Physics-Biophysics I. 2015/2016

Lecture…:

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Topics of the lecture:

1. Definition of oscillation

1.1 Definition of harmonic oscillation

2. Measuring angles in radian

3. Angular velocity

4. Hooke’s law

5. Kinetics of periodic motion

6. Dynamics of periodic motion

7. Superposition-interference

8. Damped oscillations

9. Resonance-driven oscillation

Oscillations (1) Definition of oscillation

A physical quantity changes periodically in space and/or time. Movement, state or change that has

a periodic component. If a particle in periodic motion moves back and forth over the same path, we

call the motion oscillatory or vibratory.

1.1 Harmonic oscillations

Oscillation with a single frequency. It can be described with sine (or cosine) function. Constant

amplitude and period time. (Any motion that repeats itself in equal intervals of time is called

periodic or harmonic motion.)

(2) Measuring angles in radian

Physics-Biophysics I. 2015/2016

Problem (1)

𝜃 =𝑠

𝑟=

200𝑚

75𝑚= 2.67 𝑟𝑎𝑑

Problem (2)

How many radians in 180°?

length of arc s = half of circumference of circle = (1/2) (2πr) = πr

𝜃 =𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ

𝑟𝑎𝑑𝑖𝑢𝑠=

𝑠

𝑟=

𝜋𝑟

𝑟 = π radians 180°=3.14159 rad

(3) Angular velocity

Measuring angles in degrees: 1°=1/360 of a full circle. Angles are

also measured in a unit called the radian. Arc length and the radius

are the same length. The angle for which this occurs is called one

radian. Number of radians in an angle=number of times radius fits

into the arc. 𝜃 = 𝑠

𝑟

s

r

r

Θ

200 m

75 m

ϴ

A small object P (=particle) is moving at a constant speed

of v (meters/second) in an anticlockwise direction around

the circle. At any moment P is moving along a tangent to

the circle (= tangential speed/linear speed).

The direction of the velocity of P is continually changing

P is accelerating. The magnitude of P’s velocity (its

speed) is constant.

P moves from X to the position shown in t seconds; s in

meters

v=s/t.

t (time) increases angle ϴ increases

Angular velocity: the rate of change of angle with respect to time.

𝑎𝑛𝑔𝑙𝑒 𝑡𝑟𝑎𝑐𝑒𝑑 𝑜𝑢𝑡

𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛

If an object is moving in a circular path of radius r with constant angular velocity ω (rad/s) and

constant linear speed v (m/s) then: v=rω

𝜔 =𝜃

𝑡 [rad/s]

𝑣 = 𝑟𝜔

Physics-Biophysics I. 2015/2016

Problem (3)

If the angle traced out by P in 4 seconds is 10 radians, find its angular velocity.

ω=𝜃

𝑡=

10

4= 2.5

𝑟𝑎𝑑

𝑠𝑒𝑐

Problem (4)

A small object moving in a circle with a steady speed does 3000 revolutions of the circle per minute.

Find its angular velocity (in 𝑟𝑎𝑑

𝑠𝑒𝑐).

1 revolution= 2π radians 3000 rpm =3000x2π radians per minute

ω = 3000𝑥2𝜋

60

𝑟𝑎𝑑

𝑠𝑒𝑐= 314.16

𝑟𝑎𝑑

𝑠𝑒𝑐

(4) Hooke’s law

(5) Kinetics of periodic motion

Oscillation: vertical projection of circular motion

When an object is bent, stretched or compressed by a

displacement s, the restoring force F is directly proportional to

the displacement (provided the elastic limit is not exceeded).

F~-s 𝐹 = −𝑘𝑠 (restoring force ~ displacement)

𝑘: elastic constant

Negative sign: displacement and the restoring force are always in

the opposite direction.

Any system that obeys Hooke’s law will execute simple harmonic

motion (SHM)

A body is moving with SHM if:

1. its acceleration is directly proportional to its

distance from a fixed point on its path and

2. its acceleration is always directed towards

that point.

