We’ll deal mainly with simple harmonic oscillations where the position of the object is specified by a sinusoidal (sine, cos) function. Chapter 15: Oscillatory motion part 2 Reading assignment: Chapter 16 Homework:(due Wednesday, Nov. 9, 2005): Problems:Q5, 1, 3, 6, 11, 13, 27, 33, 35, 39

An mass oscillates with an amplitude of 4.00 m, a frequency of 0.5 Hz and a phase angle of  /4. (a)What is the period T? (b)Write an equation for the displacement of the particle. (c)Calculate the velocity and acceleration of the object at any time t. Black board example 13.1 (d) Determine the position, velocity and acceleration of the object at time t = 1.00s. (e) Calculate the maximum velocity and acceleration of the object.

The block-spring system The frequency depends only on: - the mass of the block - the force constant of the spring. The frequency does not depend on the amplitude.

A spring stretches by 3.90 cm when a 10.0 g mass is hung from it. A 25.0 g mass attached to this spring oscillates in simple harmonic motion. Black board example 13.2 (Problem 13.7) (a)Calculate the period of the motion. (b)Calculate frequency and the angular velocity of the motion.

Energy of harmonic oscillator Kinetic energy: Potential energy: Total energy:

txvaKU 0A0 -2A-2A 0½kA 2 T/40 -A-A 0½kA 2 0 T/2-A0 -2A-2A 0½kA 2 3T/40 -A-A 0½kA 2 0 TA0 -2A-2A 0 Table: Summary for harmonic oscillation

A kg mass is attached to a spring and undergoes simple harmonic motion with a period of s. The total energy of the system is 2.00 J. Black board example 13.3 (a)What is the force constant of the spring? (b)What is the amplitude of the motion? (c)What is the velocity of the mass when the displacement is 8.00 cm? (d)What is the kinetic and potential energy of the system when the displacement is 8.00 cm?

A person swings on a swing. When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, two people sit on the swing, the natural frequency of the swing is 1. greater. 2. the same. 3. smaller.

The pendulum For small motion (less than about 10°).

A person swings on a swing.When the person sits still, the swing oscillates back and forth at its natural frequency. If, instead, the person stands on the swing, the natural frequency of the swing is 1. greater. 2. the same. 3. smaller.

The physical pendulum For small motion (less than about 10°

Black board example 13.4 Find the period of a 14.7 inch (0.37 m) long stick that is pivoted about one end and is oscillating in a vertical plane.

Simple harmonic motion and uniform circular motion

Damped, simple harmonic motion b is damping constant

Forced Oscillations and Resonance A damped, harmonic oscillator (ang. frequency  ) is driven by an outside, sinusoidal force with ang. frequency  d  Resonance when  d =  (can get very large amplitudes) b is damping constant

A lead weight is fastened to a large solid piece of Styrofoam that floats in a container of water. Because of the weight of the lead, the water line is flush with the top surface of the Styrofoam. If the piece of Styrofoam is turned upside down, so that the weight is now suspended underneath it, the water level in the container 1. rises. 2. drops. 3. remains the same.

Consider an object floating in a container of water. If the container is placed in an elevator that accelerates upward, 1. more of the object is below water. 2. less of the object is below water. 3. there is no difference.

Consider an object that floats in water but sinks in oil.When the object floats in water, half of it is submerged. If we slowly pour oil on top of the water so it completely covers the object, the object 1. moves up. 2. stays in the same place. 3. moves down.

In the following section we assume: - the flow of fluids is laminar (not turbulent)  There are now vortices, eddies, turbulences. Water layers flow smoothly over each other. - the fluid has no viscosity (no friction).  (Honey has high viscosity, water has low viscosity)

Equation of continuity For fluids flowing in a “pipe”, the product of area and velocity is constant (big area  small velocity). Why does the water emerging from a faucet “neck down” as it falls?

A circular hoop sits in a stream of water, oriented perpendicular to the current. If the area of the hoop is doubled, the flux (volume of water per unit time) through it 1. decreases by a factor of decreases by a factor of remains the same. 4. increases by a factor of increases by a factor of 4.

Bernoulli’s equation Conservation of energy

Black board example 15.7 Homework Bernoulli’s law Water flows through a horizontal pipe, and then out into the atmosphere at a speed of 15 m/s. The diameters of the left and right sections of the pipe are 5.0 cm and 3.0 cm, respectively. (a)What volume of water flows into the atmosphere during a 10 min period? (b)What is the flow speed of the water in the left section of the pipe? (c)What is the gauge pressure in the left section of the pipe?

Two hoses, one of 20-mm diameter, the other of 15-mm diameter are connected one behind the other to a faucet. At the open end of the hose, the flow of water measures 10 liters per minute. Through which pipe does the water flow faster? 1. the 20-mm hose 2. the 15-mm hose 3. The flow rate is the same in both cases. 4. The answer depends on which of the two hoses comes first in the flow.