Two important examples of
s.h.m.
© D Hoult 2010
The mass / spring oscillator
The mass / spring oscillator
net force acting on the mass = zero
A graph of force against extension for a spring is a
straight line passing through the origin:
force a extension
A graph of force against extension for a spring is a
straight line passing through the origin:
force a extension
The constant of proportionality, k is called the
elastic constant of the spring (usually considered
to be a positive number)
When the displacement is downwards, the net
force is upwards and vice versa
So a better way to draw the graph would be
In this case, F represents the force with which the
spring “pulls back” when it is given an extension x
by some external agency
slope = - k
F = -kx
F = -kx
If the mass is released after having been given a
displacement, x, then its acceleration will be
a=
F = -kx
If the mass is released after having been given a
displacement, x, then its acceleration will be
a = -kx
m
F = -kx
If the mass is released after having been given a
displacement, x, then its acceleration will be
a = -kx
m
So the motion is s.h.m. and the constant of
proportionality between a and x has magnitude
F = -kx
If the mass is released after having been given a
displacement, x, then its acceleration will be
a = -kx
m
So the motion is s.h.m. and the constant of
proportionality between a and x has magnitude
k
m
F = -kx
If the mass is released after having been given a
displacement, x, then its acceleration will be
a = -kx
m
So the motion is s.h.m. and the constant of
proportionality between a and x has magnitude
k
m
which is equal to w2 in the “shm equation”
Remembering that
T = 2p
w
we can suggest that the time period of a mass
spring oscillator is given by
Remembering that
T = 2p
w
we can suggest that the time period of a mass
spring oscillator is given by
T = 2p m
k
The simple pendulum
F=
F = mg sin q
F=
F = - mg sin q
a=
a = - g sin q
q=
x
q=
L
If we make sure that q is
a small angle
If we make sure that q is
a small angle, sin q can
be replaced by q
g
x
a=L