The Physics of Everyday Phenomena (Eigth Edition) Chapter 6, 4SP is not answered

Question

The Physics of Everyday Phenomena (Eigth Edition) Chapter 6, 4SP is not answered at all.

The question is:

Suppose that a 200-g mass (0.20 kg) is oscillating at the end of a spring upon a horizontal surface that is essentially friction-free. The spring can be both stretched and compressed and has a spring constant of 360 N/m. It was originally stretched a distance of 14 cm (14m) from its equilibrium (unstretched) position prior to release.

a) What is its initial potential energy?

b) What is the maximum velocity that the mass will reach in its oscillation? Where in the motion is this maximum reached?

c) Ignoring friction, what are the values of the potential energy, kinetic energy, and velocity of the mass when the mass is 7 cm from the equilibrium position?

d) How does the value of velocity computed in part c compare to that computerd in part b? (What is the ratio of the values?)

Explanation / Answer

a) POtential energy stored in spring = k x^2 / 2

so initial PE = 360 x (0.14^2) /2 = 3.53 J

b) m w^2 = k

w = sqrt(360 / 0.20) = 42.43 rad/s

and amplitude A = 0.14 m

Vmax = Aw = 0.14 x 42.43 = 5.94 m/s

c) Total energy of this SHM = kA^2 /2 = 3.53 J

when x = 0.07 m

PE = 360 (0.07^2) /2 = 0.882 J

and total energy = PE + KE

3.53 = 0.882 + KE

KE = 2.65 J

and KE = m v^2 /2

2.65 = 0.2v^2 /2

v = 5.14 m/s

d) v / Vmax = 5.14 / 5.94 = 0.866

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