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thx! 3. 10 Marks The dynamics of a plane pendulum subject to some external force is...
3. [10 Marks The dynamics of a plane pendulum subject to some external force is described...
3. [10 Marks The dynamics of a plane pendulum subject to some external force is described by si u(t), vhere is the angular displacement, l is the length of its link, and is the standard gravity (a) 2 Marks] What is the equilibrium of the pendulum with u(t) = F being a constant? (b) [2 Marks] Write down the linearized equation for small angular displacement (e) 6 Marks Derive the harmonic response r(t) X sin(wt) subject to u(t) Fsin(wt) from...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
[132 2 2 3 4 17 marks] Question 4 A plane wave is travelling in a...
[132 2 2 3 4 17 marks] Question 4 A plane wave is travelling in a vacuum in the +z-direction with wavenumber k and angular frequency . It is linearly polarised in the x-direction, and has electric field given by E(t, z) Eo Cos(kz - wt)f This wave is normally incident on a perfectly electrically conducting, semi-infinite slab in the region z > 0 and the resulting field in vacuum (z < 0) is a superposition of the incident and...
2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with...
2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with we obtain the linearized shallow-water equations Ot on O a) For the special non-rotating case (f -0 ) with constant depth (Ho - const.) show that the speed of gravity waves is c-VgHo Hint: set v-0 and derive a wave equation for the sea level η b) Given a harmonic wave η(x,t)=Asin(k-or) with amplitude A (again for f-0 and Ho= const.), derive the equation...
please solve both. thank you! A mass of 1.25 kg stretches a spring 0.06 m. The...
please solve both. thank
you!
A mass of 1.25 kg stretches a spring 0.06 m. The mass is in a medium that exerts a viscous resistance of 56 N when the mass has a velocity of 2 . The viscous resistance is proportional to the speed of the object. Suppose the object is displaced an additional 0.03 m and released. Find an function to express the object's displacement from the spring's equilibrium position, in m after t seconds. Let positive...
[10] (2) Consider a rocket traveling in a straight-line subject to an external force Fext acting...
[10] (2) Consider a rocket traveling in a straight-line subject to an external force Fext acting along the same line. The equation of motion is my =-mvex + Fext where (-mvex) is the thrust. [4] (2.a) Specialize to the case of a rocket taking off vertically (from rest; and positive y is upward) in a gravitational field g, so the only external force is the weight mg. Assume that the rocket ejects mass at a constant rate, m=-k (where k...
1. The angular displacement e(t) of an undamped pendulum of length l swinging in a vertical...
1. The angular displacement e(t) of an undamped pendulum of length l swinging in a vertical plane under the influence of gravity can be modelled with the nonlinear second order ODE 0"(t) + "$(0(e) – Pef) = = 0, (1) where w2 = { (a) Re-write equation (1) as a system of first-order ODEs by defining (1 = 0(t) and 02 = 't). (b) Find the nullclines and equilibrium points for the system of first-order ODEs. (c) The first order...
B1. As shown in Figure Bl, a horizontally-directed force P is applied to a homogenous disk...
B1. As shown in Figure Bl, a horizontally-directed force P is applied to a homogenous disk weighted W with radius ofr. P y G Figure B1 (a) Draw the free body diagram of the disk. (4 marks) (b) Show the moment of inertia Ig about point G of the disk as follow: (where mis mass of the disk): mur? (4 marks) 2 (c) Considering force equilibrium in x- and y-directions and also the moment equilibrium about point G, derive the...
Part 1: (Theory) Simple Pendulum 1. Consider a mass m hanging from a string of length...
Part 1: (Theory) Simple Pendulum 1. Consider a mass m hanging from a string of length L that makes an angle with the vertical (shown below). Assume the string is massless and that the hanging object is a point mass. Use Newton's Second Law directly to show that the equation of motion for this simple pendulum can be written: (LO) = -mgsin(o), (1) dia where is the angular displacement of the pendulum from its vertical equilibrium position (and is a...
The Problem The single pendulunm Consider the single pendulum shown below. There is a bob at...
