A spring of negligible mass stretches 3.00 cm from its relaxed length when a for

Question

A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 7.60 N is applied. A 0.450-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x = 5.00 cm and released from rest at t = 0.

a) Determine the displacement x (in cm) of the particle from the equilibrium position at t = 0.500 s.
b) Determine the velocity and acceleration of the particle when t = 0.500 s. (Indicate the direction with the sign of your answer.)

Explanation / Answer

Part A)

First find the frequency

f = 1/2pi sqrt(k/m)

F = kx to find k

7.6 = k(.03)

k = 253.3 N/m

f = 1/2pi sqrt (253.3)/(.45)

f = 3.78 Hz

Then, x = A cos(2pi(ft))

x = (3)(cos (2pi)(3.78)(.5)

x = 2.31 cm

Part B)

v = sqrt[(k/m)(A^2 - x^2)]

v = sqrt[(253.3/.45)(.03^2 - .0231^2)]

v = .454 m/s

a = -kx/m

a -(253.3)(.0231)/(.45)

a = -13 m/s^2

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