Question

consider a damped pendulum where the damping force is given by F =-kv where k is the damping coefficient v is the velocity of the pendulum and F is the damping force applied to the pendulum . dertemine an expression for the period of the pendulum for its motion about the equilibrum position as function of the damping factor k.

Answer #1

The motion of a spring that is subject to a frictional force or
a damping force (such as a shock absorber on a car) is often
modeled by the product of an exponential function and a sine or
cosine function. Suppose the equation of motion of a point on such
a spring is
s(t)=7e−1.2tsin(2πt)s(t)=7e−1.2tsin(2πt)
where ss is measured in centimeters and tt is measured in
seconds. Find the velocity of the point after tt seconds.
v(t)v(t) =
Graph both the...

consider a simple pendulum in simple harmonic motion () is
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m/sec2, the increased period is T = 4.59 sec
Determine:
a. suppose that the pendulum has friction where the amplitude
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A
mass of 1kg stretches a spring by 32cm. The damping constant is
c=0. Exterbal vibrations create a force of F(t)= 4 sin 3t Netwons,
setting the spring in motion from its equilibrium position with
zero velocity. What is the coefficient of sin 3t of the
steady-state solution? Use g=9.8 m/s^2. Express your answe is two
decimal places.

Consider the driven damped harmonic oscillator
m(d^2x/dt^2)+b(dx/dt)+kx = F(t)
with driving force F(t) = FoSin(wt).
Consider the overdamped case
(b/2m)^2 < k/m
a. Find the steady state solution.
b. Find the solution with initial conditions x(0)=0,
x'(0)=0.
c. Use a plotting program to plot your solution for
m=1, k=0.1, b=1, Fo=0.25, and w=0.5.

A particle of mass m moves under a force F = −cx^3 where c is a
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A force F= -F0 e(-x/λ ) (where
F0 and λ are positive constants) acts on a particle of
mass m that is initially at x = x0 and moving with
velocity v0 (> 0). Show that the velocity of the
particle is given by,
v(x) = ± ( v02 +
(2F0λ/m)(e-x/λ - 1) ) 1/2 ,
where the upper (lower) sign corresponds to the motion in the
positive (negative) x direction. Consider first the upper sign. For
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A particle with mass 2.61 kg oscillates horizontally at the end
of a horizontal spring. A student measures an amplitude of 0.923 m
and a duration of 129 s for 65 cycles of oscillation. Find the
frequency, ?, the speed at the equilibrium position, ?max, the
spring constant, ?, the potential energy at an endpoint, ?max, the
potential energy when the particle is located 68.5% of the
amplitude away from the equiliibrium position, ?, and the kinetic
energy, ?, and...

An object of mass, m = 0.200kg, is hung from a single spring
with spring constant,
k = 80.0N/m. (Ignore the mass of
the spring.) The object is subject to a resistive force, f,
given by f =-bv
where v = the velocity of the mass in m/s.
If the damped frequency, w’ = 0.75w0, the
undamped frequency, what is the value of “b”?
What is the “Q” of the system?
By what factor is the amplitude...

An object of mass one unit is hanging from a spring. Initially
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(Simple Harmonic motion)

Consider the production function Y = F (K, L) = Ka *
L1-a, where 0 < α < 1. The national saving rate is
s, the labor force grows at a rate n, and capital depreciates at
rate δ.
(a) Show that F has constant returns to scale.
(b) What is the per-worker production function, y = f(k)?
(c) Solve for the steady-state level of capital per worker (in
terms of the parameters of the model).
(d) Solve for the...

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