A spring is oscillating so that its length is a sinusoidal function of time. Its length varies from a minimum of 10 cm to a maximum of 14 cm. At t=0 seconds, the length of the spring was 12 cm, and it was decreasing in length. It then reached a minimum length at time t= 1.2 seconds. Between time t=0 and t=8 seconds, how much of the time was the spring longer than 13.5 cm?

1 answer

Question:

A spring is oscillating so that its length is a sinusoidal function of time. Its length varies from a minimum of 10 cm to a maximum of 14 cm. At t=0 seconds, the length of the spring was 12 cm, and it was decreasing in length. It then reached a minimum length at time t= 1.2 seconds. Between time t=0 and t=8 seconds, how much of the time was the spring longer than 13.5 cm?

Let the x(t) represent the motion of the spring as a function of time, t.The length of the oscillating spring varies from a minimum of 10 cm to a maximum of 14 cm.Therefore its amplitude is A = (14 - 10)/2 = 2.When t = 0 s, x = 12 cm.Therefore the function is of the formx(t) = 2 sin(bt) + 12At t=0, x(t) is decreasing, and it reaches its minimum value when t = 1.2 s.Therefore, a quarter of the period is 1.2 s.The period is given byT/4 = 1.2T = 4.8 sThat is,b = (2π)/T = (2π)/4.8 = π/2.4 = 1.309The function isx(t) = 2 sin(1.309t) + 12A plot of x(t) is shown below.When x(t) = 13.5, obtain2 sin(1.309t) + 12 = 13.5sin(1.309t) = (13.5 - 12)/2 = 0.751.309t = sin⁻¹ 0.75 = 0.8481 or π - 0.8481t = 0.8481/1.309 or t = (π - 0.8481)/1.309 = 0.649 or 1.751The difference in t is 1.751 - 0.649 = 1.1026.This difference occurs twice between t=0 and t=8 s.Therefore the spring length is greater than 13.5 cm for 2*1.1026 = 2.205 s.Answer:Between t=0 and t=8, the spring is longer than 13.5 cm for 2.205 s.

The number of animals living in a zoo increased by 11% last year. At the beginning of last year, there were 900 total animals. How many animals lived in the zoo at the end of last year?
Select the best answer from the choices provided.
9
99
999
801...

Row (definitions below) 1. to propel a boat by means of oars 2. a number of objects arranged in a usually straight line Which of the following sentences uses the word row as it is defined in definition 2? (STAR 360)
Question 1 options:
The unlikely pair were able to row the boat at an unexpected r...

The Boeing 787 Dreamliner is billed to be 20% more fuel efficient than the comparable Boeing 767 and will fly at Mach 0.85. The midsize Boeing 767 has a range of 12,000 km, a fuel capacity of 90,000 L, and flies at Mach 0.80. For Boeing 787, assume the speed of sound is 700 mph and calculate the pro...

During the Reformation, many central European rulers supported Protestantism against the Catholic Church. Why?
Choose the best answer from the list below.
a
They did not like the pope.
b
They did not want to have to obey the Church.
c
They could increase their power by seizing lands that had belo...