Question

19.) 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 funtion of the time t in seconds is ( x = -\frac{4}{pi}sin(pi t) ). state the period of the function. how many oscillations will the weight undergo per second?
period
frequency
20.) if a neutron star emits beams of radiation as it rotates at 1000 times per second, what is the period of the pulsar?

Explanation:

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

For a function \(y = A\sin(Bt + C)+D\), the period \(T=\frac{2\pi}{|B|}\). In the given function \(x =-\frac{4}{\pi}\sin(\pi t)\), \(B = \pi\).

Step2: Calculate the period

Using the formula \(T=\frac{2\pi}{|B|}\), substitute \(B=\pi\). Then \(T=\frac{2\pi}{\pi}=2\) seconds.

Step3: Recall the formula for frequency

Frequency \(f\) is the reciprocal of the period. The formula 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