Transcript Chapter 15b

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
• We’ll deal mainly with simple
harmonic oscillations where the
position of the object is specified
by a sinusoidal (sine, cos) function.
Black board example 13.1
An mass oscillates with an amplitude
of 4.00 m, a frequency of 0.5 Hz
and a phase angle of p/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.
(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
2p
m
T
 2p

k
1
1
f  
T 2p
k
m
The frequency depends only on:
- the mass of the block
- the force constant of the spring.
The frequency does not depend on the amplitude.
Black board example 13.2
(Problem 13.7)
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.
(a) Calculate the period of the motion.
(b) Calculate frequency and the angular velocity of the motion.
Energy of harmonic oscillator
Kinetic energy:
1 2 1
K  mv  m 2 A2 sin 2 (t   )
2
2
Potential energy:
U
Total energy:
E  K U 
1 2 1 2
kx  kA cos 2 (t   )
2
2
1 2 1
kA  mvmax 2  constant
2
2
Table: Summary for harmonic oscillation
t
x
v
a
K
U
0
A
0
-2A
0
½kA2
T/4
0
-A
0
½kA2
0
T/2
-A
0
-2A
0
½kA2
3T/4
0
-A
0
½kA2
0
T
A
0
-2A
0
½kA2
Black board example 13.3
A 0.200 kg mass is attached to
a spring and undergoes
simple harmonic motion
with a period of 0.250 s.
The total energy of the
system is 2.00 J.
(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
2p
L
T
 2p

g
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
2p
I
T
 2p

mgd
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
x - component : x(t )  R cos  R cos t
y - component : y (t )  R sin   R sin t
Damped, simple harmonic motion
x(t )  A cos( ' t   ) e
k
b2
' 

m 4m 2
 bt
2m
b is damping
constant
Forced Oscillations and
Resonance
b is damping
constant
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)
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
A1v1  A2v2  constant
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 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
Conservation of energy
1 2
P  v  gy  constant
2
1
1
2
2
P1  v1  gy1  P2  v2  gy2
2
2
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 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.