A mass m = 1.50 kg oscillates on an ideal,massless horizontal spring with a constant k = 73.5 N/m.The amplitude of oscillation is 10.0 cm.The system is frictionless. The positive x-direction is to the right. What is the mass's natural period of oscillation? At t = 0, the mass is located at x = 0 and is moving to the left.What is the mass's position and velocity as functions of time? If at t = 0, the mass is located at x = -5.00 cm and is moving to the right. What is the mass's position as a function of time?
The period of oscillation is
P = 2 pi sqrt(m/k)= 0.898 s
For your other questions, choose a value of the phase angle phi such that
x = 10 cm * sin (2 pi t/P + phi)
and the direction of motion (the sign of dx/dt) is correct.
Fot the x=0 @ t=0 case, phi = pi
To find the mass's natural period of oscillation, we can use the formula:
T = 2π√(m/k)
where T represents the period, m is the mass, and k is the spring constant.
Given that m = 1.50 kg and k = 73.5 N/m, we can substitute these values into the formula:
T = 2π√(1.50 / 73.5)
Calculating this expression gives us the mass's natural period of oscillation.
To find the mass's position and velocity as functions of time, we need to use the equations of simple harmonic motion. The general equation for the position of an object undergoing simple harmonic motion is:
x(t) = A cos (ωt + φ)
where x(t) represents the position of the mass as a function of time, A is the amplitude of the oscillation, ω is the angular frequency, t is the time, and φ is the phase constant.
The angular frequency ω can be calculated using the formula:
ω = 2π / T
where T is the period of oscillation.
Substituting the calculated T value into the formula, we can determine the angular frequency ω.
Next, to find the phase constant φ, we need to consider the initial conditions given. At t = 0, the mass is located at x = 0 and is moving to the left. This initial condition implies that the phase constant φ is 0.
Therefore, the position of the mass as a function of time is:
x(t) = A cos (ωt)
To find the velocity of the mass as a function of time, we can take the derivative of the position function with respect to time:
v(t) = -Aω sin (ωt)
Finally, if at t = 0, the mass is located at x = -5.00 cm and is moving to the right, we can determine the phase constant φ. Since the mass is moving to the right, this indicates a phase shift of π (180 degrees).
Therefore, the position of the mass as a function of time when t = 0 and x = -5.00 cm is:
x(t) = A cos (ωt + π)