4.84. A particle of mass m can perform undamped harmonic oscil-
lations due to an electric force with coefficient
k.
When the particle
was in equilibrium, a permanent force
F
was applied to it for T sec-
onds. Find the oscillation amplitude that the particle acquired
after the action of the force ceased. Draw the approximate plot
x (t)
of oscillations. Investigate possible cases.
4.85. A ball of mass nz, when suspended by a spring stretches the
latter by Al. Due to external vertical force varying according to a
harmonic law with amplitude F
0
the ball performs forced oscilla-
tions. The logarithmic damping decrement is equal to X. Neglecting
the mass of the spring, find the angular frequency of the external
force at which the displacement amplitude of the ball is maximum.
What is the magnitude of that amplitude?
4.86. The forced harmonic oscillations have equal displacement
amplitudes at frequencies co
l
= 400 s -
1
and co
t
= 600 s-
1
.
Find the resonance frequency at which the displacement amplitude
is maximum.
4.87. The velocity amplitude of a particle is equal to half the maxi-
mum value at the frequencies co
l
and co
t
of external harmonic force.
Find:
(a)
the frequency corresponding to the velocity resonance;
(b)
the damping coefficient 13 and the damped oscillation frequency
co of the particle.
4.88. A certain resonance curve describes a mechanical oscillat-
ing system with logarithmic damping decrement ? = 1.60.
For
this curve find the ratio of the maximum displacement amplitude
to the displacement amplitude at a very low frequency.
4.89. Due to the external vertical force
F
x
= F
0
cos cot
a body
suspended by a spring performs forced steady-state oscillations accord-
ing to the law
x = a cos (cot โ
(T). Find the work performed by
the force
F
during one oscillation period.
4.90. A ball of mass m
.
= 50
g
is suspended by a weightless spring
with stiffness x = 20.0 N/m. Due to external vertical harmonic
force with frequency co = 25.0 s
-1
the ball performs steady-state
oscillations with amplitude
a =
1.3 cm. In this case the displace-
ment of the ball lags in phase behind the external force by cp =
27
3
3t.
Find:
(a)
the quality factor of the given oscillator;
(b)
the work performed by the external force during one oscillation
period.
4.91. A ball of mass
m
suspended by a weightless spring can per-
form vertical oscillations with damping coefficient
13.
The natural
oscillation frequency is equal to co
0
. Due to the external vertical
force varying as
F = F , cos cot
the ball performs steady-state har-
monic oscillations. Find:
(a) the mean power
(P),
developed by the force
F,
averaged over
one oscillation period;
12*
