Item 1Learning Goal:To understand the application of the general harmonic equation to the kinematics of aspring oscillator.One end of a spring with spring constant k is attached to the wall. The other end isattached to a block of mass m. The block rests on a frictionless horizontal surface. Theequilibrium position of the left side of the block is defined to be x = 0. The length of therelaxed spring is L. (Figure 1)The block is slowly pulled from its equilibrium position to some position init> 0 along thex axis. At time t = 0, the block is released with zero initial velocity.The goal is to determine the position of the block (t) as a function of time in terms of wand initIt is known that a general solution for the displacement from equilibrium of a harmonicoscillator isx(t) = C cos (wt) + S sin (wt),where C, S, and w are constants. (Figure 2)Your task, therefore, is to determine the values of C and S in terms of w and initFigure1 of 3LXinitwinx = 0Part AUsing the general equation for x (t) given in the problem introduction, express the initial position of the block init in terms of C, S, and w (Greek letter omega).► View Available Hint(s)IVE ΑΣΦ?Tinit =SubmitPart B Complete previous part(s)Part C Complete previous part(s)Now, imagine that we have exactly the same physical situation but that the x axis is translated, so that the position of the wall is now defined to be x = 0. (Figure 3)The initial position of the block is the same as before, but in the new coordinate system, the block's starting position is given by new (t = 0) = L + init-Part DFind the equation for the block's position new (t) in the new coordinate system.Express your answer in terms of L, Zinit, w (Greek letter omega), and t.► View Available Hint(s)LIVE ΑΣΦ?Tnew (t) =Submit

In the given question, Spring constant = k Length of the relaxed spring = L At t = 0, the block is…

Answered: Using the general equation for z (t)…

Using the general equation for z (t) given in the problem introduction, express the initial position of the block it in terms of C, S, and w (Greek letter omega).

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter12: Coupled Oscillations

Section: Chapter Questions

Problem 12.22P

See similar textbooks

Related questions

Q: Assuming the +x-axis is horizontal to the right for the vectors in the following figure, find the…

Q: Find a vector (v) with an initial point <-5,-5,-4> and a terminal point of <4,-10,4>.…

A: The objective of the question is to find the vector v with given initial and terminal points and…

Q: Write the given vectors in terms of its components and the corresponding unit vector and then verify…

Q: For the cartesian coordinates shown in the figure, the product of the unit vectors a xa =a_.a y O az…

A: In the Cartesian coordinate system, Cross-products are defined as, 1) ax x ax= 0, ay x ay =0,…

Q: E The components of a vector A>satisfy Ax 6 0 and Ay 6 0.Is the direction angle of A>between…

Q: Given two vectors A = - 2.00 + 3.00ĵ + 4.00 k and B = 3.00 î+ 1.00 – 3.00 k, (a) find the magnitude…

A: We have to calculate the magnitude of each vector. Given Data, A→=-2.0 i^+3.0 j^+4.0 k^B→=3.0 i^+1.0…

Q: Problem 3: Find unit vector of a force. Find the unit vector corresponding to a 100 N force at 60°…

Q: If the dot product of two vector A and B is negative, what statements reguarding A and B are true…

A: In this question we are given that the dot product of two vectors A and B is negative and we have to…

Q: Eij and k are the usual unit vectors in the x, y and z directions, find aixi c.

A: As we know that i, j and k are unit vectors along the x, y and z directions. These unit vectors are…

Q: If a vector W has components Wx--3.0 and Wy-4.0, what is the magnitude of the vector? Make sure your…

A: Given vector W has x-component wx = -3.0 And y-component wy = 4.0

Q: 9. Given three vectors A = -2i - 3j + 4k, B = i + 4j – 3k and C = 2j + k, Find the scalar product of…

A: 10. Scalar product,…

Q: : 3 2.00j, (a) find the magnitude of each vector; (b) use unit vectors to write an expression for…

A: magnitude of a vector is the square root of the some of the squares of components.

Q: A vector has equal, positive x and y components and zero z componet. What is the angle between this…

A: The expression for the required angle is.

Q: Consider the following 2 vectors: A =3.42 î–6.44ĵ B =2.33 î+4.26} What is the magnitude of Á × B.

