A 0.67-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 , (c) the spring constant k, (d) the speed of the object at t = 1.0 s, and (e) the magnitude of the object's acceleration at t = 1.0 s.
To determine the values requested, we will use the given displacement graph of the object as a function of time. Here's how we can find each value step-by-step:
(a) Amplitude (A) of the motion:
The amplitude refers to the maximum displacement from equilibrium. Looking at the graph, we can see that the maximum positive and negative displacements are 0.06 m and -0.06 m, respectively. Therefore, the amplitude is the absolute value of either of these values, which gives us A = 0.06 m.
(b) Angular frequency (ω):
The angular frequency represents the rate at which the object oscillates and is related to the period (T) of the motion. From the graph, we can visually determine the period as the time it takes for one complete cycle, which appears to be approximately 2 seconds. Thus, the period is T = 2 s. The angular frequency is then given by ω = 2π / T = 2π / 2 = π rad/s.
(c) Spring constant (k):
The spring constant describes the stiffness of the spring and determines how strong its restoring force is. To find k, we need to use the formula for angular frequency: ω = √(k / m), where m is the mass of the object attached to the spring. Given that m = 0.67 kg and ω = π rad/s, we can rearrange the formula to solve for k: k = m ω². Plugging in the values, we get k = (0.67 kg) × (π rad/s)².
(d) Speed of the object at t = 1.0 s:
To determine the speed of the object, we need to find the derivative of displacement with respect to time, which gives us the velocity function. Precisely at t = 1.0 s, we need to evaluate this velocity function. It is worth mentioning that the gradient on the displacement graph represents the instantaneous velocity at any given point. Based on the graph, the slope appears to be steepest at t = 1.0 s, indicating maximum velocity. Therefore, the speed of the object is at its maximum at t = 1.0 s.
(e) Magnitude of the object's acceleration at t = 1.0 s:
Similar to finding the speed, we need to find the derivative of velocity with respect to time, resulting in the acceleration function. Specifically, at t = 1.0 s, we evaluate this acceleration function to obtain the magnitude of acceleration.
Please provide the actual displacement graph, and I will help you calculate the numerical values for parts (b), (c), (d), and (e) using the information provided on the graph.