𝑎 = −𝜔2𝑠(= −𝜔2r)

𝑎: acceleration of a particle

𝑠 (or can be labelled with 𝑟): displacement of the

particle from the fixed point O

𝜔2: is a constant

negative sign: a and s are always in opposite directions

Physics-Biophysics I. 2015/2016

Problem (5)

A particle moving with SHM makes 10 full oscillations in 4 seconds. Find the period and the

frequency of the motion.

Period = time for one oscillation=𝑡𝑖𝑚𝑒 𝑓𝑜𝑟 10 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛𝑠

10=

4

10= 0.4𝑠𝑒𝑐

Frequency: 𝑓 =1

𝑇=

1

0.4= 2.5 𝐻𝑧.

Problem (6)

Physics-Biophysics I. 2015/2016

The time between two consecutive heart beat is 0.83 second. How much is the heart rate (HR;

heartbeats in 1 minute)?

T=0.83sec

𝑓 =1

𝑇=

1

0.83𝑠𝑒𝑐= 1.2048 𝐻𝑧 (

1

𝑠)

HR=60x1.20481/s=72.291

𝑚𝑖𝑛

Problem (7)

The period of a particle executing SHM is 2 seconds. What is the acceleration when it is 15 cm from

the equilibrium position?

𝑇 =2𝜋

𝜔𝜔 =

2𝜋

𝑇=

2𝜋

2𝑠= 3.142

1

𝑠𝑒𝑐

𝑎 = 𝜔2𝑟 = 3.1422(0.15𝑚) = 1.481𝑚

𝑠2

Problem (8)

A particle executing SHM has a maximum acceleration of 3 𝑚

𝑠2. Find the peroid of the motion if the

total distance travelled in one oscillation is 0.5 m.

Total distance travelled in one oscillation=0.5mamplitude =0.5

4= 0.125𝑚

𝑎 = 𝜔2𝑟𝑎𝑚𝑎𝑥 = 𝜔2𝑟𝑚𝑎𝑥 = 𝜔2(𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒)3=𝜔20.125𝑚𝜔 = √3

𝑚

𝑠2

0.125𝑚= 4.9

𝑇 =2𝜋

𝜔=

2𝜋

4.9= 1.28 𝑠𝑒𝑐

(6) Dynamics of periodic motion

Physics-Biophysics I. 2015/2016

Problem (9)

A spring with a spring constant of 120 N/m is loaded with a mass and lowered to reach an

equilibrium position. Here the extension of the spring is 15 cm. What will be the oscillation

frequency if the system is pushed away from equilibrium?

In equilibrium: ∑ 𝐹 = 0 é𝑠 𝑚𝑔 = 𝐷𝑥 𝑚 =𝐷𝑥

𝑔=

120𝑁

𝑚 0.15𝑚

9.81𝑚

𝑠2

= 1.83 𝑘𝑔

𝑇 = 2𝜋√𝑚

𝐷= 2𝜋√

1.83𝑘𝑔

120𝑁

𝑚

=0.77s

𝑓 =1

𝑇=

1

0.77𝑠= 1.28

1

𝑠(𝐻𝑧)

Physics-Biophysics I. 2015/2016

As a body moves with SHM its energy changes continually from potential to kinetic and back to

potential.

(7) Superposition-interference

All kind of periodic and non-periodic oscillation could be described as the sum (or integral) of

individual sinusoidal oscillations (with different frequency, amplitude or phase)

Combinations of Harmonic Motions-Interference:

The phenomenon in which two (or more) waves superpose each other to form a resultant wave.

Two linear simple harmonic motions combined (same and perpendicular directions).

The resulting motion is the sum of two independent oscillations.

(8) Damped oscillations

Some bodies’ periodic motion is stopped because other forces (e.g. friction) dissipate the energy of

motion. Damped motion is an oscillatory motion which decays with time.

In damped harmonic motion the mechanical energy approaches zero as time increases, being

transformed into internal thermal energy associated with the damping mechanism.

Physics-Biophysics I. 2015/2016

(9) Resonance-driven oscillation

Free (self-) oscillation: oscillating system without external force, oscillate with its self-frequency

Driven oscillation: the oscillation is driven by external, periodic force; oscillation frequency =

external excitation frequency; amplitude, phase could be different

Resonance: driving frequency is in the nearby range of the self-frequency

friction force only drag force