Q: 50.0⁰ 37.0⁰ A (12.0 m) 40.0⁰ -X

A: A = 12 m angle = 37 degrees

Q: The magnitude of vector A is 22.5 units and points in the direction 335° counterclockwise from the…

A: Given that the magnitude of the given A vector is 22.5units, and the vector is pointed…

Q: Assuming the +x-axis is horizontal to the right for the vectors in the following figure, find the…

Q: Imagine a force T, with a magnitude other than 0, which makes an angle of 60° with the vertex Find…

Q: Given two vectors, Land D where L=275ⅈ-64j and D=161.7ⅈ+44j what is the resultant, R, of these two…

A: Resultant of a vector is given by, R=L2+D2 Substitute the given values in above equation,…

Q: Try and solve for the resultant of the two vectors. Also, determine the direction cosines of the…

A: Let n1 and n2 denote the unit vectors along the forces F and P. Therefore, from the given…

Q: If a vector W has components W=-3.0 and Wy=4.0, what is the magnitude of the vector? Make sure your…

A: the vector W has the horizontal and vertical components which can be given as Wx=3.0 Wy=4.0…

Q: F, = 50 N F,= 30 N %3D 15° F= 80 N

A: Forces on the support in cartesian form. Given, F1=50 NF2=80 NF3=30 Nθ2=15o

Q: Problem no 1: Find sum, difference and scalar and vetor product of vectors u = [-4,6], w = [0,4].…

A: Given vectors:u = [-4, 6] w = [0, 4]

Q: Vector C has magnitude 6.50 and is at an angle of 55.0° measured counterclockwise from the +x-axis…

Q: The vector -5.2 A→ has a magnitude of 37 mm and points in the positive x direction. Find the x…

Q: Determine the magnitude of the vertical reaction (N) at A if P = 355 N and Q = 2100 N. Let x = 2.45…

Q: Problem 2: Given vectors: A = 3 i+j- k, B = - i+ 2 j+ 5 k, C = 2 j - 3 k. Find: 1. C. (A + B) 2. Cx…

Q: Given 3 vectors, A = x-2y-z B = 3x+2y-z C = -6x-y+2z Find A×B, BxC, A.B, and B.C?

A: Vectors A = x-2y-z B = 3x+2y-z C = -6x-y+2z

Q: Express the following vectors in competent form using unit vectors F₁ = 10 N, -35° F₂ = 5 N, 45° F₁…

Q: Vectors C and D have magnitudes of 3 units and 4 units, respectively. What is the angle between the…

A: ""Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Consider two vectors, A = 16.5 m at 48.0°, and B = (-12.7Î + 2.30j) m. %3D Determine the x and y…

A: The given vectors, A→=16.5 m at 48°B→=-12.7 i^+2.30 j^ m The x and y components of A vector,…

Concept explainers

Newton’s Third Law of Motion

The horse pulls the cart and the cart pulls the horse in response as per Newton third law. So the big question is, “Don’t the two forces cancel each other? How does the cart accelerate?”

Question

Item 1
Learning Goal:
To understand the application of the general harmonic equation to the kinematics of a
spring oscillator.
One end of a spring with spring constant k is attached to the wall. The other end is
attached to a block of mass m. The block rests on a frictionless horizontal surface. The
equilibrium position of the left side of the block is defined to be x = 0. The length of the
relaxed spring is L. (Figure 1)
The block is slowly pulled from its equilibrium position to some position init> 0 along the
x axis. At time t = 0, the block is released with zero initial velocity.
The goal is to determine the position of the block (t) as a function of time in terms of w
and init
It is known that a general solution for the displacement from equilibrium of a harmonic
oscillator is
x(t) = C cos (wt) + S sin (wt),
where C, S, and w are constants. (Figure 2)
Your task, therefore, is to determine the values of C and S in terms of w and init
Figure
1 of 3
L
Xinit
win
x = 0
Part A
Using the general equation for x (t) given in the problem introduction, express the initial position of the block init in terms of C, S, and w (Greek letter omega).
► View Available Hint(s)
IVE ΑΣΦ
?
Tinit =
Submit
Part B Complete previous part(s)
Part C Complete previous part(s)
Now, imagine that we have exactly the same physical situation but that the x axis is translated, so that the position of the wall is now defined to be x = 0. (Figure 3)
The initial position of the block is the same as before, but in the new coordinate system, the block's starting position is given by new (t = 0) = L + init-
Part D
Find the equation for the block's position new (t) in the new coordinate system.
Express your answer in terms of L, Zinit, w (Greek letter omega), and t.
► View Available Hint(s)
LIVE ΑΣΦ
?
Tnew (t) =
Submit

