2. A pendulum is a mechanical system in which a mass mis attached to a massless incxtensible string of lengthwhich is in turn connected to a frictionless pivot, asshown at right.The swing angle (t) of the pendulum satisfies thesecond-order nonlinear differential equationd'e22²+ sin 0-0.2sineThe transformationv --allows us to transform thisequation into the separable first-order differential equa-tion(1)U +sin 0-0.sinopivot1v(0)'mass 77(a) If the mass m is initally at rest at an angle of — T/6 degrees, solve the differ-ential equation (1) to find v = v(8).(b) Given that(2)deintegrate both sides of this separable differential equation to find the generalsolution tt(0) in the form of an integral (with respect to 0).(c) Hence, write down a definite integral for the period 7' of the pendulum (that is,the time for the mass m to complete one full swing and return to its startingposition. (Note: v <0 on this section because is decreasing.)(Hint: The period is four times the amount of time it takes for the mass to reachthe "vertically down" position from the starting position. You should integrate(2) from the starting point (- 0, 0 = π/6) to the "vertically down" position(t-T/4, 0-0). Note: v <0 on this section because is decreasing.)(d) If = 1.6 metres, use Simpson's rule with four strips to estimate T. (Note: Youshould assume, for simplicity, that g = 10.)

vdvdθ+glsinθ=0 above equation can be re written as vdv= -glsinθdθintegrating on both side∫vdv=…

2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length which is in turn connected to a frictionless pivot, as shown at right. The swing angle (t) of the pendulum satisfies the second-order nonlinear differential equation de + sin 0-0. The transformation v = allows us to transform this equation into the separable first-order differential equa- tion (1) 1- sin 0-0. pivot mass m

2. A pendulum is a mechanical system in which a mass m is attached to a massless inextensible string of length which is in turn connected to a frictionless pivot, as shown at right. The swing angle (t) of the pendulum satisfies the second-order nonlinear differential equation de + sin 0-0. The transformation v = allows us to transform this equation into the separable first-order differential equa- tion (1) 1- sin 0-0. pivot mass m

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

See similar textbooks

Related questions

Q: suspension or You are requested to design an aut shock absorber system. In order to simplify the…

A: Given: M1=2500 kgM2=320 kgK1=80000 N/mK2=500000 N/mB=350 Ns/m To find: Transfer function Plot

Q: Consider the dynamic system shown below. The input f(t) is a force applied to mass m1. The…

Q: Consider the following rotational mechanical a. Apply the "by inspection" method in Laplace domain…

A: Given: To find: Equation of motion Solve for q1(s) Transfer function

Q: Prob. # 4] a) Derive the differential equation of motion for the system, as shown below; b) Compute…

Q: For the undamped linear system with two degree-of-freedom system, the initial conditions are shown…

A: The system is a two degree-of-freedom system with two masses and two springs. The first mass, m, is…

Q: The 4-kg cylinder is attached to a viscous damper and to the spring of stiffness k-800 N/m. Write…

A: Given data:- Mass of the block =4 kg spring of stiffness = 800 N/m. c=124 N.s/m c=80 N.s/m To…

Q: 1a. Consider the problem 2-15 from Ozisik (1993). Starting with the heat diffusion equation, Eq.…

Q: Q4) Consider a mechanical system that satisfies the following differential equation:…

Q: 3.5 Consider the nonlinear equation U where u is a constant. a. Is this equation in conservative…

A: The following equation is not conservative since it includes both the variable's derivative term and…

Q: Consider a mass of 10 kg. attached to a spring with (K=40 N/m) and to a dashpot with damping…

A: First we have drawn schematic of given system. Then we setup the model by writing the differential…

Q: the pump and motor assembly stops work when you find the sum of the product of vibration intensity…

A: First, we'll denote the vibration intensity as V, acceleration as a, speed as v, and distance moved…

Q: Determine the rotational speed of link 3 of the mechanism given in figure P4.8 for the position…

Q: Recall the two coupled masses undergoing linear oscillations in Example 8, p.22-25. Solve a simple…

A: For solution refer below images.

Q: Translational mechanical systems with the image values I need to get free body diagram Get the…

A: To Find : The input and output model get the transfer function, Free Body Diagram : Here input…

Q: 2. Duffing's equation is a model for a dynamic system that includes a damping term and a nonlinear…

A: Hamilton's Principle states that the dynamics of a physical system can be derived from minimizing…

Q: ki= 15 N/m; k2 = 10 N/m; m, k3 = 12 N/m; %3D C1 Ci =1 N.s/m; m2 K3 c2= 2 N.s/m; C3 = 2 N.s/m; m3 mi…

A: Given : To find : Equation of motion

Concept explainers

Conduction and Convection

When energy travels from one object to another it is said to have undergone heat transfer. Heat transfer is nothing but the transfer of thermal energy or internal energy from one thing to another, like e.g., when a vessel with water, kept on top of a burner, heats up because of the flame underneath it.

