• 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 1

Reading assignment: Chapter 15.4-15.7

Homework : (due Monday, Nov 7, 2005):

Problems: 32, 33, 43, 45

Two cups are filled to the same level withwater. One of the two cups has ice cubesfloating in it. When the ice cubes melt, inwhich cup is the level of the water higher?

1. The cup without ice cubes.2. The glass with ice cubes.3. It is the same in both.

Two cups are filled to the same level withwater. One of the two cups has plastic ballsfloating in it. If the density of the plastic ballsis less than that of ice, which of the two cupsweighs more?

1. The cup without plastic balls.2. The cup with plastic balls.3. The two weigh the same.

A boat carrying a large boulder is floatingon a lake.The boulder is thrown overboardand sinks. The water level in the lake (withrespect to the shore)

1. rises.2. drops.3. remains the same.

A lead weight is fastened on top of a largesolid piece of Styrofoam that floats in a containerof water. Because of the weight of thelead, the water line is flush with the top surfaceof the Styrofoam. If the piece of Styrofoamis turned upside down so that theweight is now suspended underneath it,

1. the arrangement sinks.2. the water line is below the top surface ofthe Styrofoam.3. the water line is still flush with the topsurface of the Styrofoam.

A lead weight is fastened to a large solidpiece of Styrofoam that floats in a containerof water. Because of the weight of thelead, the water line is flush with the top surfaceof the Styrofoam. If the piece of Styrofoamis turned upside down, so that theweight is now suspended underneath it, thewater level in the container

1. rises.2. drops.3. remains the same.

Consider an object floating in a container ofwater. If the container is placed in an elevatorthat 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 butsinks in oil.When the object floats in water,half of it is submerged. If we slowly pour oilon top of the water so it completely coversthe 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

constant2211 vAvA

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, orientedperpendicular to the current. If thearea of the hoop is doubled, the flux (volumeof water per unit time) through it

1. decreases by a factor of 4.2. decreases by a factor of 2.3. remains the same.4. increases by a factor of 2.5. increases by a factor of 4.

Bernoulli’s equation

constant2

1 2 gyvP

22

2212

11 2

1

2

1gyvPgyvP

Conservation of energy

Black board example 15.7

Homework 15.51

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 otherof 15-mm diameter are connected one behindthe other to a faucet. At the open endof the hose, the flow of water measures 10liters per minute. Through which pipe doesthe water flow faster?

1. the 20-mm hose2. the 15-mm hose3. The flow rate is the same in both cases.4. The answer depends on which of the twohoses comes first in the flow.

Simple harmonic motion/oscillation

- Block attached to a spring

- Motion of a swing

- Motion of a pendulum

- Vibrations of a stringed musical instrument

- Motion of a cantilever

- Oscillations of houses, bridges, …

- Clocks

An object hangs motionless from a spring.When the object is pulled down, the sum ofthe elastic potential energy of the spring andthe gravitational potential energy of the objectand Earth.

1. increases.2. stays the same.3. decreases.

Simple harmonic motion/oscillation

Restoring force:

F = - k·x

Acceleration and restoring force:

- proportional to x

- directed toward the equilibrium position

Acceleration:

xm

ka

An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely directed.

Simple harmonic motion/oscillation

xdt

xd

xa

22

2

is angular frequency

An object hangs motionless from a spring.When the object is pulled down, the sum ofthe elastic potential energy of the spring andthe gravitational potential energy of the objectand Earth.

1. increases.2. stays the same.3. decreases.

Properties of simple harmonic motion

)cos()( tAtx

…Angular frequency

A…amplitude

…phase constant, phase angle

Position of particle at time t:

)cos()( tAtx

T…period

t …phase

Properties of simple harmonic motion

The period T of the motion is the time it takes for the particle to complete one full cycle

Periodic motion

The value of x at time t is equal to the value of x at time t +T

A person swings on a swing.When the personsits still, the swing oscillates back andforth at its natural frequency. If, instead, twopeople sit on the swing, the natural frequencyof the swing is

1. greater.2. the same.3. smaller.

A person swings on a swing.When the personsits still, the swing oscillates back andforth at its natural frequency. If, instead, theperson stands on the swing, the natural frequencyof the swing is

1. greater.2. the same.3. smaller.

Properties of simple harmonic motion

Displacement: )cos()( tAtx

)sin()( tAtv

)cos()( 2 tAta

2

T2

1

Tf

Tf

22

Period T: Frequency: Angular frequency:

Units: 1/s = 1 Hz

Velocity:

Acceleration:

Properties of simple harmonic motion

Ax max

Av max

Aa 2max

Phase of velocity differs by – /2 or 90° from phase of displacement.

Phase of acceleration differs by – or 180° from phase of displacement.

Properties of simple harmonic motion

• Acceleration of particle is proportional to the displacement, but is in the opposite direction (a = - 2·x).

• Displacement, velocity and acceleration vary sinusoidally.

• The frequency and period of the motion are independent of the amplitude. (demo).

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

k

mT

22

m

k

Tf

2

11

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.