Answer to Question #163402 in Trigonometry for Elai

Question #163402

A weight on the end of a spring oscillating in harmonic motion. The equation model for oscillation is d (t) = 6 sin ((pi/2)(t)) where d is the distance (in centimeter) from the equilibrium point in t sec.

a. What is the period of the motion? What is the frequency of the motion?

b. What is the placement from the equilibrium at t = 2.5? Is the weight moving toward the equilibrium at this time?

c. What is the displacement from equilibrium at t = 3.5? Is the weight moving toward the equilibrium point or away from the equilibrium at this time?

d. How far does the weight move between t = 1 and t = 1.5 sec?

e. What is the average velocity for this interval? 

f. Do you expect a greater or lesser velocity for t = 1.75 to t = 2? Explain why.

 




1
Expert's answer
2021-02-24T06:49:55-0500

"d(t) =6\\sin(\\frac{\\pi}{2}t)"

"a)\\omega= \\frac{\\pi}{2};\\omega-\\text{\u03c9 is the angular frequency}"

"T=\\frac{2\\pi}{\\omega}=4;T\\text{ is period of the motion}"

"f=\\frac{1}{T}=0.25;f \\text{ is the frequency of the motion}"


"b)d(2.5)=6\\sin(\\frac{\\pi}{2}*2.5)\\approx-4,24"

"v(t) = d'(t);v(t)\\text{velocity object}"

"v(t)=\\frac{6\\pi}{2}\\cos{(\\frac{\\pi}{2}t)}"

"\\text{if }v(t)>0\\text{ weight moving from the equilibrium}"

"\\text{if }v(t)<0\\text{ weight moving toward the equilibrium}"

"v(2.5)=\\frac{6\\pi}{2}\\cos{(\\frac{\\pi}{2}*2.5)}=-6.6<0"

"\\text{then the weight moving toward the equilibrium}"


"c)d(3.5)=6\\sin(\\frac{\\pi}{2}*3.5)\\approx-4,24"

"v(3.5)=\\frac{6\\pi}{2}\\cos{(\\frac{\\pi}{2}*3.5)}=6.6>0"

"\\text{then the weight moving from the equilibrium}"


"d)\\ d(1)=6\\sin(\\frac{\\pi}{2})\\approx=6"

"\\ d(1.5)=6\\sin(\\frac{\\pi}{2})\\approx4,24"

"e) v_{avg}=\\frac{\\Delta{s}}{\\Delta{t}}=\\frac{4.24-6}{1.5-1}=-3.52"


"f)v(t)=\\frac{6\\pi}{2}\\cos{(\\frac{\\pi}{2}t)}"

"v(1.75)=\\frac{6\\pi}{2}\\cos{(\\frac{\\pi}{2}*1.75)}\\approx-8.7"

"v(2)=\\frac{6\\pi}{2}\\cos{(\\frac{\\pi}{2}*2)}\\approx-9.4"

"|v(2)|>|v(1.75)|"












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