Question

A 2 kg object is attached to a horizontal spring with k = 200 N m . The object slides freely on a horizontal frictionless surface. You pull the object to an initial displacement of 5 cm and launch it with an initial velocity of −2 m s .

(a) Find the total energy of the motion.

(b) Write the equation for the position of the object as a function of time. Make sure you put values in for any constants that can be determined from the given information.

Answer 1

a) Total energy, TE = potentail energy + kinetic energy'

= 0.5*k*x^2 + 0.5*m*v^2

= 0.5*200*0.05^2 + 0.5*2*2^2

= 4.25 J

b)let A is the amplitude and w is angular frequency.

TE = 0.5*k*A^2

==> A = sqrt(2*KE/k)

= sqrt(2*4.25/200)

= 0.206 m

w = sqrt(k/m)

= sqrt(200/2)

= 10 rad/s

we can write the equation for position,

x = A*cos(w*t + phi) (here phi is initial phase)

at t = 0

0.05 = 0.206*cos(10*0 + phi)

cos(phi) = 0.05/0.206

phi = cos^-1(0.2427)

= 1.326 rad

x = 0.206*cos(10*t + 1.326)