Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
![Spring-Mass Systems
Harmonic Motion Basics -
Spring Constant (k in N/m) – The spring constant tells you
how strong (stiff) a spring is. A stiffer spring has a bigger k.
Amplitude (A) - maximum displacement from the
equilibrium position. The amount of energy
in a spring-mass system is determined ONLY
y the amplitude.
Period (T)- time for one complete cycle.
Frequency (f) - number of cycles in one second.
Amplitude - 14 cm
Period (T) = 4.2 sec
k-
k=
0:00.0
start
20 N/m
20 N/m
k = 20 N/m
..
faster T
k-
faster T
M MM M
10 N/m
20 N/m
0:04.2
stop
With the same spring constant
more mass causes a slower
vibration (larger T).
With the same mass the
stronger spring (bigger k) will
vibrate faster (maller T).
28 cm
A spring-mass system is called Simple Harmonic Motion (SHM)
because the force is directly proportional to the displacement of
the spring. Anything that follows this rule is known as Simple
Harmonic Motion. (A pendulum is only close to simple harmonic
motion.)
Hooke's Law:
Spring Constant (in N/m):
bigger k= stiffer spring.
Force (in N)
of the
spring
»F=-kx
displacement (in m) from
the equilibrium position.
Equilibrium position (where it will come
to rest; where it was before disturbed)
The restoring force
(F) tries to return it
to its equilibrium
position.
Force and Position: As seen above in Hooke's Law
F and x always oppose each other. If F is positive, x is
negative, etc. [k, though, is always positive]).
Stretched
wwww
X-+A; F=-max
Stretched Spring–The spring gets bigger,
so x is positive and F is negative.
Compressed Spring-The spring gets smaller,
so x is negative and F is positive.
At equilibrium: Regardless if it is moving or not, at
the equilibrium position the spring does not apply
a force. x=0 and F=0.
M
a--max; v -0
PE = max; KE =0
Relaxed
[www
x = 0; F =0
a = 0; v=-max
PE = 0; KE = max
M
Acceleration (a), Velocity (v), and Energy (E)
It should be obvious that where the force is zero
(equilibrium position), the acceleration is zero.
At the ends the mass stops for an instant, so v=0
and KE - 0, but PE is a maximum (PE- Vlox).
The mass speeds up as it moves toward the center,
where it has maximum v and KE and minimum PE,
Compressed
x=-A; F= +max
a =+max; v= 0
PE = max; KE = 0
+F
Period of a Spring-Mass System:
Ex. A 350 g mass is attached to a spring that has
a spring constant of 12 N/m. What is the period of
vibration?
Notice that amplitude
is not in the equation.
Just as in all harmonic
motion-amplitude
does not affect the
period or frequency of
a spring-mass system.
Period
Mass
(in sec)
(in kg)
m
T = 2n,
Variables:
m = 0.35 kg
(1000 g =1 kg)
k= 12 N/m
T = 2x.
Spring
constant
(in N/m)
T= 6.28.0292
k
T= 6.28(.1709)
35
T =1.07 seç
T = 6.2812
T =
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