Restoring and frictional forces of magnitudes kx and , respectively act simultaneously on an object of mass
attached to a spring. Under the influence of these forces, the mass oscillates with a frequency
and its amplitude reduces to half in
. Calculate the damping constant
, the force constant k and the damping factor b. Also write the differential equation for the system.
The motion of a damped harmonic oscillator is governed by the differential equation: [ m \frac{d^2 x}{dt^2} + \gamma \frac{dx}{dt} + k x = 0 ] where (m = 0.2 , \text{kg}), (\gamma) is the damping constant, and (k) is the spring constant. The damped frequency (f_d = 1.5 , \text{Hz}), ____ __________ _________ _______ ______ _____ _________ ____ ___ __________ _______ ___.
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एक प्रगामी तरंग को नीचे दिए गए समीकरण द्वारा निरूपित किया जाता है :$
इस तरंग की संचरण दिशा निर्धारित करें तथा इसका आयाम, तरंग संख्या, तरंगदैर्घ्य एवं आवृत्ति परिकलित करें।
द्रव्यमान के एक खंड को एक खुरदरे नत तल पर, आधार से
की प्रारंभिक चाल से ऊपर की ओर गतिमान किया जाता है। नत ताल क्षैतिज के साथ
का कोण बनाती है। खंड और नत तल के बीच गतिक घर्षण गुणांक 0.2 है। गणना करें :
i) विराम अवस्था में आने से पहले खंड द्वारा नत तल पर तय की गई अधिकतम दूरी।
ii) नत तल के आधार पर वापस आने पर खंड की चाल।
iii) इस सम्पूर्ण गति में क्षयित ऊर्जा। लें।
Three vectors ,
and
satisfy the condition
. If
and
determine the value of
.
Two collinear harmonic oscillations, each of frequency , have amplitudes, a1 and a2 and initial phases,
and
, respectively. Show that, when these oscillations are superposed, the amplitude of the resultant motion is equal to (a1 - a2). What will be the value of the resultant amplitude when
and
.
A block of mass is moved up from the base a rough inclined plane making an angle of
with the horizontal with initial speed
. The coefficient of kinetic friction between the block and the plane is 0.2. Calculate
i) the maximum distance the block travels up the incline before coming to rest.
ii) the speed of the block when it returns to the base of the inclined plane.
iii) the energy lost over the entire motion. Take .
An object undergoes SHM with frequency . The initial displacement is
and the initial velocity is
. Calculate the amplitude, maximum velocity and maximum acceleration of the object.