Final answer:
The displacement, velocity, and acceleration of a mass in a spring-mass system with given values of mass, spring constant, initial displacement, and initial velocity can be determined using the formulas related to simple harmonic motion (SHM). The equations of motion for SHM are: displacement = amplitude * cos(angular frequency * time + phase constant), velocity = -amplitude * angular frequency * sin(angular frequency * time + phase constant), and acceleration = -amplitude * angular frequency^2 * cos(angular frequency * time + phase constant). By plugging in the given values and values calculated from the formulas, we can find the displacement, velocity, and acceleration at any given time.
Step-by-step explanation:
The displacement, velocity, and acceleration of a mass in a spring-mass system can be determined using the formulas related to simple harmonic motion (SHM). For this system, we are given:
Mass (m) = 2 kg
Spring constant (k) = 500 N/m
Initial displacement (x₀) = 0.1 m
Initial velocity (v₀) = 5 m/s
The equations of motion for SHM are:
-
where:
x(t) = displacement at time t
v(t) = velocity at time t
a(t) = acceleration at time t
A = amplitude of oscillation (maximum displacement from equilibrium)
ω = angular frequency, given by ω = sqrt(k/m)
φ = phase constant
In this case, A = 0.1 m, ω = √(500/2) = 10√5 rad/s, and φ = 0 (assuming the mass is released from equilibrium). Now we can calculate the displacement, velocity, and acceleration at any given time using the formulas above.