QUESTION 4 The vertical vibration at position r and at time t of a stretched homogeneous...
QUESTION 2 The vertical vibration at position i and at time t of a stretched homogeneous and infinitely long string is determined by the function u = u(x,y). Suppose that when the string is straight, it has a linear density of 2 and the tension at any given point of the string is 8. a. Derive the partial differential equation satisfied by u. b. If the initial position is 1 – r and the initial velocity is e 1 write...
5. Imagine a string that is fixed at both ends (e.g. a guitar string). When plucked, the string forms a standing wave. The vertical displacement u of the string varies with position r and time t. Suppose u(x,t) = 2 sin(nx) sin(mt/2), for 0 x 1 and t 0. Convince yourself of the following: If we freeze the string in time, it will form a sine curve. Alternatively, if we instead focus on a single position, we will see the...
Partial Differential
Equation
- Wave equation : Vibrating spring
Question 2 A plucked string, Figure 2 shows the initial position function f (x) for a stretched string (of length L) that is set in motion by moving t at midpoint x =-aside the distance-bL and releasing it from rest timet- 0. f (x) bL Figure 2 (a) If the length of string is 10cm with amplitude 5cm was set initially, state the initial condition and the boundary conditions for the...
6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy" 10y' + 8y = 100 cos 3t + 4et assuming that the mass is released from rest at the equilibrium position. (This forcing function is not very realistic.) (b) Assume the equation from part (a) describes a mass-spring-dashpot system with a dashpot containing honey. Imagine that the honey is changed...
a-d please
6 (10) Spring Problems: (a) Find the displacement, y(t), (in arbitrary units) as a function of time for the mass in a mass-spring system described by the differential equatiorn Zy" 10y' + 8y = 100 cos 3t + 4et assuming that the mass is released from rest at the equilibrium position. (This forcing function is not very realistic.) (b) Assume the equation from part (a) describes a mass-spring-dashpot system with a dashpot containing honey. Imagine that the honey...
Question 9 You should be able to answer this question after studying Unit 8. An object moves along a straight line, and its speed v (in metres per second) when at a position r (in metres) from its starting point can be modelled by the differential equation 10 marks 0 dr0, 20), where k is a positive constant (a) Find the general solution of this differential equation in explicit form (b) The speed of the object at its starting point...
what is the answer for number
4
1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j +...
Question 2 ul lu (a) Find the solution u(x,t) for the 1-D wave equationfor -oo < x < oo with initial conditions u (x,0)-A(x) , where A(x) s presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and t, somewhat similar to f(x) on page 85 of the Notes Part 2. di+10 dı+15di+20 (b) Check for the wave equation in (a) that if f(xtct) (use appropriate value...
d1= 3 & d2= 2
Question 2 Find the solution 11(x, 1) for the 1-D wave equation aT = (a) 25-for -o <x < oo with initial conditions it (x,0) = A (x) , where A(x) is presented in the diagram below, and zero initial velocity. For full marks u(x,t) needs to be expressed as an equation involving x and 1, somewhat similar to fex) on page 8s of the Notes Part 2. 2 d2+5 r-0 di+10 di+15 di+20 3...
question 4-7
4. Travelling Waves and Their Characteristics A rope wave travels in the positive x -direction. You are also told that the speed of the wave is 1000 cm/s, its frequency is 200 Hz, and that the wave is subject to the following initial conditions: at x 0 and t = 0: y =-1 cm, and, at x = 0 and t : ar = +20 cm/s (this is the velocity of the point on the rope at horizontal...