医学物理学-chapter 4a_14
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4.2.1 The damped harmonic motion
• Real vibrating systems have damped force or friction.
How do they swing so high?
(4.8)
Squaring both sides of the above equations, the amplitude of the SHM can be found
x A cos
2 0 2 2
v A sin
2 0 2 2 2
x
2 0
v
2 0 2
A (cos sin ) A
(1). the angular frequency, we have
k 50.0 50.0 2 m 2.0010
(rad/s)
(2). The initial phase of the vibration can be found using the initial displacement and initial velocity. At t = 0, we know x0 = A cos = 3.00 × 10-2 m v0 = - A sin = -1.32 m/s The can be obtained by solving above equations. On the other hand, it can be calculate directly by
x
Find its amplitude and period.
t
•The frequency, denoted by f, is the number of complete vibrations per second, it is the reciprocal (倒数) of the period
Solution: the known quantities are: m = 2.00 × 10-2 kg k = 50.0 N/m x0 = 3.00 × 10-2 m v0 = –1.32 m/s Now using the formulae we have learned, the problem can be solved easily.
Energy is conserved in the system!
E Ep Ek
x Fig. 4.3 Total energy of the vibrating system
Example 4.2 simple pendulum
Self study
F ma
2
m gsin
m g cos
(4.2)
This is the differential equation of the SHM. Its solution can be expressed as
x A cos(t )
(4.3)
The motion described by a cosine or sine function of time is also called Simple Harmonic Motion. Differentiating the equation (4.3) with respect to t, the velocity and acceleration of the SHM can be obtained
l
f
h
d m l 2 m g sin dt
d m l 2 m g dt
2
Discuss: how to swing higher 如何把秋千 荡的更高？
需要考虑的问题：
1. 2. 3. 4. 5…
§4.2 Damped (阻尼的) harmonic motion, forced vibration (受迫振动) and resonance
The angular frequency or angular velocity is defined as
1 f T
(4.6)
2 2f T
∴
f 2
(4.7)
3. Phase and initial phase （初位相）
t is called the phase of SHM,
v
2 0 2
4.0010 m
2
(4). The period can be found through the relation between the angular frequency and the period;
2 T 0.126 s 50.0
2
(5)
1 f T
2
x A cos(t )
is indeed the solution of diff-equation
4.1.2 The characteristic quantities of SHM
In the equation of SHM, A, and are constants and but they are so important.
d x dv d a 2 A (sin ) dt dt dt d d A (sin ) d dt A (cos ) A cos(t )
2
2
x
2
(4.5)
So we get: This shows
d x 2 x 0 2 dt
Chapter Four
Oscillation, wave motion and sound
New words and expressions
•
simple harmonic motion (简谐运动); spring (弹簧)，elastic (弹性的)， Phase (位相) reciprocal (倒数) ， unavoidable (不可避免的) amplitude (振幅); damp (vt.阻尼,n.湿气) ; deduce (vt. 推论), deduction vibration (振动)； oscillation (振动) Resonance (共振）
• The period, denoted by T, is the time taken for a complete vibration which is independent of the position chosen for the starting point of the complete vibration.
= t +
dx d v [ A cos(t )] d(sin x)/dx = cos x dt dt d d d A (cos ) A (cos ) dt d dt A( sin ) A sin(t )
(4.4)
d(cos x)/dx = -sin x
4.1.3 The referencearound a fixed point O
M
MP
O
A
t+
P
M0 P0
x = A cos ( t + )
The equation of SHM.
Fig. 4.2 the circle of reference of SHM.
4.1.4 The energy of SHM
Energy = kinetic + potential.
1 2 1 dx 1 Ek mv m m 2 A2 sin 2 (t ) 2 2 dt 2
2
∴
1 2 1 2 2 2 E p kx m A cos (t ) 2 2 1 1 2 2 2 (4.11) Ek E p m A kA 2 2
F=-kx k is the elastic constant. (4.1)
• SHM If a body moves in a straight line under the simple harmonic force, the motion of the body is called simple harmonic motion. 2. Equation of SHM If a body’s mass is m and is exerted by a simple harmonic force, its equation of motion can be obtained by using Newton’s second law of motion.
d x F ma m 2 dt
2
On the other hand, considering eq. (4.1), we have
d x m 2 kx dt
2
2
Or
d x k x 2 dt m
2
Define 2 = k/m and we have
d x 2 x 0 2 dt
x A cos(t )
1. A is called Amplitude (振幅). It is the maximum displacement of a vibrating body from equilibrium position.
2. Period (周期) and frequency (频率)
2 2 2
2
∴
mv A x x k
2 0 2 0
v
2 0 2
2 0
(4.9)
On the other hand, the initial phase can also be worked out from equation (4.8), so we have
∴
v0 arctan x 0
is the phase at t = 0, called initial phase. At t = 0, equations (4,3) and (4.4) becomes respectively
x0 A cos v0 A sin
This is called initial condition.
v0 tan x0
(4.10)
Example 1. A particle with mass m = 2.00× 10-2 kg is in SHM at the end of a spring with spring constant k = 50.0 N/m. The initial displacement and velocity of the particle is 3.00 × 10-2 m and –1.32 m/s respectively. Calculate: (1) the angular frequency; (2) the initial phase; (3) the amplitude of the vibration; (4) the period; (5) the frequency.
• • • • • •
§4.1 Simple harmonic motion (SHM)
4.1.1 Equation of SHM
1. Definition of SHM
o x
• Simple harmonic force (简谐力): The force F on a body is proportional to its displacement x from the origin and always directed towards the origin.
v0 1.32 arctan 41.3 2 x arctan 3.0010 50.0 0
(3). The amplitude can be calculate by the formula
A x
2 0
• Real vibrating systems have damped force or friction.
