Type:
Double
Objective:
Date:_________
Simple Harmonic Motion II
Simple Harmonic Motion III
Homework: Read 10.3, Do CONCEPT QUEST #(7)
Do PROBLEMS #(5, 19, 28) Ch. 10
AP Physics “B”
Mr. Mirro
Date: ________
Simple Harmonic Motion II
Spiders detect prey by the vibrations of their webs, cars oscillate up and down when they hit a bump,
buildings and bridges sway when the wind is fierce.
As we look around us, we find many other examples of objects that vibrate or oscillate –
‰
‰
‰
‰
A
A
A
A
pendulum on a grandfather clock
mass on the end of a spring
guitar string, head of a drum, reed of a saxophone or tuning fork
ruler held firmly over the edge of a table – gently struck…etc
Since most solids are “elastic”, most objects vibrate when given an “impulse.” That is, once they are
distorted, their shape tends to be restored to some equilibrium configuration. The repetitive nature of
distorting and reshaping follows a pattern. These patterns, often referred to as Circadian Motion exist
at many levels of the natural and physical world.
We can better understand and visualize many aspects of simple harmonic motion along a straight line by
looking at the relationship to uniform circular motion. By analyzing the an object rotating with UCM, we
can “ M-O-D-E-L ” the periodic behavior and make conclusions about the similarity wrt SHM later
on …
vo
θ
v
√ A2 – x2
A
Note:
As object rotates with
Uniform Circular Motion
side (v) ∼ side (√A – x )
2
Top View
θ
2
side (v0) ∼ A
x
v
shadow
-x
+x
A
Side View
Object’s shadow “appears” to
move back and forth with
Simple Harmonic Motion
By similar triangles we can develop an expression relating the tangential speed to the speed of the
horizontal projection.
v = √ A2 – x2
A
vo
So that
v = vo √ A2 – x2
A
= vo
1 - x2
√
A
[ Eq. v ]
where, as [x → A]
then,
[v
→ vo]
AP Physics “B”
Mr. Mirro
Date: ________
Simple Harmonic Motion II
Ex 1: Consider the displacement equation x = 0.20 cos π t
[Serway13.7]
8
for a pendulum in simple harmonic motion.
a. Determine the amplitude, frequency and period of the motion for this oscillator.
b. What is the position of the object after two seconds have elapsed ?
c. What is the length of the arm of this pendulum ?
Ex 2: Consider a cart attached to two identical springs as shown. The cart has a mass of 1 kg and is
displaced by 5 cm (m = ??) to the right of equilibrium by someone reaching down to move with
a horizontal force of 10 N. [Hecht10.8]
a. What is the spring constant of the spring(s) ?
b. What is the resulting period of oscillation ?
c. Where will the cart be 0.2 sec after release ?
AP Physics “B”
Mr. Mirro
Date: ________
Simple Harmonic Motion III
When dealing with forces that are not constant, such as with simple harmonic oscillators – it is often
convenient and useful to use the ENERGY approach to solving problems.
To stretch or compress a spring, “work” has to be done. Hence, potential energy is stored in a
stretched or compressed spring as:
PEs = ½ kx2
Thus, the total mechanical energy ET of a mass-spring system is the sum of the kinetic and potential
energies:
k
ET = KE + PEs
2
m
ET = ½ mv + ½ kx
2
A
Note: As long as there is no friction, the total mechanical energy (ET) remains constant.
While the mass oscillates back and forth, energy continuously changes from potential energy to kinetic
energy and back again.
‰
At the extreme points, x = A and x = - A, all the energy is stored in the spring as potential energy.
‰
At these extreme points, the mass stops momentarily as it changes direction so v = 0 !
Therefore,
ET = ½ m(0)2 + ½ k(A)2 ⇒
½ k A2
Thus, the total mechanical energy of a simple harmonic oscillator is proportional to the
square of the “amplitude.”
‰
At the equilibrium point, x = 0 all the energy is kinetic:
ET = ½ m(v)2 + ½ k(0)2 ⇒
‰
½ mv2
However, at the other intermediary points, the conservation of energy applies as follows:
Estretch = Erelease
½ m(0)2 + ½ k(A)2 = ½ m(v)2 + ½ k(x)2
So,
½ k(A)2 = ½ m(v)2 + ½ k(x)2
If we continue the previous expression for the total mechanical energy (ET) of the mass-spring system,
we can solve for the velocity (v) of the object at any position in terms of - that position (x), the spring
constant (k), mass (m) and amplitude (A) as was done previously using uniform circular motion.
k(A)2 = m(v)2 + k(x)2
[ divide each term by (½) ]
k(A)2 - k(x)2 = m(v)2
[ transpose (k x2) ]
v2 = kA2 - kx2
m
m
[ divide each term by (m) ]
k A2 1 - x2
m
A2
v2 = k [A2 - x2 ] ⇒
m
Star this ***
We’ll get back to it !
Recalling that the maximum velocity (vmax = v0) occurs at the instant that the object passes
through the equilibrium position (Δx = 0), we can express the gcf factor KA2 in terms of v0
by examining the conservation of energy.
m
0
0
½ m v02 + ½ k Δx2 = ½ m vmin2 + ½ k (A)2
max
min
(½) m v0
min
2
max
= (½) k A2
m v02 = k A2
v02 = k A2
m
Since
Therefore,
v2 = k A2
m
v = v0
1 - x2
A2
1 - x2
A2
By substitution of v0
***
Which looks just like Eq. (v)
We have since come full circle – relating both UCM and SHM to the same expression for velocity for a
particular position (x).
On a final note…
‰
The word “harmonic” refers to the motion being “sinusoidal”.
‰
The motion is “simple” when there is pure sinusoidal motion of a “single” frequency
such as one tuning fork, rather than a mixture of frequencies such as our voice.
AP Physics “B”
Mr. Mirro
Date: ________
Simple Harmonic Motion III
Ex 1: The cone of a loudspeaker vibrates in SHM at a frequency of 262 Hz (middle “C”). The amplitude at
the center of the cone is A = 1.5 x 10-4 m, and at t = 0, x = A. [Giancolli11.7]
a. What is equation describing the motion of the center of this cone in terms of time (t) ?
b. What is the maximum velocity at the center of the cone ?
x=0
-x
c. What is the maximum acceleration at the center of the cone ?
d. What is the position of the cone at t = 1 millisecond ⇒ 1 x 10-3 s ?
+x
A
Ex 2: A spring stretches 0.15 m when a 0.30 kg mass is hung from it. The spring is then stretched an
additional 0.10 m from its equilibrium point and released. [Giancoli11.3]
a. Determine the spring constant of the spring.
Equilibrium
b. What is the amplitude of oscillation ?
_____________
c. Using an energy approach, develop an expression for the maximum velocity v0 of the mass in
terms of amplitude (A), spring constant (k) and mass (m).
d. Compute the maximum velocity for the mass in m/s.
e. What is the magnitude of the velocity, v, when the mass is 0.050 m from equilibrium ?
f. Compute the maximum acceleration of the mass.
g. What is the total mechanical energy of the mass-spring system in joules ?