Question:
A mass-spring system is undergoing simple harmonic motion with a period of 2 seconds. The mass has an amplitude of 0.5 meters and a maximum speed of 2 m/s.
a) Determine the angular frequency (ω) of the harmonic motion. b) Calculate the mass of the object. c) Find the maximum acceleration of the object. d) What is the equation of motion for the mass-spring system?
Answer:
a) The angular frequency (ω) of a simple harmonic motion is given by the formula:
ω = 2π / T
where T is the period of the motion.
Given that the period of the motion is 2 seconds, we can use the formula to find the angular frequency:
ω = 2π / 2 = π rad/s
Therefore, the angular frequency (ω) of the harmonic motion is π rad/s.
b) The formula for the angular frequency (ω) of a mass-spring system is:
ω = √(k / m)
where k is the spring constant and m is the mass of the object.
To find the mass of the object, we need to rearrange the formula:
m = k / ω^2
Since we already know the angular frequency (ω) is π rad/s and the formula needs the square of ω, we can calculate the mass as:
m = k / (π^2)
c) The maximum acceleration (a_max) of an object undergoing simple harmonic motion is given by the formula:
a_max = ω^2 * A
where A is the amplitude of the motion.
Given that the amplitude of the motion is 0.5 meters and the angular frequency (ω) is π rad/s, we can calculate the maximum acceleration as:
a_max = (π^2) * 0.5 = π^2 / 2 m/s^2
Therefore, the maximum acceleration of the object is π^2 / 2 m/s^2.
d) The equation of motion for a mass-spring system undergoing simple harmonic motion is:
x(t) = A * cos(ωt + Ø)
where x(t) is the displacement of the object at time t, A is the amplitude of the motion, ω is the angular frequency, and Ø is the phase angle.
In this case, the amplitude (A) is given as 0.5 meters and the angular frequency (ω) is π rad/s. Therefore, the equation of motion for the mass-spring system is:
x(t) = 0.5 * cos(πt + Ø)
Please note that the exact phase angle (Ø) cannot be determined without additional information.