How do they swing so high?
(4.8)
Squaring both sides of the above equations, the amplitude of the SHM can be found
x A cos
2 0 2 2
v A sin
2 0 2 2 2
x
2 0
v
2 0 2
A (cos sin ) A
(1). the angular frequency, we have
k 50.0 50.0 2 m 2.0010
(rad/s)
(2). The initial phase of the vibration can be found using the initial displacement and initial velocity. At t = 0, we know x0 = A cos = 3.00 × 10-2 m v0 = - A sin = -1.32 m/s The can be obtained by solving above equations. On the other hand, it can be calculate directly by
x
Find its amplitude and period.
t
•The frequency, denoted by f, is the number of complete vibrations per second, it is the reciprocal (倒数) of the period
Solution: the known quantities are: m = 2.00 × 10-2 kg k = 50.0 N/m x0 = 3.00 × 10-2 m v0 = –1.32 m/s Now using the formulae we have learned, the problem can be solved easily.
Energy is conserved in the system!
E Ep Ek
x Fig. 4.3 Total energy of the vibrating system
Example 4.2 simple pendulum
Self study
F ma
2
m gsin
m g cos
(4.2)
This is the differential equation of the SHM. Its solution can be expressed as
x A cos(t )
(4.3)
The motion described by a cosine or sine function of time is also called Simple Harmonic Motion. Differentiating the equation (4.3) with respect to t, the velocity and acceleration of the SHM can be obtained
l
f
h
d m l 2 m g sin dt
d m l 2 m g dt
2
Discuss: how to swing higher 如何把秋千 荡的更高？
需要考虑的问题：
1. 2. 3. 4. 5…
§4.2 Damped (阻尼的) harmonic motion, forced vibration (受迫振动) and resonance
The angular frequency or angular velocity is defined as
1 f T
(4.6)
2 2f T
∴
f 2
(4.7)
3. Phase and initial phase （初位相）
t is called the phase of SHM,
v
2 0 2
4.0010 m
2
(4). The period can be found through the relation between the angular frequency and the period;
2 T 0.126 s 50.0
2
(5)
1 f T
2
x A cos(t )
is indeed the solution of diff-equation
4.1.2 The characteristic quantities of SHM
In the equation of SHM, A, and are constants and but they are so important.
d x dv d a 2 A (sin ) dt dt dt d d A (sin ) d dt A (cos ) A cos(t )
2
2
x
2
(4.5)
So we get: This shows
d x 2 x 0 2 dt
Chapter Four
Oscillation, wave motion and sound
New words and expressions
•
simple harmonic motion (简谐运动); spring (弹簧)，elastic (弹性的)， Phase (位相) reciprocal (倒数) ， unavoidable (不可避免的) amplitude (振幅); damp (vt.阻尼,n.湿气) ; deduce (vt. 推论), deduction vibration (振动)； oscillation (振动) Resonance (共振）
• The period, denoted by T, is the time taken for a complete vibration which is independent of the position chosen for the starting point of the complete vibration.
= t +
dx d v [ A cos(t )] d(sin x)/dx = cos x dt dt d d d A (cos ) A (cos ) dt d dt A( sin ) A sin(t )
(4.4)
d(cos x)/dx = -sin x
4.1.3 The referencearound a fixed point O
M
MP
O
A
t+
P
M0 P0
x = A cos ( t + )
The equation of SHM.
Fig. 4.2 the circle of reference of SHM.
4.1.4 The energy of SHM
Energy = kinetic + potential.
1 2 1 dx 1 Ek mv m m 2 A2 sin 2 (t ) 2 2 dt 2
2
∴
1 2 1 2 2 2 E p kx m A cos (t ) 2 2 1 1 2 2 2 (4.11) Ek E p m A kA 2 2
F=-kx k is the elastic constant. (4.1)
• SHM If a body moves in a straight line under the simple harmonic force, the motion of the body is called simple harmonic motion. 2. Equation of SHM If a body’s mass is m and is exerted by a simple harmonic force, its equation of motion can be obtained by using Newton’s second law of motion.
d x F ma m 2 dt
2
On the other hand, considering eq. (4.1), we have
d x m 2 kx dt
2
2
Or
d x k x 2 dt m
2
Define 2 = k/m and we have
d x 2 x 0 2 dt
x A cos(t )
1. A is called Amplitude (振幅). It is the maximum displacement of a vibrating body from equilibrium position.
2. Period (周期) and frequency (频率)
2 2 2
2
∴
mv A x x k
2 0 2 0
v
2 0 2
2 0
(4.9)
On the other hand, the initial phase can also be worked out from equation (4.8), so we have
∴
v0 arctan x 0
is the phase at t = 0, called initial phase. At t = 0, equations (4,3) and (4.4) becomes respectively
x0 A cos v0 A sin
This is called initial condition.
v0 tan x0
(4.10)
Example 1. A particle with mass m = 2.00× 10-2 kg is in SHM at the end of a spring with spring constant k = 50.0 N/m. The initial displacement and velocity of the particle is 3.00 × 10-2 m and –1.32 m/s respectively. Calculate: (1) the angular frequency; (2) the initial phase; (3) the amplitude of the vibration; (4) the period; (5) the frequency.
• • • • • •
§4.1 Simple harmonic motion (SHM)
4.1.1 Equation of SHM
1. Definition of SHM
o x
• Simple harmonic force (简谐力): The force F on a body is proportional to its displacement x from the origin and always directed towards the origin.
v0 1.32 arctan 41.3 2 x arctan 3.0010 50.0 0
(3). The amplitude can be calculate by the formula
A x
2 0