Download Energy And SHM

Energy And SHM
Energy of Spring
Spring has elastic potential energy
PE = ½ kx2
 If assuming no friction, the total energy
at any point is the sum of its KE and PE
E = ½ mv2 + ½ kx2

At Extreme
Stops moving before starts back, so all
energy is PE and x is max extension (A)
E = ½ k A2
 At equilibrium, all energy is KE and vo is
the max velocity
E = ½ m vo2

Algebraic Manipulation

½ mv2 + ½ kx2 = ½ k A2
Example

If a spring is stretched to 2A what
happens to a) the energy of the
system? B) maximum velocity? C)
maximum acceleration?
Example

A spring stretches .150m when a .300kg
mass is hung from it. The spring is
stretched and additional .100m from its
equilibrium point then released.
Determine a) k b) the amplitude c)
the max velocity d) the velocity when
.050 m from equilibrium e) the max
acceleration f) the total energy
Period of Vibration
Time for one oscillation depends on the
stiffness of the spring
 Does not depend on the A
 SHM can be thought of similar to an
object moving around a circle
 Time for one oscillation is the time for
one revolution

v = 2r/ T
 At max displacement r = A
 vo = 2A/T
 T = 2A/ vo
 ½ kA2 = ½ mvo2
 A/vo = (m/k)
 T = 2(m/k)

Period of oscillation depends on m and
k but not on the amplitude
 Greater mass means more inertia so a
slower response time and longer period
 Greater k means more force required,
more force causes greater acceleration
and shorter period

Example

How long will it take an oscillating spring
(k = 25 N/m) to make one complete
cycle when:
a) a 10g mass is attached
b) a 100g mass is attached
Pendulum
Small object (the bob) suspended from
the end of a lightweight cord
 Motion of pendulum very close to SHM
if the amplitude of oscillation is fairly
small
 Restoring force is the component of the
bobs weight – depends on the weight
and the angle

Period of Pendulum

T = 2√(L/ g)
Period does not depend on the mass
 Period does not depend on the
amplitude

Example

Estimate the length of the pendulum in
a grandfather clock that ticks once
every second. B) what would the period
of a clock with a 1.0m length be?
Example

Will a grandfather clock keep the same
time everywhere? What will a clock be
off if taken to the moon where gravity is
1/6 that of the earth’s?