Question

Hi, can you solve the question for me step by step, I will rate up if the working is correct. I will post the answer together with the question.

Answer:

Question 5 A particle of mass m rests on a smooth horizontal track. It is connected by two springs to fixed points at A and B, which are a distance 2lo apart as shown in Figure Q5. The left-hand spring has natural length 2lo and stiffness k, whilst the right-hand spring has natural length lo and stiffness 2k. Let x be the distance of the particle from A 21 0 2k 7m Figure Q5 (a) Draw a diagram of the system indicating clearly the forces acting on the particle and the positive coordinates for the motion. Then express the force each spring exerts on the particle in terms of magnitude and a unit vector (5 marks) (b) Apply Newton's second law, derive the equation of motion (3 marks) Write down the general solution of the differential equation obtained in part (b) (5 marks) (c) and thus find the equilibrium position of the particle measured from A (d) Calculate the period of the resulting oscillations (2 marks) Now consider afresh the given system described earlier in Figure Q5 (e) Find an expression for the total mechanical energy of the system when the (3 marks) (f) The particle is initially midway between A and B, and is set in motion with particle is a distance x from the fixed point A velocity v. By using the conservation of energy, show that the amplitude of the resulting oscillations is mv 3k 3 (4 marks) (g) Show that in order for the particle to remain between the supports in its subsequent motion, without hitting either of them, the magnitude of v must not exceed I,y 1m (3 marks) 3k 4kl 3k (c) Bcos(cor) + Csin(ω) + 10 where ω x 0 τ = 2π 3k (f) & (g) to be shown

Answer 1

2k a 1 2Ls 0 กา s01

e have 3K rm) ㄒ 3 K nangy