Question

2. A pendulum is a mechanical system in which a mass m
is attached to a massless incxtensible string of length
which is in turn connected to a frictionless pivot, as
shown at right.
The swing angle (t) of the pendulum satisfies the
second-order nonlinear differential equation
d'e
22²
+ sin 0-0.
2sine
The transformation
v --
allows us to transform this
equation into the separable first-order differential equa-
tion
(1)
U +
sin 0-0.
sino
pivot
1
v(0)'
mass 77
(a) If the mass m is initally at rest at an angle of — T/6 degrees, solve the differ-
ential equation (1) to find v = v(8).
(b) Given that
(2)
de
integrate both sides of this separable differential equation to find the general
solution tt(0) in the form of an integral (with respect to 0).
(c) Hence, write down a definite integral for the period 7' of the pendulum (that is,
the time for the mass m to complete one full swing and return to its starting
position. (Note: v <0 on this section because is decreasing.)
(Hint: The period is four times the amount of time it takes for the mass to reach
the "vertically down" position from the starting position. You should integrate
(2) from the starting point (- 0, 0 = π/6) to the "vertically down" position
(t-T/4, 0-0). Note: v <0 on this section because is decreasing.)
(d) If = 1.6 metres, use Simpson's rule with four strips to estimate T. (Note: You
should assume, for simplicity, that g = 10.)

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 (a)

Step 2 (b)

Step 3 (c)

Step 4 (d)

Trending now

This is a popular solution!

Step by step

Solved in 4 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, mechanical-engineering and related others by exploring similar questions and additional content below.

Similar questions

The following figure shows a machine of mass m mounted on a vibration isolator. The machine (starting at rest at t=0) is subjected to a sinusoidal excitation force p(t) = P sin wt. Solve for the transfer function X(s)/P(s). Then state in words the process you would then follow to determine the force transmissibility, TR. p(t) = P sin wi m b₂
Need help with this engineering problem.
The physical system shown below consists of a mass, viscous damping, and two parallel springs. Do the following: a) Neatly draw a proper free body diagram b) Find the differential equation of motion that describes the system. c) Find the transfer function X(s) / F(s). x(t) ki k2 m f(t) b
identify the differential equations defining the motion of the masses from the figure, via Newton’s 2nd law, hence construct the system matrix, A, where the state vector is:x=[x;x ̇;y;y ̇ ]
The relative displacement u(t) of a single-storey shear building subjected to an earthquake ground motion is represented by the following second-order linear ordinary differential equation: d?u dt2 du + c + ku = a, (t) m dt where m, c, and k are the mass (kg), damping constant (Ns/m), and stiffness of the structure (N/m), respectively. Meanwhile, a,(t) is the function of earthquake ground acceleration. Suppose the building with a mass of 2000 kg and supported by columns of combined stiffness of 32 x 103 N/m is subjected to earthquake with ground acceleration given by the following function: ag(t) = 36000 cos 2t Find the equation of the displacement, u(t), given the damping is not installed to the building. Then, find how much the building is displaced from t =30 seconds to t =60 seconds of earthquake.
An object attached to a spring undergoes simple harmonic motion modeled by the differential equation d²x = 0 where x (t) is the displacement of the mass (relative to equilibrium) at time t, m is the mass of the object, and k is the spring constant. A mass of 3 kilograms stretches the spring 0.2 meters. dt² Use this information to find the spring constant. (Use g = 9.8 meters/second²) m k = + kx The previous mass is detached from the spring and a mass of 17 kilograms is attached. This mass is displaced 0.45 meters below equilibrium and then launched with an initial velocity of 2 meters/second. Write the equation of motion in the form x(t) = c₁ cos(wt) + c₂ sin(wt). Do not leave unknown constants in your equation. x(t) = Rewrite the equation of motion in the form ä(t) = A sin(wt + ), where 0 ≤ ¢ < 2π. Do not leave unknown constants in your equation. x(t) =
Differential equations: mechanical vibrations Show one example. Summarize the method. How would you solve these applications without using a linear system?
My question and answer is in the image. Can you please check my work? A 2 kg mass is attached to a spring with spring constant 50 N/m. The mass is driven by an external force equal tof(t) = 2 sin(5t). The mass is initially released from rest from a point 1 m below the equilibrium position. (Use theconvention that displacements measured below the equilibrium position are positive.)(a) Write the initial-value problem which describes the position of the mass. 2y"+50y=2cos(5t) (b) Find the solution to your initial-value problem from part (a). (1+(1/2)tcos(t))cos(5t)-(1/2)tcos(t) (c) Circle the letter of the graph below that could correspond to the solution. B (d) What is the name for the phenomena this system displays? Resonance
A weighing scale can be modelled as a dynamic system, as shown in Figure. The transfer function of the system, using Laplace transforms, is: F(t) m Oms² + cs+k=F(s) mx+cx+kx = F(t) X (s) F(s) 1 mx + cx + k X (s) F(s) ms 2 1 +cs + k 2
For the following water liquid level system, 9mi(t) is the mass flow rate input deviation to the system and the restrictions can be considered linear resistances about the equilibrium: 9mi h2 Ro R1 a) Using basic fluids principles, develop the Dynamic Equations for the deviation of the two liquid level heights from their equilibrium height as a function of the input mass flow rate, Imi-
Problem Derive the differential equation governing the motion of the one degree-of-freedom system by applying the appropriate form(s) of Newton's laws to the appropriate free-body diagrams. Use the generalized coordinate shown in Figure P2.427. Linearize nonlinear differential equations by assuming small displacements. 2k - Rigid massless link Identical slender bars of mass m, length L. FIGURE P 2.47
(k - 1)/k P2 Tds relationship: Tds = dh -vdp. Start from the Tds function, please derive that . Please T1 P1 list clearly the assumptions/conditions applied for the derivation