Question

A block attached to a spring undergoes simple harmonic motion, sliding back and forth along a straight line on a horizontal, frictionless surface. The amplitude of the block's motion is
16
cm, the frequency of the block's motion is Hz, and the mass of the block is kg.

a) Determine the spring's stiffness constant.

N/m

b) The block is initially stretched and then released at time . Determine a formula for the position function of the block, where the position is measured in cm.

=  

c) Determine a formula for the velocity function of the block, where the velocity is measured in cm/s.

=  

d) Determine the position of the block after s:  cm.
Determine the velocity of the block after s:  cm/s.
Determine the kinetic energy of the block after s:  J.
Determine the elastic potential energy of the block after s:  J.
Determine the total mechanical energy of the block after s:  J.

e) Determine the position of the block after s:  cm.
Determine the velocity of the block after s:  cm/s.
Determine the kinetic energy of the block after s:  J.
Determine the elastic potential energy of the block after s:  J.
Determine the total mechanical energy of the block after s:  J.

f) Determine the position of the block after s:  cm.
Determine the velocity of the block after s:  cm/s.
Determine the kinetic energy of the block after s:  J.
Determine the elastic potential energy of the block after s:  J.
Determine the total mechanical energy of the block after s:  J.

g) Compare the results for the total mechanical energy after s, s, and s. Is this consistent with what you know about simple harmonic motion with no dissipative forces acting? Explain.

Answer 1

A = 16cm: f = 1.9 Hz: m = 1.3 kg (a) spring constant k = 4 pi^2 mf^2 or, k = 4 pi^2 (1.3) times (1.9)^2 = 185.2722 N/m (b) The equation for position of the system x = A Cos (2 pi ft) or, x = (16 cm) Cos (2 pi times (1.9) t] (c) velocity function is v = -2 pi f A sin (2 pi ft) or v = -(2 pi times 1.9 times 16) sin [2 pi times (1.9) times t] (d) position at t = 1 sec is x = 16 Cos [2 pi times 1.9 times 1] = 12.9447 cm (e) velocity at t = 1 sec is v = -(2 pi times 1.9 times 16) sin (2 pi times 1.9 times 1) = 112.265 cm/s