Important Reminders
 Last Lecture
Power, Impulse, Center of Mass
 Pset # 8 due tomorrow.
 Today
 Problem Solving session in class tomorrow.
Simple Harmonic Motion
 MasteringPhysics due next Monday.
 Important Concepts
The physics of the motion is in the mass and spring
constant which determine the period of each oscillation.
The amplitude does not affect the period.
NOTE: Class grading guidelines clearly allow discussion
of MasterPhysics problems but also clearly prohibit
directly working together or copying the answers of others.
 No Pset due next week.
Energy oscillates between Kinetic and Potential.
 No formal tutoring sessions next week.
8.01L Fall 2005
11/17/2005
8.01L Fall 2005
Center of Mass Velocity
!
 Definition: vC.M . = 1 M
TOT
11/17/2005
Simple Harmonic Motion - I
!
!
!
!m v
!
!
 Start with Force equation: FSpring = !k(l ! l0 )
i i
 Define x axis along direction spring is stretched and
put x=0 at the point the spring is unstretched:
!
 Connection to momentum: M TOT vC.M . = pTOT
(
2
Fx = !kx = max = m d x
dt 2
)
d2x
k
" 2 =! x
dt
m
 So, if momentum is conserved, the velocity of the
center of mass is constant.
 So, what is the solution to the differential equation?
8.01L Fall 2005
11/17/2005
8.01L Fall 2005
11/17/2005
1
Simple Harmonic Motion - II
Simple Harmonic Motion - III
 The answer is sine and/or cosine function with three
mathematically equivalent ways to write it:
x = A cos(! t + " )
x = Asin(! t + " )
x = A cos(! t) + Bsin(! t)
ω is angular frequency in radians per second:
k
m
8.01L Fall 2005
A is the amplitude, the maximum displacement from zero
φ is an arbitrary constant that depends only on when you
define t=0, no real connection to the nature of the motion
 In all cases, A, B, and φ are constants determined
by the initial conditions.
 ω is given by the physics: ! =
 Connections to the physical motion:
11/17/2005
Frequency in cycles/second f =
!
1
=
2" 2"
Period (time for one cycle) T =
1 2"
m
=
= 2"
f
!
k
8.01L Fall 2005
11/17/2005
Velocity/Acceleration in SHM
Energy in SHM
 Spring PE: PEspring =
Also sine/cosine functions:
x = A cos(! t + " )
2
dvx
2
dt = #A! cos(! t + " )
k
= #! 2 x = # x
m
 Total:
m ( A! )
kA 2
( cos(! t + " ))2 +
(sin(! t + " ))2
2
2
2
ETotal = KE + PE =
!2 = k m
kA 2
kA 2 1 2
ETotal =
= mvMax
( cos(! t + " ))2 + (sin(! t + " ))2 =
2
2
2
 Note that: Maximum speed vMax = A!
8.01L Fall 2005
1 2 kA 2
kx =
( cos(! t + " ))2
2
2
1 2 m ( A! )
(sin(! t + " ))2
 Kinetic energy: KE = mv =
2
2
vx = dx dt = #A! sin(! t + " )
ax =
k
m
(
11/17/2005
8.01L Fall 2005
)
11/17/2005
2