Answer:
0.390 s
Explanation:
The spring and mass undergo simple harmonic motion. Using Hooke's law, which says that the spring force is equal to the spring constant times the displacement, and Newton's second law of motion, which says the net force on the object is equal to its mass times acceleration, the motion of the object can be described with a differential equation. The period of the motion can be found by solving this equation.
Applying Newton's second law:
F = ma
-kx = m d²x/dt²
The object is in simple harmonic motion. If x = A cos(ωt) is a solution, then the acceleration is:
dx/dt = -Aω sin(ωt)
d²x/dt² = -Aω² cos(ωt)
Substituting:
-kA cos(ωt) = -mAω² cos(ωt)
k = mω²
ω = √(k/m)
Period is 2π divided by the angular velocity.
T = 2π / ω
T = 2π √(m/k)
Plugging in values:
T = 2π √(0.200 kg / 52.0 N/m)
T = 0.390 s
