Lec Harmonic Motion.notebook
January 24, 2013
Brief reminder:
Springs: As a spring is compressed or stretched, the force required increases.
The force can be calculated using Hooke's Law:
F = ks where k is the spring constant for that particular spring
and s is the displacement from the rest position
Brief reminder:
Springs: As Work is change in energy, which is F*d, and this can be calculated finding the area under a F vs D graph, plotting the force on a spring gives the following graph:
F
s
Lec Harmonic Motion.notebook
January 24, 2013
Brief reminder:
Springs: As Work is change in energy, which is F*d, and this can be calculated finding the area under a F vs D graph, plotting the force on a spring gives the following graph:
W = area = 1/2 F * s
F = ks
F
2
W = 1/2 ks
PEs = W = 1/2 ks2
s
Frequency and Period
f = 1/T
Orbit of Earth = 365 days ­­­ this is period
The frequency of the Earth's orbit is 1/365 What is the frequency & period of a clock's second hand?
Lec Harmonic Motion.notebook
January 24, 2013
We know ω as angular velocity.
ω = 2π/T (think as distance over time)
ω = 2π/T = 2πf à ω is called angular frequency
To get a linear velocity, multiply by A (amplitude) as we would for R.
v = 2πfA (similar to converting v to ω in radians/sec)
Oscillating spring We know E = ½ mv2
and E = ½ kA2
so ½ mv2 = ½ kA2
so vmax = A(k/m)1/2
since v = 2πfA
we get v = 2πfA = A(k/m)1/2
f = 1/2π(k/m)1/2
Lec Harmonic Motion.notebook
January 24, 2013
Pendulum: important formulas
F = ­mgsinθ
Eq 1: fo = 1/2π sqrt(g/L)
Eq 2: T = 2π sqrt(L/g)
Concepts:
Damping: under (some damping), critically­(more damping) stops quickest, over damping­no overshooting (most damping)
Natural frequency
Forced Oscillation (pumping in energy)
How far will the spring compress (x) when the ball is placed on it?
2kg
x
k=100
Lec Harmonic Motion.notebook
2kg
x
k=100
January 24, 2013
How far will the spring compress (x) when the ball is placed on it?
.
year
t
s
a
PEg = mgx
om l ut the r
2
f
s
o
PEs = 1/2 kx
i
This , how ab
Now lation?
2
l
mgx = 1/2 kx
osci
2
0 = 1/2 kx ­ mgx
0 = x (1/2kx ­ mg)
0 = 1/2kx ­ mg
1/2 kx = mg
x = 2mg / k
x = .4meters
How far will the spring compress (x) when the ball is placed on it?
When the ball is placed on the spring,
is it at equilibrium (DEFINE!)?
2kg
What is the net force?
When is it at equilibrium?
x
k=100
At max compression, how much E is there?
Lec Harmonic Motion.notebook
January 24, 2013
How far will the spring compress (x) when the ball is placed on it?
2kg
x
k=100
When the ball is placed on the spring,
is it at equilibrium?
No
What is the net force?
mg ­ kx = ma (kx = 0!)
When is it at equilibrium?
mg = kx ­­> x = .2meters
At max compression (.4m), how much E is there?
mgx = 1/2 kx2 = 8J
How far will the spring compress (x) when the ball is placed on it?
If the ball is dropped onto the spring,
will is have the same amplitude?
2kg
What happens when the ball hits the spring?
x
k=100
Lec Harmonic Motion.notebook
January 24, 2013
How far will the spring compress (x) when the ball is placed on it?
If the ball is dropped onto the spring,
will is have the same amplitude?
2kg
What happens when the ball hits the spring?
x
k=100
This is harmonic motion...
but only part of the motion could be considered SHM. (when f=­kx)
Another view of SHM and springs.
http://ia600200.us.archive.org/20/items/AP_Physics_C_Lesson_19/Container.html
Lec Harmonic Motion.notebook
Displacement in SHM
x = A cos θ
x = A cos ωt
(x is maximum amplitude)
x = A cos ωt + ε
(if 'out of phase')
Velocity in SHM
if x = A cos ωt then
v = dx/dt = ­Aω sin ωt
January 24, 2013
Lec Harmonic Motion.notebook
January 24, 2013
Acceleration in SHM
if and
x = A cos ωt v = dx/dt = ­Aω sin ωt
the
a = dv/dt = ­Aω2 cos ωt
HEY!
