Transcript Physics 106P: Lecture 1 Notes

Lecture 20:
Ideal Spring and Simple Harmonic Motion

New Material: Textbook Chapters 10.1 and 10.2
Physics 101: Lecture 20, Pg 1
Ideal Springs

Hooke’s Law: The force exerted by a spring is
proportional to the distance the spring is stretched or
compressed from its relaxed position.
FX = -k x
Where x is the displacement from
the relaxed position and k is the
constant of proportionality.
(often called “spring constant”)
relaxed position
FX = 0
x
x=0
Physics 101: Lecture 20, Pg 2
Ideal Springs

Hooke’s Law: The force exerted by a spring is
proportional to the distance the spring is stretched or
compressed from its relaxed position.
FX = -k x
Where x is the displacement from
the relaxed position and k is the
constant of proportionality.
(often called “spring constant”)
relaxed position
FX = -kx > 0
x
x0
x=0
Physics 101: Lecture 20, Pg 3
Ideal Springs

Hooke’s Law: The force exerted by a spring is
proportional to the distance the spring is stretched or
compressed from its relaxed position.
FX = -k x
Where x is the displacement from
the relaxed position and k is the
constant of proportionality.
(often called “spring constant”)
relaxed position
FX = - kx < 0
x
x>0
x=0
Physics 101: Lecture 20, Pg 4
Concept Question
In Case 1 two people pull on the same end of a spring whose other end is
attached to a wall. In Case 2, the same two people pull with the same
forces, but this time on opposite ends of the spring. In which case does
the spring stretch the most?
1. Case 1
2. Case 2
3. Same
CORRECT
Case 1
Case 2
Physics 101: Lecture 20, Pg 5
What does moving along a circular path have to do with
moving back & forth in a straight line (oscillation about
equilibrium) ??
x = R cos  = R cos (wt)
since  = w t
x
x
1
1
2
R
3
R
8

2
8
7
3
y
7
4
6
5
0
-R

2

4

3
2
6
5
Physics 101: Lecture 20, Pg 6
Concept Question
A mass on a spring oscillates back & forth with simple harmonic motion
of amplitude A. A plot of displacement (x) versus time (t) is shown
below. At what points during its oscillation is the speed of the block
biggest?
1. When x = +A or -A (i.e. maximum displacement)
2. When x = 0 (i.e. zero displacement)
CORRECT
3. The speed of the mass is constant
x
+A
t
-A
Physics 101: Lecture 20, Pg 7
Concept Question
A mass on a spring oscillates back & forth with simple harmonic motion
of amplitude A. A plot of displacement (x) versus time (t) is shown
below. At what points during its oscillation is the magnitude of the
acceleration of the block biggest?
1. When x = +A or -A (i.e. maximum displacement)
CORRECT
2. When x = 0 (i.e. zero displacement)
3. The acceleration of the mass is constant
x
+A
t
-A
Physics 101: Lecture 20, Pg 8
Springs and Simple Harmonic Motion
X=0
X=A; v=0; a=-amax
X=0; v=-vmax; a=0
X=-A; v=0; a=amax
X=0; v=vmax; a=0
X=A; v=0; a=-amax
X=-A
X=A
Physics 101: Lecture 20, Pg 9
Simple Harmonic Motion:
x(t) = [A]cos(wt)
v(t) = -[Aw]sin(wt)
a(t) = -[Aw2]cos(wt)
x(t) = [A]sin(wt)
OR
v(t) = [Aw]cos(wt)
a(t) = -[Aw2]sin(wt)
xmax = A
Period = T (seconds per cycle)
vmax = Aw
Frequency = f = 1/T (cycles per second)
amax = Aw2
Angular frequency = w = 2f = 2/T
For spring: w2 = k/m
Physics 101: Lecture 20, Pg 10