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

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

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

Similar questions

A particle of mass m moving in one dimension has potential energy U(x) = U0[2(x/a)2 (x/a)4], where U0 and a are positive constants. (a) Find the force F(x), which acts on the particle. (b) Sketch U(x). Find the positions of stable and unstable equilibrium. (c) What is the angular frequency of oscillations about the point of stable equilibrium? (d) What is the minimum speed the particle must have at the origin to escape to infinity? (e) At t = 0 the particle is at the origin and its velocity is positive and equal in magnitude to the escape speed of part (d). Find x(t) and sketch the result.
Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit τ → 0) for overdamping. Sketch the response functions.
Problems 5 & 6 refer to the mass-spring oscillator depicted in the figure on the right. The block has a mass of 350 g, and is attached to a spring with a spring constant of k pushed 20 cm to left from C, its equilibrium position, before being released and allowed to move horizon- tally on a frictionless surface. . Ax = 20 cm 45 N. The mass is initially m A B DE 5. What is the frequency of oscillation for the block? A. 0.13 Hz В. 0.35 Hz С. 1.8 Hz D. 7.8 Hz 5. 6. What is the speed of the block as it passes point B? A. 0.50 m/s В. 1.0 m/s C. 1.5 m/s D. 2.0 m/s 6.
MärkxsfH) Problem 1: A mass of 20 grams stretches a spring by 10/169 meters. (a) Find the spring constant k, the angular frequency w, as well as the period T and frequency f of free undamped motion for this spring-mass system. (b) Find the general solution x(t) of the DE for the free spring-mass system. (c) Suppose that an exterior force of F(t) = 2.5 sin(12t) in Newtons acts on the spring-mass system. Find the equation of motion (the solution x(t)) of the system if the mass initially is at rest in its equilibrium position.
Needs Complete typed solution with 100 % accuracy.
A mass of 3kg stretches a spring 40cm. Suppose the mass is displaced an additional 12em in the positive (downward) direction and then ruleased. Suppose that the damping constant is 2N/m and assume g9.8 m/s is the gravitational accelaration. Set up a differential equation that describes this system. Let a to denote the displacement, in metern, of the mass from its equilbrium position, and give your answer in terms of z, , a". 3"+ 2 + 245r=0 b) Enter the intial conditions (0) 12 m. (0) o m/s is this wyotem under damped, over damped, or critically damped? under damped
An undamped harmonic oscillator of mass m and spring constant k oscillates with an amplitude A. In terms of A, at what postion x is the speed of the oscillator, v, is half of its maximum speed v, max Hint: Set-up the energy equation at any postion. What is the maximum energy of the oscillator?
For a simple harmonic oscillator with x = A sin ot write down an expression for the velocity. Please use II* II for products (e.g. B*A), "/" for ratios (e.g. B/A) and the usual "+" and "-" signs as appropriate (without the quotes). For trigonometric functions use the usual sin and cos, while for Greek letters such as w, use omega. Please use the "Display response" button to check you entered the answer you expect.
Problem 2 (Estimating the Damping Constant). Recall that we can experimentally mea- sure a spring constant using Hooke's law-we measure the force F required to stretch the spring by a certain y from its natural length, and then we solve the equation F = ky for the spring constant k. Presumably we would have to determine the damping coefficient of a dashpot empirically as well, but how would we do so? As a warm-up, suppose we have a underdamped, unforced spring-mass system with mass 0.8 kg, spring constant 18 N/m, and damping coefficient 5 kg/s. We pull the mass 0.3 m from its rest position and let it go while imparting an initial velocity of 0.7 m/s. %3D (a) Set up and solve the initial value problem for this spring-mass system. (b) Write your answer from part (a) in phase-amplitude form, i.e. as y(t) = Aeºt sin(ßt – 4) and graph the result. Compare with a graph of your answer from (a) to check that you have the correct amplitude and phase shift. (c) Find the values of t at which y(t)…
Q. 2: Derive a relation for the time period of a simple pendulum using dimensional analysis. The various possible factors on which the time period T may depend upon a) Length of pendulum (I) b) Mass of the bob (m) c) Angle 0. d) Accerlation due to gravity. Find the relation for the time period. · RIGID SUPPORT . BOB в (a) (b) Figure 2
It is known that a general solution for the position of a harmonic oscillator is x(t) = C cos (wt) + S sin (wt), where C, S, and ware constants. (Figure 2) Your task, therefore, is to determine the values of C, S, and w in terms of k, m,and init and then use the connection between x (t) and a (t) to find the acceleration. Figure L min x=0 Xinit 1 of 2 > Part A Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block a (t) as a function of time. Express your answer in terms of k, m, and the coordinate of the block x (t). ► View Available Hint(s) a (t) = Submit 17 ΑΣΦ Part B Complete previous part(s) Review | Constants ?
(a) Find the amplitude, period, and horizontal shift. (Assume the absolute value of the horizontal shift is less than the period.) (b)Write an equation that represents the curve in the form