Download Phy 202: General Physics II

Phy 202: General Physics II
Ch 10: Simple Harmonic Motion &
Elasticity
The Ideal Spring & Hooke’s Law
• Springs are objects that exhibit elastic behavior
• An ideal spring is:
– Massless (the mass of the spring is negligible compared to
– The applied force (Fapplied) required to compress/stretch is
proportional to the displacement of the spring from its
unstrained length (x) or
Fapplied = kx
where k is called the spring constant (or stiffness of the spring)
– To stretch/compress a spring, the spring exerts a restoring
force of equal & opposite magnitude (reaction force, F)
against the stretching/compressing force or
F = -kx
{this is referred to as Hooke’s Law!}
The (Elastic) Restoring Force
(& Newton’s 3rd Law)
Action:
– Applied force is
proportional to
displacement of the
spring:
Fapplied = kx
Reaction:
– Restoring force is
equal/opposite to
applied force:
F = -Fapplied = -kx
Work Performed by an Ideal Spring
• When an ideal spring is stretched a displacement x
by an applied force, the average force applied to
the spring is
Favg = ½ kx
• The work performed to stretch the spring is
W = (Favgcos q)x = ½ kx2
• The strained spring therefore gains an elastic
potential energy
PEelastic = ½ kx2
Displacing an Ideal Spring results
in Simple Harmonic Motion
• Releasing a strained a spring results in oscillating
motion due to the spring restoring force
• Motion oscillates between (+x and –x)
• This type of motion is called Simple Harmonic Motion
Newton’s
nd
2
Law & Ideal Springs
• Applying Newton’s 2nd Law to a stretched ideal spring:
SF = ma = -kx
The acceleration of the spring is
a = - (k/m).x
• The acceleration of the spring at any point in the motion
is proportional to the displacement of the spring
• For motions of this type, the angular frequency (w) of
the motion is
General form:
w =(k/m)½
a = w2 x (when a ~ x)
• The period of the motion (T) is
T = 1/f = 2p/w = 2p.(m/k)½
Simple Harmonic Motion
• When the restoring force of a spring obeys Hooke’s
Law (F=-kx), the resulting motion is called Simple
Harmonic Motion
• Consider a mass attached to a stretched spring that is
released at to=0:
– The displacement (x) of the mass due to the spring’s restoring
force will be
x = A cos wt
where A is the amplitude of the strained spring
– The velocity (v) of the mass will be
v = -Aw sin wt = -vmax sin wt
where vmax = Aw
– The acceleration (a) of the mass will be
a = -Aw2 cos wt = -amax cos wt
where amax = Aw2
Graphical Perspective of SHO
Displacement (x)
x = Acos(wt)
Velocity (v)
v = -vmaxsin(wt)
Acceleration (a)
a = -amaxcos(wt)
Conservation of Energy
& Simple Harmonic Motion
• When work is performed on a spring due to stretch/compression by an
applied force the spring gains potential energy equal to
PEelastic = ½ kx2
• As the spring is released and the restoring force w/in the spring drives
the motion of the spring (assuming no friction)
– PEelastic is converted to KE as the spring force does work
– When the spring’s length equals its unstretched length, all of the PEelastic is
converted to KE
• Applying conservation of Energy to the spring:
(PEelastic)stretched = KEunstretched
or
½ kx2 = ½ mv2
• Therefore, the speed of the spring at its unstretched length is related to
the length of the original displacement of the spring: v = (k/m)½
The Pendulum
•
Consider the motion of a mass (m) attached to
a string (length, l):
–
–
The gravitational force (mg) exerts a torque on the
mass (at all but the bottom point of the swing)
q
t = Ia = mgl sinq
l
2
since I = ml and sin q ~ q
a = (mgl/I) sinq = (mgl/ml2) sinq
m
a = (g/l).sinq
or
a = (g/l).q
mg
The period of the motion (T) is therefore
T = 2p(l/g)½
Note: a ~ q similar to a ~x (for a spring) therefore
this motion looks like the form:
a = w2q
Elastic Deformation
Types of deformation:
– Stretching/compression
Fstretch/compress = Y(DL /Lo)A
– Shear deformation
Fshear = S (DX/Lo)A
– Volume deformation
DP = -B (DV/Vo)
Stress, Strain & Hooke’s Law
• We can consider a strained mass
as though it were a collection of
small masses attached by a system
of springs
• Since deformation is related to the
applied force:
Fstretch/compress = (YA/Lo )DL
• The effective spring constant (keffective) for the mass is
keffective = YA/Lo
• Note that keffective is inversely proportional to Lo
Question: Consider a long spring (spring constant=k). How
do the spring constants of the smaller pieces (k’) compare
to the original k?