since x = A cos ωt,
a = ­ω2x
we can substitute and get
(acceleration is proportional to displacement)
(showed up in 2009)
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
a. The potential energy in the spring at the instant the mass is released
Lec Harmonic Motion.notebook
January 24, 2013
Use calculus to find Energy when F=kx
W = F d
(d = x)
W = F dx
W = ∫kx dx
x
W = k x2 /2 |0
W = 1/2 kx2
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
a. The potential energy in the spring at the instant the mass is released
Lec Harmonic Motion.notebook
January 24, 2013
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
b. The maximum speed of the mass
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
b. The maximum speed of the mass
Lec Harmonic Motion.notebook
January 24, 2013
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
c. The displacement of the mass at the point where the potential energy of the spring and the kinetic energy of the mass are equal
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
c. The displacement of the mass at the point where the potential energy of the spring and the kinetic energy of the mass are equal
Lec Harmonic Motion.notebook
January 24, 2013
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
The amplitude of the oscillation is now increased:
d. State whether the period of the oscillation increases, decreases, or remains the same. Justify your answer.
1986M3. A special spring is constructed in which the restoring force is in the opposite direction to the displacement, but is proportional to the cube of the displacement; i.e., F = ­kx3
This spring is placed on a horizontal frictionless surface. One end of the spring is fixed, and the other end
is fastened to a mass M. The mass is moved so that the spring is stretched a distance A and then released.
Determine each of the following in terms of k, A, and M.
The amplitude of the oscillation is now increased:
d. State whether the period of the oscillation increases, decreases, or remains the same. Justify your answer.
Lec Harmonic Motion.notebook
January 24, 2013
An ideal massless spring is fixed to the wall at one end, as shown above. A block of mass 2kg attached to the other end of the spring oscillates with amplitude .05m on a frictionless, horizontal surface. The maximum speed of the block is 5m/s. What is the force constant of the spring?
An ideal massless spring is fixed to the wall at one end, as shown above. A block of mass 2kg attached to the other end of the spring oscillates with amplitude .05m on a frictionless, horizontal surface. The maximum speed of the block is 5m/s. What is the force constant of the spring?
f = 1/(2π)(k/m)1/2
Lec Harmonic Motion.notebook
January 24, 2013
An ideal massless spring is fixed to the wall at one end, as shown above. A block of mass 2kg attached to the other end of the spring oscillates with amplitude .05m on a frictionless, horizontal surface. The maximum speed of the block is 5m/s. What is the force constant of the spring?
f = 1/(2π)(k/m)1/2
k = f 2m4π2 = m (2 π f)2
ω = 2π/T = 2πf
k = m (ω)2
ω = v / A
so: k = m(v/A)2
2kg*(5m/s)2 / (.05m)2 = 20000 N/m
An ideal massless spring is fixed to the wall at one end, as shown above. A block of mass 2kg attached to the other end of the spring oscillates with amplitude .05m on a frictionless, horizontal surface. The maximum speed of the block is 5m/s. What is the force constant of the spring?
f = 1/(2π)(k/m)1/2
k = f 2m4π2 = m (2 π f)2
ω = 2π/T = 2πf
k = m (ω)2
ω = v / A
so: k = m(v/A)2
2kg*(5m/s)2 / (.05m)2 = 20000 N/m
Or Energy:
KE = 1/2 mv2
PE = 1/2 kx2
1/2 mv2 = 1/2 kx2
mv2 = kx2
mv2/x2 = k
k = 20000 N/m
Lec Harmonic Motion.notebook
January 24, 2013
2003 Mech. 2.
An ideal spring is hung from the ceiling and a pan of mass M is suspended from the end of the spring, stretching it a distance D as shown above. A piece of clay, also of mass M, is then dropped from a height H onto the pan
and sticks to it. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Determine the speed of the clay at the instant it hits the pan.
(b) Determine the speed of the pan just after the clay strikes it.
(c) Determine the period of the simple harmonic motion that ensues.
(d) Determine the distance the spring is stretched (from its initial unstretched length) at the moment the speed of the pan is a maximum. Justify your answer.
(e) The clay is now removed from the pan and the pan is returned to equilibrium at the end of the spring. A rubber ball, also of mass M, is dropped from the same height H onto the pan, and after the collision is caught in midair before hitting anything else.
Indicate below whether the period of the resulting simple harmonic motion of the pan is greater than, less than, or the same as it was in part (c).
_______Greater than _______Less than _______The same as
Justify your answer.
m
θ
pivot
m
2L
1978M3. A stick of length 2L and negligible mass has a point mass m affixed to each end. The stick is arranged so that it pivots in a horizontal plane about a frictionless vertical axis through its center. A spring of force constant k is connected to one of the masses as shown above. The system is in equilibrium when the spring and stick are perpendicular. The stick is displaced through a small angle θo as shown and then released from rest at t = 0 . a. Determine the restoring torque when the stick is displaced from equilibrium through the small angle θo.
