simple harmonic motion harmonic motion due to a net...

Simple Harmonic Motion

Harmonic motion due to a net restoring forcedirectly proportional to the displacement

Example: Spring motion: F = -kx

2

2

2

2

Net Force:

0

F ma

d xkx

d x kx

dt

mdt

m

=

+

! =

=

"

Equation of motion for the simple harmonic oscillator


2

2

2

2

Net Force:

0

F ma

d xkx

d x kx

dt

mdt

m

=

+

! =

=

"2

2

2

2

2

2

2

cos( )

( cos( )) sin( )

[ sin( )]

cos( )

Now substitute in the SHM equation:

d 0

cos( ) cos( ) 0

( ) cos(

x A tdx d

A t A tdt dtd x d dx d

A tdt dt dt dt

A t

x kx

dt mk

A t A tm

kA t

m

! "

! " ! ! "

! ! "

! ! "

! ! " ! "

! !

= +

= + = # +

= = # +

= # +

+ =

# + + + =

# ) 0

km

"

!

+ =

$ =

Angular Frequency :

Frequency, f :

Period, T:

2

2=

T

km

f

!

!

"!

!

"

=

=

22

22

22

22

1)

2)

2( )3)

4)

m Mv kAmmv kAMm Mv kAM

mv kAM

+=

=

+=

=

1. Nikita devised the following method of measuring the muzzle velocity of arifle. She fires a bullet into a wooden block (mass M) resting on a smoothsurface, and attached to a spring with a spring constant k. The bullet, whosemass is m, remains embedded in the wooden block. She measures the distancethat the block recoils and compresses the spring to be A. What is the speed ofthe bullet?

2. The block is going to execute simple harmonic motion. What is thefrequency of oscillation?

!

1)1

2"k

m + M

2) 12"

kM

3)1

2"km

4) 12"

kA2

m + M5) None of the above

3. A mass (m) is connected to two springs with spring constants k1 and k2.The mass is displaced from the equilibrium position and released. (a) Show that the mass executes simple harmonic motion. (b) Find the frequency of oscillation.


a) To show that it executes SHM,we need the restoring forceequation.



x If we move the mass to the rightby an amount x, then the restoring force is…




!

F1 = "k1x

!

F2 = "k2xSpring 1

Spring 2




!

F1 = "k1x

!

F2 = "k2xSpring 1

Spring 2

!

F = F1 + F2 = "(k1 + k2)x = "keff x




!

F1 = "k1x

!

F2 = "k2xSpring 1

Spring 2

!

F = F1 + F2 = "(k1 + k2)x = "keff x

!

keffective = k1 + k2 f = 2" k1 + k2

m


Δx1 Δx2

If we apply a force Fto stretch the springs,then the total displacement

!

"x = "x1 + "x2


Δx1 Δx2


!

"x = "x1 + "x2

= # Fk1

#Fk2

!

F = F1 = F2


Δx1 Δx2


!

"x = "x1 + "x2

= # Fk1

#Fk2

= # F 1k1

+1k2

$

% &

'

( )

= # Fkeff

!

F = F1 = F2


Δx1 Δx2


!

"x = "x1 + "x2

= # Fk1

#Fk2

= # F 1k1

+1k2

$

% &

'

( )

= # Fkeff

!

F = F1 = F2

!

1keff

=1k1

+1k2

5. A mass (m) is supported by two springs with spring constants k1 and k2.The mass is displaced from the equilibrium position and released.Find the frequency of oscillation.


When the mass is pulled down by a distance x,

!

F = F1 + F2 = "k1x " k2x


When the mass is pulled down by a distance x,

!

F = F1 + F2 = "k1x " k2x = "(k1 + k2)x = "keff x

6. A uniform stick of length L and mass M is pivoted on a hinge at one end and held horizontal by a spring with a spring constant k. If the stick oscillates up and down slightly, then show its motion is simple harmonic.What is its frequency?

L


L

In equilibrium:Stick: horizontal positionSpring: extended by x0


L


!

"# = Mgl2$

% & '

( ) * kx0l = 0


L


!

"# = Mgl2$

% & '

( ) * kx0l = 0θ

If the stick is displaced througha small angle θ, and the spring isextended by an amount x

!

"# = Mgl2$

% & '

( ) * k(x + x0)l = I+


L


!

"# = Mgl2$

% & '

( ) * kx0l = 0θ


!

"# = Mgl2$

% & '

( ) * k(x + x0)l = I+

* kxl = Id2,dt 2

* kl2, = Id2,dt 2


L


!

"# = Mgl2$

% & '

( ) * kx0l = 0θ


!

