Transcript Q05. Using Newtons Laws - National Cheng Kung University

Q13. Oscillatory Motion
Q14. Wave Motion
1. Two identical undamped oscillators have the same amplitude
of oscillation only if they are started with the same :
1. displacement x0
2. velocity v0
3. phase
4.
2 x02  v02
5.
x02  2 v02
1
1 2 1 2 1
2
E  m v  k x  k A  m  2 A2
2
2
2
2
 same A requires same E.
1. displacement x0
Different v0 gives different A.
2. velocity v0
Different x0 gives different A.
3. phase
A is indifferent to phase.
4. 2 x02  v02
Same E gives same A.
5.
x02  2 v02
Sum is meaningless.
2. The amplitude of any oscillator can be doubled by: :
1.
doubling only the initial displacement
2.
doubling only the initial speed
3.
doubling the initial displacement and halving the initial speed
4.
doubling the initial speed and halving the initial displacement
5.
doubling both the initial displacement and the initial speed
1
1 2 1 2 1
2
E  m v  k x  k A  m  2 A2
2
2
2
2

1
1
1
2
2
2
m  2v0   k  2 x0   4 E  k  2 A
2
2
2
1.
doubling only the initial displacement
2.
doubling only the initial speed
3.
doubling the initial displacement and halving the initial speed
4.
doubling the initial speed and halving the initial displacement
5. doubling both the initial displacement and the initial speed
3. A certain spring elongates 9 mm when it is suspended
vertically and a block of mass M is hung on it. The natural
frequency of this mass-spring system is :
1.
is 0.088 rad/s
2. is 33 rad/s
3.
is 200 rad/s
4.
is 1140 rad/s
5. cannot be computed unless the value of M is given
M g  kx


k

M
g

x
9.8 m / s 2
 33 rad / s
0.009 m
4. A particle is in simple harmonic motion along the x axis.
The amplitude of the motion is xm.
When it is at x = x1, its
kinetic energy is K = 5J and its potential energy (measured
with U = 0 at x = 0) is U = 3J.
When it is at x = –1/2xm, the
kinetic and potential energies are :
1.
K = 2J and U = 6J
2.
K = 6J and U = 2J
3.
K = 2J and U = – 6J
4.
K = 6J and U = – 2J
5.
K = 5 and U = 3J
E
1 2
k xm  K  U  5 J  3 J  8 J
2
1
U k
2
2
 xm  1  1 2 
     k xm   2 J
 2  4 2

K  E  U  8J  2 J  6 J
5. Five hoops are each pivoted at a point on the rim and allowed
to swing as physical pendulums. The masses and radii are :
hoop 1: M = 150g and R = 50 cm
hoop 2: M = 200g and R = 40 cm
hoop 3: M = 250g and R = 30 cm
hoop 4: M = 300g and R = 20 cm
hoop 5: M = 350g and R = 10 cm
Order the hoops according to the periods of their motions, smallest to largest.
1. 1, 2, 3, 4, 5
2.
5, 4, 3, 2, 1
3.
1, 2, 3, 5, 4
4. 1, 2, 5, 4, 3
5.
5, 4, 1, 2, 3
5.
Five hoops are each pivoted at a point on the rim and allowed to swing as physical
pendulums.
The masses and radii are :
hoop 1: M = 150g and R = 50 cm
hoop 2: M = 200g and R = 40 cm
hoop 3: M = 250g and R = 30 cm
hoop 4: M = 300g and R = 20 cm
hoop 5: M = 350g and R = 10 cm
Order the hoops according to the periods of their motions, smallest to largest.
1.
1, 2, 3, 4, 5
2.
5, 4, 3, 2, 1
3.
1, 2, 3, 5, 4
4.
1, 2, 5, 4, 3
5.
5, 4, 1, 2, 3
For small  :
R
I    M g R 
I  I cm  MR2  MR2  MR2  2MR2


