A block of mass m1 = 0.1 kg attached to an ideal spring moves on a frictionless

Question

A block of mass m1 = 0.1 kg attached to an ideal spring moves on a frictionless surface. Let x(t) be its displacement as a function of time. (a) At time t = 0, the block passes through the point x = 0 moving to the right (i.e., toward positive x). At time t = 0.5 s, the block reaches its maximum excursion of Xm = 10 cm (i) What is the period T of oscillation? (Ans: t = 2 s) (ii) What is the spring constant k? (Ans: k = 0.1 2 N/m) (iii) What is the maximum velocity vm? (Ans: 0.314 m/s) (iv) Write the full expression for x(t). (b) At time ti = 5T/8, a particle of mass m and moving with velocity u = 3vn/ 2 to the right collides completely inelastically with the block (i.e., collides and sticks to the block) of the oscillator in (a) (i) What is the position of the block as a function of time after the collision? (Ans: x() =-V3/2)x,n cos[ (t-ti) + ] where = tan-1 V2 (ii) At what time does the block pass the origin again? (Ans: 1.53 s)

Explanation / Answer

i)

for one fourth of the oscillations the time taken is t= 0.5 sec

for full oscillation ,the time taken is time period is T = 0.5*4 = 2 sec

ii)

spring constant is k = m*w^2 = m*(2*pi/T)^2 = 0.1*(2*pi/2)^2 = 0.1*pi^2 N/m

iii) maximum velocity is Vmax = A*w = A*(2*pi/T) = 10*10^-2*(2*3.142/2) = 0.314 m/s

iv) x(t) = A*cos(w*t) = A*cos(2*pi*t/T)

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