A 0.49-kg object is attached to one end of a spring, as in the first drawing, and the system is set into simple harmonic motion. The displacement x of the object as a function of time is shown in the second drawing. With the aid of these data, determine (a) the amplitude A of the motion, (b) the angular frequency w, (c) the spring constant k, (d) the speed of the object at t=1.0s and (e) the magnitude of the object’s acceleration at t=1.0s
To determine the values requested, we need to analyze the given information and use the equations of simple harmonic motion.
(a) The amplitude A is the maximum displacement from the equilibrium position. Looking at the graph, we can see that the maximum displacement is 0.15m. Therefore, A = 0.15m.
(b) The angular frequency w is the rate at which the object oscillates back and forth. It can be calculated using the formula:
w = 2πf
where f is the frequency. The period T can be determined by observing the time it takes for one complete cycle of motion. From the given information, we can see that it takes 2 seconds for one complete cycle. Therefore, T = 2s. And since f = 1/T, the frequency is 0.5 Hz. So, w = 2π(0.5) = π rad/s.
(c) The spring constant k can be determined using the formula:
k = mω^2
where m is the mass of the object. In this case, m = 0.49 kg. So, k = (0.49 kg)(π rad/s)^2.
(d) To determine the speed of the object at t = 1.0s, we need to consider that the velocity of an object in simple harmonic motion is given by the equation:
v = Aωsin(ωt + φ)
where φ is the phase angle, which is not given in the information provided. However, we can use the fact that the object is at maximum displacement at t = 0 and heading towards the equilibrium position. Therefore, at t = 1.0s, the object would be moving toward the equilibrium position and would have zero velocity. So, the speed of the object at t = 1.0s is 0 m/s.
(e) The magnitude of the object's acceleration, a, at t = 1.0s can be determined using the formula:
a = -Aω^2sin(ωt + φ)
where the negative sign indicates that the acceleration is in the opposite direction of the displacement. Since the object is at maximum displacement at t = 0, and acceleration is zero at that point, we can conclude that the object is also at maximum acceleration when t = 1.0s. Therefore, the magnitude of the object's acceleration at t = 1.0s is Aω^2.
Note: Without the phase angle information, we can only determine the magnitude of the acceleration at t = 1.0s.