M gR

I
T R
g
2R
6. The displacement of a string carrying a traveling sinusoidal
y  x, t   ym sin  k x  t   
wave is given by
At time t  0 the point at x  0 has velocity v0 and displacement
y0 .
The phase constant  is given by tan   :
1.
v0 /  y0
2.
 y0 / v0
3.
 v0 / y0
4.
y0 /  v0
5.
 v0 y0
6.
The displacement of a string carrying a traveling sinusoidal wave is given by
y  x, t   ym sin  k x  t   
At time t  0 the point at x  0 has velocity v0 and displacement y0.
The phase constant  is given by tan   :
y  x, t   ym sin  k x  t   
y0   ym sin 
v  x, t    ym cos  k x  t   
v0   ym cos
tan  
 y0
v0
7. The diagram shows three identical strings that have been put
under tension by suspending masses of 5 kg each.
which is the wave speed the greatest ?
1.
1
2.
2
3.
3
4.
1 and 3 tie
5.
2 and 3 tie
For
v
T
T  Mg

 Larger T
1
T  Mg
2
Ans: 1 & 3 tied
 larger v
T  Mg
8. Suppose the maximum speed of a string carrying a sinusoidal
wave is vs.
When the displacement of a point on the string
is half its maximum, the speed of the point is :
1.
vs / 2
2.
2 vs
3.
vs / 4
4.
3 vs / 4
5.
3 vs / 2
8.
Suppose the maximum speed of a string carrying a sinusoidal wave is vs.
When the displacement of a point on the string is half its maximum, the
speed of the point is :
y  ym sin  k x   t   

y
1
ym
2
v   ym cos  k x   t   
vs   ym

1
 sin  k x   t   
2
cos  k x   t    
v   ym
2
3
1
1   
2
2
3
3

vs
2
2
9. The sinusoidal wave
y(x,t)  ym sin( k x –  t ) is incident
on the fixed end of a string at x  L.
The reflected wave is
given by :
1.
ym sin( k x +  t )
2.
–ym sin( k x +  t )
3.
ym sin( k x +  t – k L )
4.
ym sin( k x +  t – 2 k L )
5.
–ym sin( k x +  t + 2 k L )
9. The sinusoidal wave y(x,t)  ym sin( k x –  t ) is incident
on the fixed end of a string at x  L.
The reflected wave is
given by :
Let the time of incidence be t0
yin  L, t0   ym sin  k L   t0   0  yrefl  L, t0 

k L   t0
yrefl  x, t   ym sin  k  x  L     t  t0    
 ym sin  k x   t  k L   t0 
 ym sin  k x   t  2k L
10. Standing waves are produced by the interference of two
traveling sinusoidal waves, each of frequency 100 Hz.
distance from the 2nd node to the 5th node is 60 cm.
wavelength of each of the two original waves is :
1.
50 cm
2.
40 cm
3.
30 cm
4.
20 cm
5.
15 cm
The
The
10. Standing waves are produced by the interference of two traveling sinusoidal
waves, each of frequency 100 Hz.
5th node is 60 cm.
The distance from the 2nd node to the
The wavelength of each of the two original waves is :
In order to have a standing wave, these waves must travel in opposite directions.
y1  A sin  k x   t 
y2  A sin  k x   t   
 


y1  y2  A sin   t   cos  k x  
2
2


Distance from the 2nd node to the 5th node is 60 cm :
2

 60 cm    5  2  

  40 cm
11. A 40-cm long string, with one end clamped and the other free
to move transversely, is vibrating in its fundamental standing
wave mode.
If the wave speed is 320 cm/s, the frequency is :
1.
32 Hz
2.
16 Hz
3.
8 Hz
4.
4 Hz
5.
2 Hz
11. A 40-cm long string, with one end clamped and the other free
to move transversely, is vibrating in its fundamental standing
wave mode.
If the wave speed is 320 cm/s, the frequency is :
One end clamped and the other free to move transversely.
Fundamental standing wave mode
f

  4 L.
320 cm / s 

 
 2 Hz
  4  40 cm
v