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...

See similar textbooks

Related questions

Q: b. y - y'cosx = y² (1 - senx)cosx

A: Introduction: The Bernoulli's equation is an extension of linear first order differential equation…

Q: Part 3. Find the anti derivative of the exponential and trigonometric function. Write your provided…

A: According to our guidelines we can answer only one question or three subparts of a question. Other…

Q: 3.45. Find all solutions to the following equations: (a) Log(z) = (e) cos(z) = 0 (b) Log(z) = i Злі…

Q: Solve 8(t-nt)cos () dt.

A: A delta function is defined as δ(t-a)=0 for t≠a∞ for t=a. The fundamental property of the…

Q: A system of pulleys will be needed to assemble the cable tower on site. A differential equation…

Q: a - Find h (Stet sinh t sint dt)

A: The objective is to find ∫te-3tsintsinhtdt It is known that, sint=et-e-t2isinht=eit-e-it2…

Q: Solve the HDE: z'y = ry+3xev*. Your Answer:

A: To solve the homogenous differential equation given.

Q: ㅈ Sa 0 tan²4t dt = 2

A: The given integral is:∫0π4 tan4t dt

Q: the differential equati on 21?y -dy-34=5€ dx Solve

Q: Solve the following Clairaut equation y=xy'+(y')^2

A: Firstly we will differentiate this equation to get simpler differential equation

Q: A bungee jumper dives from a tower at time t = 0. Her height h (measured in feet) at time t (in…

A: The given height function

Q: y = | 9t cos (t10) dt O A. 9/x cos (x°) B. 10/x cos (x) Oc. cos (x) 2 OD. 9 cos (x) - 9/ã sir D. 9…

A: on using the test we get option (D)

Q: Illustration 10.8 Form the differential equation having y = (sin ¹x)² + A cos¯¹ x + B, where A and B…

A: Since given that y=sin-1x2+Acos-1x+B…

Q: A simple pendulum consists of a light wire of length 1 at the end of which is suspended a ball of…

A: Given that equation of motion of simple pendulum when it displace from verticle line about an angle…

Q: d'a b. = -ka dt? Solve the differential equation and Express the x as a function of trigonometric…

Q: Take x = 2.5 as a first approximation to the root of the equation e* – 4x = 0 and apply Newton's…

Q: Find the second derivative of the following in its simplified form.

A: To find: The second derivative of y=4cos-1x2+24-x2. Formula used: Quotient rule of derivative:…

Q: Consider the function P = 0.2x2 - 12x- 13. Find the differentlal for this function. %3D

A: The first order derivative of the function will take us to the result.

Q: Find the limit of tan^-1(e^(x^2)+x^2+1/x) as x approaches negative infinity

Q: A particles velocity is 7 cos(2t) meters per second. at time t=0 its gone 2meters, where is it at…

A: The velocity of the function is given as:

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)

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Trending now

This is a popular solution!

Step by step

Solved in 5 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

Solve dx dt = = et cos(2t) + t with x(0) = 3.
The acceleration of an object (in m/s²) is given by the function a(t) = 9 sin(t). The initial velocity o object is v(0) = -6 m/s. Round your answers to four decimal places. a) Find an equation v(t) for the object velocity. v(t) = −9 cos (t) + 3 b) Find the object's displacement (in meters) from time 0 to time 3. 5.8584 X meters c) Find the total distance traveled by the object from time 0 to time 3. 16.7176 X meters
Use Newton's method to approximate a root of the equation cos( s(x² + 3) = x³ as follows. Let x₁ = 2 be the initial approximation. The second approximation is
do fast i will 10 upvotes.
Solve plz
Near the grounds of a building is a windmill h(t)=2 cos (30t)+6, where h(t) is the height of the blade above the ground and t is the time in seconds. If the rotation begins at the blade with the highest possible height, at what time(s) will the blade be exactly 7.8m above the ground within the second rotation?