oscillations

27 Critical Damping Question: How do you damp the system to return to equilibrium as fast as possible? Answer: Use critical damping.

28 Driven Oscillations & Resonance An oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency f 0 of the system.

Driven Oscillations & Resonance If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the

29 Driven Oscillations & Resonance If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance. Small damping: sharp peak Large damping: broad peak The resonant frequency f 0 is not shifted by damping.

30 Driven Oscillations & Resonance The Q-factor characterizes the sharpness of the peak

OSCILLATIONS ABOUT EQUILIBRIUM

OSCILLATIONS ABOUT EQUILIBRIUM Chapter 13 Units of Chapter 13 Periodic Motion Simple Harmonic Motion Connections between Uniform Circular Motion and Simple Harmonic Motion The Period of a Mass on a Spring