Question

19.1 a weight hanging on a vertical spring is set in motion with an upward velocity of 4 cm/sec from its equilibrium position. a formula that gives the location of the weight in centimeters as a function of the time t in seconds is ( x = -\frac{4}{2}sin(pi t) ). state the period of the function. how many oscillations will the weight undergo per second? period frequency

Explanation:

Step1: Recall the formula for the period of a sine function

The general form of a sine function is \(y = A\sin(Bt + C)+D\). The period \(T\) is given by \(T=\frac{2\pi}{|B|}\). For the function \(x =-\frac{1}{2}\sin(\pi t)\), here \(B = \pi\).

Step2: Calculate the period

Substitute \(B=\pi\) into the period formula \(T=\frac{2\pi}{|B|}\). So \(T=\frac{2\pi}{\pi}=2\) seconds.

Step3: Recall the formula for frequency

The frequency \(f\) (number of oscillations per second) is the reciprocal of the period. That is \(f=\frac{1}{T}\).

Step4: Calculate the frequency

Since \(T = 2\) seconds, then \(f=\frac{1}{2}=0.5\) oscillations per second.

Answer:

Period: \(2\) seconds; Frequency: \(0.5\) oscillations per second.