Ablock with a mass of 0.400 kg is connected to a spring, displaced in the positive direction a distance of 50.0 cm from equilibrium, and released from rest at t = 0. the block then oscillates without friction on a horizontal surface. after being released, the first time the block is a distance of 25.0 cm from equilibrium is at t = 0.200 s.

(a) what is the block's period of oscillation?

(b) what is the the value of the spring constant?

a)  T = 1,199 s b)  k = 11.0 N / m

Explanation:

The oscillatory movement of a mass-spring system is described by the equation

    x (t) = A cos (w t)

Where A is the amplitude of the movement. W is the angular velocity, which is given by

    w = √ k / m

The amplitude of the system is the maximum elongation that the spring has before releasing it, in this case it is A = 50 cm = 0.50 m, the equation is

     x = 0.50 cos (wt)

We can calculate the angular velocity with the point given by x = 0.250 m for t = 0.200s

     w t = cos⁻¹ (x / A)

     w = 1 / t cos⁻¹ (x / A)

     w = 1/0.200  cos⁻¹(0.25 / 0.50)

Let's be careful because the angle is in radians

     w = 5.24 rad / s

Angular velocity is

    w = 2 π f = 2π / T

    T = 2π / w

    T = 2π / 5.24

    T = 1,199 s

With the angular velocity equation we can take off the spring constant

    w² = k / m

    k = m w²

    k = 0.400 5.24²

    k = 11.0 N / m

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