Download lecture 2 simple harmonic motion and spring

LECTURE 2
SIMPLE HARMONIC MOTION AND SPRING
Instructor: Kazumi Tolich
Lecture 2
2
¨
Reading chapter 13.3 – 13.4
Circular motion and simple harmonic motion
¤ Period of a mass on a spring
¤ Vertical spring
¤
Quiz: 1
3
¨
A particle is going around in a uniform circular motion with a
fixed angular speed of 𝜔 and a radius of 𝐴. The particle’s
position vector from the center of the circle points in the +𝑥direction at 𝑡 = 0. What is the 𝑥-component of this vector as a
function of time?
A.
B.
C.
𝐴𝜔𝑡
𝐴 sin 𝜔𝑡
𝐴 cos 𝜔𝑡
Quiz: 1-1 answer
4
¨
¨
¨
¨
¨
𝐴 cos 𝜔𝑡
The 𝑥-component of any vector with a magnitude of 𝐴
that makes an angle 𝜃 above the 𝑥-axis is 𝐴 cos 𝜃.
The angular position of the vector is given by
𝜃 = 𝜔𝑡.
The 𝑥-component of the position is given by
𝑥 = 𝐴 cos 𝜔𝑡 .
Simple harmonic motion can be described as the 𝑥component of the position vector for an object in a
uniform circular motion.
Angular frequency
5
¨
Position of a particle in a simple harmonic motion is
𝑥 = 𝐴 cos 𝜔𝑡
where 𝜔 is the angular frequency of the oscillation:
𝜔 = 2𝜋𝑓 =
¤
2𝜋
𝑇
The frequency is a physical characteristic of the system and independent of the
amplitude, 𝐴.
Demo: 1
6
¨
Circular vs. Simple Harmonic Motion (Spring)
¤ Comparison
of shadow of a circular motion and simple harmonic motion.
Sinusoidal pattern and uniform circular motion
7
¨
Bubbles foaming off the edge of a rotating propeller that is moving
through water produce a sinusoidal pattern.
Quiz: 2
8
¨
The graph shows displacement of an object undergoing simple harmonic
motion as a function of time. What is the angular frequency in
radians/second?
𝑥[m]
𝑡[s]
Quiz: 2-2 answer
9
¨
𝜔=
23
4
=
23
5 7
= 0.7 rad/s
x[m]
𝑇
t[s]
Quiz: 3
10
¨
The graph shows displacement of an object undergoing simple harmonic
motion as a function of time. Around what time is the velocity most negative?
Indicate the time by placing a dot on the time axis.
𝑥[m]
𝑡[s]
Quiz: 2-3 answer
11
¨
¨
𝑡 ~ 4.5 s
Remember that the velocity is represented by the slope. At the point
indicated, we have the largest negative slope, so the velocity is minimum.
𝑥[m]
𝑡[s]
Velocity and acceleration
12
¨
When the position of a particle
in a simple harmonic motion is
given by 𝑥 = 𝐴 cos 𝜔𝑡 , its
velocity and acceleration are
given by
𝑣 = −𝐴𝜔 sin 𝜔𝑡
𝑎 = −𝐴𝜔2 cos 𝜔𝑡
Period of a mass on a spring
13
For an object with a mass 𝑚 on a horizontal spring with a force
constant 𝑘, the restoring force of the spring is responsible for its
oscillation: 𝐹 = 𝑚𝑎 = −𝑘𝑥.
¨ The period of the oscillation is given by
¨
𝑚
𝑇 = 2𝜋
𝑘
Quiz: 4
¨
A student makes a simple harmonic oscillator using two springs, each with a
force constant 𝑘. The student connects the unstressed springs to a mass 𝑚, in
the two ways shown, and then they displace the mass by 𝑥 to the right.
Which block has a larger oscillation period?
A.
B.
C.
D.
Block 1
Block 2
They are the same.
Not enough information is given.
Quiz: 2-4 answer
¨
¨
¨
They are the same.
The force from a spring is always towards
the equilibrium position. In both cases, the
net force to the left is 𝐹 = 𝑚𝑎 = −2𝑘𝑥.
𝑇 = 2𝜋
H
2I
Example: 1
16
¨
An object with a mass 𝑚 = 3.0 kg on a
frictionless horizontal surface is attached
to one end of a horizontal spring and
oscillates with an amplitude 𝐴 = 10 cm
and a frequency 𝑓 = 2.4 Hz.
a)
What is the period of the motion?
b)
What is the force constant of the
spring?
c)
What is the maximum speed of the
object?
d)
What is the maximum acceleration of
the object?
Vertical spring
17
A mass on a vertical spring
oscillates about the equilibrium
position with the mass attached.
¨ This oscillation has the same period
as the same mass on the same
spring placed horizontally.
¨
Example: 2 (Walker Ch. 13-45)
18
¨
When an object with a mass of
𝑚 = 0.184 kg is attached to a
vertical spring, it causes the
spring to stretch a distance 𝑑. If
the mass is now displaced slightly
from equilibrium, it is found to
make 25.0 oscillations in 12.6 s.
Find the stretch distance 𝑑.