A mass of 3.25 kg is attached to the end of a spring that is stretched 20 cm by a force of 14 N. It is set in motion with an initial position x_0 = 0 and initial velocity (in m/s) v_0 = -11. Find the amplitude, period, and frequency of the resulting motion. You should assume that the system is frictionless
To find the amplitude, period, and frequency of the resulting motion, we can use the equation of motion for a mass-spring system given by:
x(t) = A * cos(ωt + φ)
Where:
x(t) is the displacement of the mass at time t,
A is the amplitude of the motion,
ω is the angular frequency,
t is the time,
φ is the phase constant.
First, let's find the angular frequency (ω) of the motion using the equation:
ω = √(k / m)
Where:
k is the spring constant (force per unit displacement),
m is the mass.
The spring constant can be determined using Hooke's Law:
F = k * x
Where:
F is the force,
x is the displacement.
In this case, the displacement is 20 cm = 0.2 m and the force is 14 N.
k = F / x = 14 N / 0.2 m = 70 N/m
Now we can calculate the angular frequency:
ω = √(k / m) = √(70 N/m / 3.25 kg) = 3.771 rad/s
Next, let's find the amplitude (A) using the initial conditions given. We know that at time t = 0, the position (x_0) is 0 and the velocity (v_0) is -11 m/s. The equation of motion can be rewritten as:
x(t) = A * cos(ωt)
At t = 0, x(0) = A * cos(0) = A
Therefore, the initial displacement (x_0) is equal to the amplitude (A). So, A = 0.
Since the amplitude is 0, the motion is a simple harmonic motion along the equilibrium position without any oscillation.
Now, let's find the period (T) and frequency (f) of the motion. The period is the time taken for one complete cycle of the motion, and the frequency is the number of cycles per unit time.
The period (T) can be calculated as:
T = 2π / ω
T = 2π / 3.771 rad/s ≈ 1.671 s
The frequency (f) can be calculated as:
f = 1 / T
f = 1 / 1.671 s ≈ 0.598 Hz
Therefore, the amplitude of the motion is 0 m, the period is approximately 1.671 seconds, and the frequency is approximately 0.598 Hz.