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Oscillations II-Classical Physics-Handouts, Lecture notes of Classical Physics

University of AllahabadClassical Physics

This course includes alternating current, collisions, electric potential energy, electromagnetic induction and waves, momentum, electrostatics, gravity, kinematic, light, oscillation and wave motion. Physics of fluids, sun, materials, sound, thermal, atom are also included. This lecture includes: Oscillation, Simple, Harmonic, Motion, Mass, String, Equilibrium, Pendulum, Angular, Momentum, Moment, Inertia, Irregular, Object

Typology: Lecture notes

2011/2012

Uploaded on 08/12/2012

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bg1
PHYSICS –PHY101 VU
© Copyright Virtual University of Pakistan
44
M
g
P
C
θ
θ
Summary of Lecture 16 – OSCILLATIONS: II
1. In this chapter we shall continue with the concepts developed in the previous chapter
that relate to simple harmonic motion and the simple harmonic oscillator (SHO). It is
really very amazing that the SHO occurs again and again in physics, and in so many
different branches.
2. As an example illustrating the above, consider a mass suspended
a string. From the diagram, you can see that sin . For small
values of we know that sin . Using (length of arc),
Fmg
xL
θ
θθθθ
=−
≈= we
have . So now we have a restoring
force that is proportional to the distance away from the equilibrium
point. Hence we have a SHO with / . What if we had not
xmg
Fmg mg x
LL
gL
θ
ω
⎛⎞
=− =− =−
⎜⎟
⎝⎠
=
made the small approximation? We would still have an oscillator
(i.e. the motion would be self repeating) but the solutions of the
differential equation would be too complicated to discuss
θ
here.
3. If you take a common object (like a piece of cardboard) and pivot
it at some point, it will oscillate when disturbed. But this is not the
simple pendulum discussed above because all the mass is not
concentrated at one point. So now let us use the ideas of torque and
angular momentum discussed earlier for many particle systems.
You can see that sin . For small , Mgd
τθθ
=−
2
2
22
22
sin and so
. But we also know that where I is the moment of
inertia and is the angular acceleration, . Hence, we have
, or, . From
Mgd I
d
dt
ddMgd
IMgd
dt dt I
θθ
τθ τα
θ
αα
θθ
θθ
≈
=− =
=
⎛⎞
=− =−
⎜⎟
⎝⎠ this we immediately
see that the oscillation frequency is . Of course, we have
used the small angle approximation over here again. Since all variables
except are known, we can u
Mgd
I
I
ω
=
se this formula to tell us what is about
any point. Note that we can choose to put the pivot at any point on the
body. However, if you put the pivot exactly at the centre of mass then
i
I
t will not oscillate. Why? Because there is no restoring force and the
torque vanishes at the cm position, as we saw earlier.
m
θ
L
x
L
θ
=
T
θ
sinmg
θ
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Mg

P

θ C

θ

Summary of Lecture 16 – OSCILLATIONS: II

  1. In this chapter we shall continue with the concepts developed in the previous chapter that relate to simple harmonic motion and the simple harmonic oscillator (SHO). It is really very amazing that the SHO occurs again and again in physics, and in so many different branches.
  2. As an example illustrating the above, consider a mass suspended a string. From the diagram, you can see that sin. For small values of we know that sin. Using (length of arc),

F mg x L

θ θ θ θ θ

≈ = we have. So now we have a restoring force that is proportional to the distance away from the equilibrium point. Hence we have a SHO with /. What if we had not

F mg mg x^ mg x L L

g L

θ

ω

= − = − = −⎛^ ⎞

made the small approximation? We would still have an oscillator (i.e. the motion would be self repeating) but the solutions of the differential equation would be too complicated to discuss

θ

here.

  1. If you take a common object (like a piece of cardboard) and pivot it at some point, it will oscillate when disturbed. But this is not the simple pendulum discussed above because all the mass is not concentrated at one point. So now let us use the ideas of torque and angular momentum discussed earlier for many particle systems. You can see that τ= − Mgd sin θ. For small θ,

2 2 2 2 2 2

sin and so

. But we also know that where I is the moment of inertia and is the angular acceleration,. Hence, we have

, or,. From

Mgd I d dt I d^ Mgd d^ Mgd dt dt I

θ θ τ θ τ α α α θ θ θ θ θ

= − = −⎛^ ⎞

⎜⎝ ⎟⎠ this we immediately

see that the oscillation frequency is. Of course, we have used the small angle approximation over here again. Since all variables except are known, we can u

Mgd I

I

ω =

se this formula to tell us what is about any point. Note that we can choose to put the pivot at any point on the body. However, if you put the pivot exactly at the centre of mass then i

I

t will not oscillate. Why? Because there is no restoring force and the torque vanishes at the cm position, as we saw earlier.

m

θ

L

x = L θ

T

θ mg sin θ

  1. Suppose you were to put the pivot at point P which is at a distance from the centre of mass of the irregular object above. What should be so that you get the same formula as for a simple

L

L

pendulum?

Answer: 2 2

P is then called the centre of gyration - when suspended from this point it appears as if all the mass is concentrated at the cm position.

  1. Sum of two sim

T L^ I^ L I

g Mgd Md

1 1 2 2 (^ ) 1 2 1 2 1

ple harmonic motions of the same period along the same line: sin and sin Let us look at the sum of and , s

x A t x A t x x x x x A

= + = (^) ( )

( ) ( )

2 1 2 2 1 2 2 1 2 2

in sin sin sin cos sin cos sin cos cos sin Let cos cos and sin sin. Using some simple trigonometry, you can put

t A t A t A t A t t A A t A A A R A R x

( )

( ) ( )

(^2 2 ) 1 2 1 2 1 2 2 2 2 1 2 1 2 1 2 1 2 1 2

in the form, sin. It is easy to find R and : cos and tan sin. cos Note that if 0 then and tan 0

  1. So we get sin.

x R t R A A A A A A A R A A A A A A A A x A A t

( ) ( )

2 2 2 1 2 1 2 1 2 1 2 1 2

This is an example of If then and tan 0

  1. Now we get sin. This is.
  2. Composition of two simple ha

constructive interference. R A A A A A A A A x A A t destructive interference

( ) 2 2

rmonic motions of the same period but now at right angles to each other: Suppose sin and sin. These are two independent motions. We can write sin and cos 1 /.

Fr

x A t y B t t x t x A A

2 2

(^2 ) 2 2

om this, sin cos sin cos cos sin 1 /. Now square and rearrange terms to find: 2 cos sin This is the equation for an ell

y (^) t t x x A B A

x y xy A B AB

ipse (see questions at the end of this section).