Download Problem Solving Strategies: Simple Harmonic Oscillator and Mechanical Energy 8.01t

Problem Solving Strategies:
Simple Harmonic Oscillator
and Mechanical Energy
8.01t
Oct 22, 2004
Modeling the Motion: Newton ‘s Second Law
• Equation of Motion:
2
d
ˆi : −kx = m x
dt 2
• Possible Solution: Period of
oscillation is T
• Position
2π
x = Acos( t)
T
• Initial Position t = 0:
• Velocity:
A = x0
dx
2π
2π
v=
=−
Asin( t)
dt
T
T
• Velocity at t = T/4
2π
2π
veq = −
A=−
x0
T
T
Mechanical Energy is Constant
1
1 2
2
Eeq − E0 = mveq − kx0 = 0
2
2
veq = −
Solve for velocity at equilibrium position
Period T: Condition from Newton’s Second Law
−
2π
x0 = veq
T
Solve for Period
2π
k
=
T
m
T = 2π
m
k
k
x0
m
The Gravitational Field of a Spherical Shell of Matter • The gravitational force on a mass placed
outside a spherical shell of matter of
uniform surface mass density is the same
force that would arise if all the mass of the
shell were placed at the center of the
sphere.
• The gravitational force on a mass placed inside a spherical shell of matter is zero.
The Gravitational Field of a Spherical Shell of Matter
mms
⎧
G
⎪−G 2 r̂, r > R
Fm , s ( r ) = ⎨
r
G
⎪
0,
r<R
⎩
where r̂ is the unit vector located at the
position of mass and pointing radially
away from the center of the shell.
Gravitational Force Inside Uniform Sphere
• Place mass m at distance r < RE from
center
• Force only due mass of earth inside
sphere of radius r
m = ρ (4 / 3)π r 3
r
• Density ρ = me /(4 / 3)π R
e3
• Force law
K
Gmmr
Fgrav = −
rˆ =
2
r