Lec Harmonic Motion.notebook
January 24, 2013
m
θ
F = ­kx
x = ­Lsinθ
τ = Fd
d = Lcosθ
pivot
m
2L
m
θ
pivot
m
a. Determine the restoring torque when the stick is displaced from equilibrium through the small angle θo.
2L
(small angle approximation)
cosθ = 1
sinθ = θ
a. Determine the restoring torque when the stick is displaced from equilibrium through the small angle θo.
F = ­kx
x = ­Lsinθ
τ = Fd
d = Lcosθ
τ = ­kLsinθ * Lcosθ
2
τ = ­kL sinθcosθ
(small angle approximation optional)
2
τ = ­kL θ
Lec Harmonic Motion.notebook
January 24, 2013
m
θ
b. Determine the magnitude of the angular acceleration of the stick just after it has been released. pivot
m
2L
m
θ
pivot
m
2L
b. Determine the magnitude of the angular acceleration of the stick just after it has been released. I = 2mL2
τ = Iα
2 τ = 2mL α
2 α = τ/2mL
Lec Harmonic Motion.notebook
January 24, 2013
m
θ
pivot
m
2L
τ = Iα
2 τ = 2mL α
2 α = τ/2mL
2
τ = ­kL θ
2 2
α = (­kL θ)/2mL
α = ­kθ/2m
m
θ
pivot
m
2L
b. Determine the magnitude of the angular acceleration of the stick just after it has been released. I = 2mL2
c. Write the differential equation whose solution gives the behavior of the system after it has been released. Lec Harmonic Motion.notebook
January 24, 2013
m
θ
pivot
c. Write the differential equation whose solution gives the behavior of the system after it has been released. α = ­kθ/2m
m
2L
2
α = d θ / dt
2
d2θ / dt2 = ­kθ/2m
m
θ
pivot
m
d. Write the expression for the angular displacement θ of the stick as a function of time t after it has been released from rest. 2L
(answer key does NOT show calculus solution....
how's your calculus?)
Lec Harmonic Motion.notebook
January 24, 2013
m
θ
pivot
m
2L
d. Write the expression for the angular displacement θ of the stick as a function of time t after it has been released from rest. x = xo cos ωt
Lsinθ = Lsinθo cos ωt
(small angle)
θ = θo cos ωt
(answer key does NOT show calculus solution....
how's your calculus?)
m
θ
pivot
m
2L
a = ­ω2x
α = ­ω2θ
­kθ/2m = ­ω2θ
k/2m = ω2
√k/2m = ω
d. Write the expression for the angular displacement θ of the stick as a function of time t after it has been released from rest. x = xo cos ωt
Lsinθ = Lsinθo cos ωt
(small angle)
θ = θo cos ωt
θ = θo cos (√k/2m)t
(answer key does NOT show calculus solution....
how's your calculus?)
Lec Harmonic Motion.notebook
January 24, 2013
2009 M2
You are given a long, thin rectangular bar of known mass M and length L with a pivot attached to one end. The bar has a nonuniform mass density, and the center of mass is located a known distance x from the end with the pivot. You are to determine the rotational inertia Ib of the bar about the pivot by suspending the bar from the pivot, as shown above, and allowing it to swing. Express all algebraic answers in terms of Ib, the given quantities, and fundamental constants.
(a) i.
By applying the appropriate equation of motion to the bar, write the differential equation for the angle θ the bar makes with the vertical.
2009 M2
Remember the odd Force equation I mentioned, and couldn't find a use for... HERE IT IS.
F = ­mgsinθ
τ = Fx = ­mgxsinθ = Ibα
α = ­mgxsinθ / Ib
d2θ = ­mgxsinθ
Ib
dt2
(a) i.
By applying the appropriate equation of motion to the bar, write the differential equation for the angle θ the bar makes with the vertical.
Lec Harmonic Motion.notebook
January 24, 2013
2009 M2
small­angle approximation
θ = sinθ
d2θ = ­mgxθ
dt2
Ib
α = ­ω2θ
(acceleration is proportional to displacement)
ω2θ = mgxθ ⇒ ω2 = mgx
Ib
Ib
T = 2π/ω
(a) ii.
⇒ T = 2π√(Ib/mgx) ... The eq for physical pendulum)
By applying the small­angle approximation to your differential equation, calculate the period of the bar's motion.
2009 M2
(b) Describe he experimental procedure you would use to make the additional measurements needed to determine Ib. Include how you would use your measurements to obtain Ib and how you would minimize experimental error.
Lec Harmonic Motion.notebook
January 24, 2013
2009 M2
(c) Now suppose that you were not given the location of the center of mass of the bar. Describe an experimental procedure that you could use to determine it, including the equipment that you would need.