The Problem The single pendulunm Consider the single pendulum shown below. There is a bob at the end of the pendulum, of mass For small oscillations in the case of a frictionless pivot and a rod with negligible mass, the motion of the pendulum over time can be described by the second-order linear ODE where θ is the angle between the rod and the vertical, g is gravity, t is the length of the rod and θ dag /dt2 Q1...
3. [10 Marks The dynamics of a plane pendulum subject to some external force is described by si u(t), vhere is the angular displacement, l is the length of its link, and is the standard gravity (a) 2 Marks] What is the equilibrium of the pendulum with u(t) = F being a constant? (b) [2 Marks] Write down the linearized equation for small angular displacement (e) 6 Marks Derive the harmonic response r(t) X sin(wt) subject to u(t) Fsin(wt) from...
2. The angular displacement e(t) of a damped forced pendulum of length 1 swinging in a vertical plane under the influence of gravity can be modelled with the second order non-homogeneous ODE 0"(t) + 270'(t) +w20(t) = f(t), (2) where wa = g/l. The second term in the equation represents the damping force (e.g. air resistance) for the given constant 7 > 0. The model can be used to approximate the motion of a magnetic pendulum bob being driven by...
[132 2 2 3 4 17 marks] Question 4 A plane wave is travelling in a vacuum in the +z-direction with wavenumber k and angular frequency . It is linearly polarised in the x-direction, and has electric field given by E(t, z) Eo Cos(kz - wt)f This wave is normally incident on a perfectly electrically conducting, semi-infinite slab in the region z > 0 and the resulting field in vacuum (z < 0) is a superposition of the incident and...
2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with we obtain the linearized shallow-water equations Ot on O a) For the special non-rotating case (f -0 ) with constant depth (Ho - const.) show that the speed of gravity waves is c-VgHo Hint: set v-0 and derive a wave equation for the sea level η b) Given a harmonic wave η(x,t)=Asin(k-or) with amplitude A (again for f-0 and Ho= const.), derive the equation...
please solve both. thank
you!
A mass of 1.25 kg stretches a spring 0.06 m. The mass is in a medium that exerts a viscous resistance of 56 N when the mass has a velocity of 2 . The viscous resistance is proportional to the speed of the object. Suppose the object is displaced an additional 0.03 m and released. Find an function to express the object's displacement from the spring's equilibrium position, in m after t seconds. Let positive...
[10] (2) Consider a rocket traveling in a straight-line subject to an external force Fext acting along the same line. The equation of motion is my =-mvex + Fext where (-mvex) is the thrust. [4] (2.a) Specialize to the case of a rocket taking off vertically (from rest; and positive y is upward) in a gravitational field g, so the only external force is the weight mg. Assume that the rocket ejects mass at a constant rate, m=-k (where k...
1. The angular displacement e(t) of an undamped pendulum of length l swinging in a vertical plane under the influence of gravity can be modelled with the nonlinear second order ODE 0"(t) + "$(0(e) – Pef) = = 0, (1) where w2 = { (a) Re-write equation (1) as a system of first-order ODEs by defining (1 = 0(t) and 02 = 't). (b) Find the nullclines and equilibrium points for the system of first-order ODEs. (c) The first order...
B1. As shown in Figure Bl, a horizontally-directed force P is applied to a homogenous disk weighted W with radius ofr. P y G Figure B1 (a) Draw the free body diagram of the disk. (4 marks) (b) Show the moment of inertia Ig about point G of the disk as follow: (where mis mass of the disk): mur? (4 marks) 2 (c) Considering force equilibrium in x- and y-directions and also the moment equilibrium about point G, derive the...
Part 1: (Theory) Simple Pendulum 1. Consider a mass m hanging from a string of length L that makes an angle with the vertical (shown below). Assume the string is massless and that the hanging object is a point mass. Use Newton's Second Law directly to show that the equation of motion for this simple pendulum can be written: (LO) = -mgsin(o), (1) dia where is the angular displacement of the pendulum from its vertical equilibrium position (and is a...
The Problem The single pendulunm Consider the single pendulum shown below. There is a bob at the end of the pendulum, of mass For small oscillations in the case of a frictionless pivot and a rod with negligible mass, the motion of the pendulum over time can be described by the second-order linear ODE where θ is the angle between the rod and the vertical, g is gravity, t is the length of the rod and θ dag /dt2 Q1...
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