"# = Mgl2$

% & '

( ) * k(x + x0)l = I+

* kxl = Id2,dt 2

* kl, = Id2,dt 2

!

d2"dt 2

+kl2

I" = 0

# =kl2

I=

kl2

Ml2 /3=

3kM

Force between two molecules

F

r

(Positive)Repulsive force

(Negative)Attractive force


Problem 7. In some diatomic molecules, the force each atom exerts on the othercan be approximated by

where r is the atomic separation, and C and D are positive constants.(a) Graph F versus r from r = 0 to r = 2D/C.

2 3 ,C DFr r

= ! +

F

r

(Positive)Repulsiveforce(Negative)Attractiveforce



where r is the atomic separation, and C and D are positive constants.(a) Graph F versus r from r = 0 to r = 2D/C.(b) Find the equilibrium position r0 in terms of the constants C and D.

2 3 ,C DFr r

= ! +

F

r




where r is the atomic separation, and C and D are positive constants.(a) Graph F versus r from r = 0 to r = 2D/C.(b) Find the equilibrium position r0 in terms of the constants C and D.

2 3 ,C DFr r

= ! +

F

r



!

F(r0) = "Cr0

2 +Dr0

3 = 0

r0 =DC


where r is the atomic separation, and C and D are positive constants.(a) Graph F versus r from r = 0 to r = 2D/C.(b) Find the equilibrium position r0 in terms of the constants C and D.(c) Assuming that, for small displacements Δr << r0 the motion is approximately

simple harmonic, determine the force constant k in terms of C and D.

2 3 ,C DFr r

= ! +

!

r0 =DC

!

F(ro + "r) = #C

(r0 + "r)2 +D

(r0 + "r)3

= # 1(r0 + "r)3 [Cr0 +C"r #D]

= # C"r(r0 + "r)3 $ #

C"r(r0)3 = #keff"r

8. A cylindrical block of wood of mass m and cross-sectional area Aand height h is floating in water. You give it a nudge from the top,and the block starts to oscillate in water. The water density is ρw andacceleration due to gravity is g. Show that it executes SimpleHarmonic Motion. Find the effective spring constant.

In equilibrium,

!

Fbuoyancy = mg

mg

Fbuoyancy


In equilibrium,

!

Fbuoyancy = mg

mg

Fbuoyancy

When you push the blockinto the water (by Δx), thereis an additional buoyantforce, equal to the additionalwater displaced.

!

Fnet = "#watergA$x


In equilibrium,

!

Fbuoyancy = mg

mg

Fbuoyancy

When you push the blockinto the water (by Δx), thereis an additional buoyantforce, equal to the additionalwater displaced.

!

Fnet = "#watergA$x

!

keffective = "watergA

Problem 9. In Alice in Wonderland, Alice imagines a 10-cm diameterhole drilled all the way through the center of the earth. Standing onone end of the hole, she drops an apple through to see that the applemakes simple harmonic motion about the center of the earth.

(a) At a point x from the center, what is thegravitational force experienced by the apple of mass m?Assume that the earth has a uniform mass density ρ.Indicate both magnitude and sign.(Hint: The apple at a distance x from the centerexperiences the gravitational attraction only fromthe part of the earth within a sphere of radius x.)

(b) What is the effective spring constant k?

(c) How long will the apple take to return to Alice, if all frictional effects are ignored?

Problem 9. (a) At a point x from the center, what is the gravitational forceexperienced by the apple of mass m? Assume that the earth has a uniformmass density ρ. Indicate both magnitude and sign. (Hint: The apple at adistance x from the center experiences the gravitational attraction onlyfrom the part of the earth within a sphere of radius x.)

rE

x


rE

x

!

M* = "V =ME

4#rE3 /3

$

% &

'

( ) 4#3x 3

$

% &

'

( ) =

ME

rE3 x 3


rE

x

!

M* = "V =ME

4#rE3 /3

$

% &

'

( ) 4#3x 3

$

% &

'

( ) =

ME

rE3 x 3

!

F = "GmM *x 2

= "GmME

rE3

x 3

x 2

= "GmME

rE3 x


rE

x

!

M* = "V =ME

4#rE3 /3

$

% &

'

( ) 4#3x 3

$

% &

'

( ) =

ME

rE3 x 3

!

F = "GmM *x 2

= "GmME

rE3

x 3

x 2

= "GmME

rE3 x

!

keff =GmME

rE3

A Mylar balloon (of mass m when empty) is filled with helium andoccupies a volume V. The balloon is tied up to a long string oflength l, as shown in the figure. When the balloon is pushedsideways, it starts to oscillate left and right in simple harmonicmotion like an up-side-down pendulum.

Air

Heinside

simple harmonic motion harmonic motion due to a net...

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