Suppose a pendulum of length L meters makes an angle of 0 radians with the vertical, as in the figure. It can be shown that as afunction of time, 0 satisfies the differential equationLd20-sin 0 = 0,Ldt2where g = 9.8 m/s? is the acceleration due to gravity. For 0 near zero we can use the linear approximation sin(0) - 0 to get alinear differential equationd²0+ 26dt?0 = 0.LUse the linear differential equation to answer the following questions.(a) Determine the equation of motion for a pendulum of length 1.5 meters having initial angle 0.4 radians and initial angulardovelocitydt0.1 radians per second.0(t)radians(b) What is the period of the pendulum? That is, what is the time for one swing back and forth?Period =seconds ту" + су" + kу — F(t), y(0) %3D 0, у'(0) — 0modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that2 kilograms, cm8 kilograms per second, k80 Newtons per meter, and the applied force in Newtons isif 0 T/2.a. Solve the initial value problem, using that the displacement y(t) and velocity y'(t) remain continuous when the applied force is discontinuous.For 0 < t < п/2, у(t)help (formulas)For t > T/2, y(t)help (formulas)b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter zero. If not, enter a function that approximates y(t) for very large positivevalues of t.For very large positive values of t, y(t) 2help (formulas)

Since you have asked multiple questions here and has not specified any questions in specific, I…

Suppose a pendulum of length L meters makes an angle of 0 radians with the vertical, as in the figure. It can be shown that as a function of time, 0 satisfies the differential equation sin 0 = 0, + dt2 where g = 9.8 m/s² is the acceleration due to gravity. For 0 near zero we can use the linear approximation sin(0) × 0 to get a linear differential equation d²0 0 = 0. dt? Use the linear differential equation to answer the following questions. (a) Determine the equation of motion for a pendulum of length 1.5 meters having initial angle 0.4 radians and initial angular do = 0.1 radians per second. dt velocity 0(t) radians (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? Period = seconds

Suppose a pendulum of length L meters makes an angle of 0 radians with the vertical, as in the figure. It can be shown that as a function of time, 0 satisfies the differential equation sin 0 = 0, + dt2 where g = 9.8 m/s² is the acceleration due to gravity. For 0 near zero we can use the linear approximation sin(0) × 0 to get a linear differential equation d²0 0 = 0. dt? Use the linear differential equation to answer the following questions. (a) Determine the equation of motion for a pendulum of length 1.5 meters having initial angle 0.4 radians and initial angular do = 0.1 radians per second. dt velocity 0(t) radians (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? Period = seconds

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Suppose a pendulum of length L meters makes an angle of 0 radians with the vertical, as in the figure. It can be shown that as a
function of time, 0 satisfies the differential equation
L
d20
-sin 0 = 0,
L
dt2
where g = 9.8 m/s? is the acceleration due to gravity. For 0 near zero we can use the linear approximation sin(0) - 0 to get a
linear differential equation
d²0
+ 26
dt?
0 = 0.
L
Use the linear differential equation to answer the following questions.
(a) Determine the equation of motion for a pendulum of length 1.5 meters having initial angle 0.4 radians and initial angular
do
velocity
dt
0.1 radians per second.
0(t)
radians
(b) What is the period of the pendulum? That is, what is the time for one swing back and forth?
Period =
seconds

ту" + су" + kу — F(t), y(0) %3D 0, у'(0) — 0
modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that
2 kilograms, c
m
8 kilograms per second, k
80 Newtons per meter, and the applied force in Newtons is
if 0 <t < T/2,
{
40
F(t)
if t > T/2.
a. Solve the initial value problem, using that the displacement y(t) and velocity y'(t) remain continuous when the applied force is discontinuous.
For 0 < t < п/2, у(t)
help (formulas)
For t > T/2, y(t)
help (formulas)
b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter zero. If not, enter a function that approximates y(t) for very large positive
values of t.
For very large positive values of t, y(t) 2
help (formulas)

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