A 2.30 -kg object is attached to a spring and placed on frictionless, horizontal

Question

A 2.30-kg object is attached to a spring and placed on frictionless, horizontal surface. A horizontal force of 24.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations.

a)Find the force constant of the spring

b)Find the frequency of the oscillations

c)Find the maximum speed of the object

d)Where does this maximum speed occur

e)Find the maximum acceleration of the object

f)Where does the maximum acceleration occur?

g)Find the total energy of the oscillating system

h)Find the speed of the object when its position is equal to one third of the maximum value

i)Find the magnitude of the acceleration of the object when its position is equal to one third the max value

Explanation / Answer

a) for spring... let the spring constant be K ..

so Force = k*x
24 = k* 0.2

so K = 120 N / m

b) frequency = sqrt ( k / m ) / (2*pi ) = sqrt ( 120 / 2.3 ) / ( 2*pi ) = 1.1496 Hz

c) maximum speed = 0.2 * frequency * 2 * pi = 0.2 * 1.1496 * 2 * pi = 1.4446 m / sec

d) this maximum speed cocuers at the equillibrim position

e) maximum acceleraion = 0.2 * (2*pi * freq )^2 = 10.43478 m / sc2

f) This occurs at the extreme ends .

g) total energy = 0.5 * mass * v_max^2 = 0.5 * 2.3 * 1.4446^2 = 2.399899534 J

h) speed = (2*pi*freq) * sqrt ( x_max^2 - x^2 )   = ( 2 * p i* 1. 1.1496) * 0.2 * sqrt ( 1 - 1/9 ) = 1.362 m / sec

i) accelelation = (2*pi*freq)^2 * x_max / 3 =   ( 2 * p i* 1. 1.1496)^2 * 0.2 / 3 = 3.478 m / sec2

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