Torque and Simple Harmonic Motion

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8.01

Week 13D2

Today’s Reading Assignment Young and Freedman: 14.1-14.6

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W013D3 Reading Assignment Young and Freedman: 14.1-14.6

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Simple Pendulum

Table Problem: Simple Pendulum by the Torque Method

(a) Find the equation of motion for θ(t) using the torque method.

(b) Find the equation of motion if θ is always <<1.

Table Problem: Simple Pendulum by the Energy Method

1. Find an expression for the mechanical energy when the pendulum is in motion in terms of θ(t) and its derivatives, m, l, and g as needed.

2. Find an equation of motion for θ(t) using the energy method.

Simple Pendulum: Small Angle Approximation

Equation of motion

Angle of oscillation is small

Simple harmonic oscillator

Analogy to spring equation

Angular frequency of oscillation

Period

sin

d 2dt2

g

l

d 2x

dt2

k

mx

ω0 ≅ g / l

T

0=

2πω0

≅2π l / g

−lmgsin =ml2 d2

dt2

Simple Pendulum: Approximation to Exact Period

Equation of motion:

Approximation to exact period:

Taylor Series approximation:

−lmgsin =ml2 d2

dt2

T ; T0(sin0 / 0 )

−3/8 =T0 + ΔT

ΔT ; T0

116

02

7

Concept Question: SHO and the Pendulum

Suppose the point-like object of a simple pendulum is pulled out at by an angle 0 << 1 rad. Is the angular speed of the point-like object equal to the angular frequency of the pendulum?

1.Yes.

2.No.

3.Only at bottom of the swing.

4.Not sure.8

Demonstration Pendulum: Amplitude Effect on

Period

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Table Problem: Torsional Oscillator

A disk with moment of inertia about the center of mass rotates in a horizontal plane. It is suspended by a thin, massless rod. If the disk is rotated away from its equilibrium position by an angle , the rod exerts a restoring torque given by

At t = 0 the disk is released from rest at an angular displacement of . Find the subsequent time dependence of the angular displacement .

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τ cm =−γ

0

(t)

Icm

Worked Example: Physical Pendulum

A general physical pendulum consists of a body of mass m pivoted about a point S. The center of mass is a distance dcm from the pivot point. What is the period of the pendulum.

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Concept Question: Physical Pendulum

A physical pendulum consists of a uniform rod of length l and mass m pivoted at one end. A disk of mass m1 and radius a is fixed to the other end. Suppose the disk is now mounted to the rod by a frictionless bearing so that is perfectly free to spin. Does the period of the pendulum

1. increase?2. stay the same?3. decrease? 12

Physical Pendulum

Rotational dynamical equation

Small angle approximation

Equation of motion

Angular frequency

Period

rτSS =IS

rα

sin

d 2dt2

lcm

mg

IS

ω0

lcm

mg

IS

T 2πω

0

2πI

S

lcm

mg

Demo: Identical Pendulums, Different Periods

Single pivot: body rotates about center of mass.

Double pivot: no rotation about center of mass.14

Small Oscillations

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Small Oscillations

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Potential energy function for object of mass m

Motion is limited to the region

Potential energy has a minimum at

Small displacement from minimum, approximate potential energy by

Angular frequency of small oscillation

U (x)

x1< x< x2

x =x0

U (x) ; U (x0) +

12!

(x−x0 )2 d2U

dx2(x0 )

U (x) ; U (x0 ) +12

keff (x−x0 )2

ω0 = keff / m=

d2Udx2

(x0 ) / m

Concept Question: Energy Diagram 1

A particle with total mechanical energy E has position x > 0 at t = 0

1) escapes to infinity in the – x-direction

2) approximates simple harmonic motion

3) oscillates around a

4) oscillates around b

5) periodically revisits a and b

6) not enough information 17

Concept Question: Energy Diagram 2

A particle with total mechanical energy E has position x > 0 at t = 0

1) escapes to infinity

2) approximates simple harmonic motion

3) oscillates around a

4) oscillates around b

5) periodically revisits a and b

6) not enough information18

Concept Question: Energy Diagram 3

A particle with total mechanical energy E has position x > 0 at t = 0

1) escapes to infinity

2) approximates simple harmonic motion

3) oscillates around a

4) oscillates around b

5) periodically revisits a and b

6) not enough information

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Concept Question: Energy Diagram 4

A particle with total mechanical energy E has position x > 0 at t = 0

1) escapes to infinity

2) approximates simple harmonic motion

3) oscillates around a

4) oscillates around b

5) periodically revisits a and b

6) not enough information20

Concept Question: Energy Diagram 5

A particle with total mechanical energy E has position x > 0 at t = 0

1) escapes to infinity

2) approximates simple harmonic motion

3) oscillates around a

4) oscillates around b

5) periodically revisits a and b

6) not enough information

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Table Problem: Small Oscillations

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A particle of effective mass m is acted on by a potential energy given by

where and are positive constants

a)Sketch as a function of .

b)Find the points where the force on the particle is zero. Classify them as stable or unstable. Calculate the value of at these equilibrium points.

c)If the particle is given a small displacement from an equilibrium point, find the angular frequency of small oscillation.

U (x) =U0 −2xx0

⎛

⎝⎜⎞

⎠⎟

2

+xx0

⎛

⎝⎜⎞

⎠⎟

4⎛

⎝⎜⎜

⎞

⎠⎟⎟

U0 x0

U (x) / U0 x / x

0

U (x) / U0

Appendix

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Simple Pendulum: Mechanical Energy

Velocity

Kinetic energy

Initial energy

Final energy

Conservation of energy

E0=K0 +U0 =mgl(1−cos0 )

v

tan=l

ddt

K

f=

12

mvtan2 =

12

m lddt

⎛

⎝⎜⎞

⎠⎟

2

E

f=K f +U f =

12

m lddt

⎛

⎝⎜⎞

⎠⎟

2

+ mgl(1−cos )

1

2m l

ddt

⎛

⎝⎜⎞

⎠⎟

2

+ mgl(1−cos) =mgl(1−cos0 )

Simple Pendulum: Angular Velocity Equation of Motion

Angular velocity

Integral form

Can we integrate this to get the period?

ddt

=2gl

(cos −cos0 )

d(cos −cos0 )

∫ =2gl

dt∫

Simple Pendulum: Integral Form

Change of variables

“Elliptic Integral”

Power series approximation

Solution

bsin a =sin 2( )

da

1−b2 sin2 a( )1 2∫ =

gl

dt∫

1−b2 sin2 a( )

−1 2=1+

12

b2 sin2 a+38

b4 sin4 a+⋅⋅⋅

2π +

12π sin2 0 2( ) +⋅⋅⋅=

glT

b =sin 0 2( )

Simple Pendulum: First Order Correction

Period

Approximation

First order correction

T =2π

lg

1+14

sin2 0 2( ) +⋅⋅⋅⎛

⎝⎜⎞

⎠⎟=

2πω

1+14

sin2 0 2( ) +⎛

⎝⎜⎞

⎠⎟

sin2 0 2( ) ≅0

2 4

T ≅2πlg

1+116

02⎛

⎝⎜⎞

⎠⎟=T0 1+

116

02⎛

⎝⎜⎞

⎠⎟

=T0 + ΔT1whereΔT1 ≅116

02T0