# An object moves in simple harmonic motion with period 8 minutes and amplitude 16 m. At time t = 0 minutes, its displacement d from rest is 0 m, and initially it moves in a positive direction. Give the equation modeling the displacement d as a function of time f. Question
Modeling An object moves in simple harmonic motion with period 8 minutes and amplitude 16 m. At time $$t = 0$$ minutes, its displacement d from rest is 0 m, and initially it moves in a positive direction. Give the equation modeling the displacement d as a function of time f. 2021-01-28
Step 1 Equation modeling the diplacement d as function of time = A sin wt where A = amplitude and time period = 2pi//w Step 2 $$A = amplitude = 16 m$$ $$Period = 8 minutes = 8 \times 60 = 480 seconds = 2\pi/w$$ Hence, $$w = 2pi/480 = \pi/240$$ Step 3 Hence, $$d = 16sin(\pi t/240)$$

### Relevant Questions An object moves in simple harmonic motion with period 5 seconds and amplitude 7 cm. At time $$\displaystyle{t}={0}$$ seconds, its displacement d from rest is -7 cm, and initially it moves in a positive direction.
Give the equation modeling the displacement d as a function of time t. A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 8.50 N is applied. A 0.530-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x = 5.00 cm and released from rest at t = 0. (Assume that the direction of the initial displacement is positive.)
(a) What is the force constant of the spring? 280 N/m
(b) What are the angular frequency (?), the frequency, and the period of the motion?
f = 3.6817 Hz
T = 0.27161 s
(c) What is the total energy of the system? 0.35 J
(d) What is the amplitude of the motion? 5 cm
(e) What are the maximum velocity and the maximum acceleration of the particle?
$$\displaystyle{v}_{{\max}}={1.1561}\frac{{m}}{{s}}$$
$$\displaystyle{a}_{{\max}}={26.73}\frac{{m}}{{s}^{{{2}}}}$$
(f) Determine the displacement x of the particle from the equilibrium position at t = 0.500 s.
(g) Determine the velocity and acceleration of the particle when t = 0.500 s. (Indicate the direction with the sign of your answer.)
v = _________________ $$\displaystyle\frac{{m}}{{s}}$$
a = _________________ $$\displaystyle\frac{{m}}{{s}^{{{2}}}}$$ A common assumption in modeling drug assimilation is that the blood volume in a person is a single compartment that behaves like a stirred tank. Suppose the blood volume is a four-liter tank that initially has a zero concentration of a particular drug. At time $$\displaystyle{t}={0}$$, an intravenous line is inserted into a vein (into the tank) that carries a drug solution with a concentration of 500 mg/L. The inflow rate is 0.06 L/min. Assume the drug is quickly mixed thoroughly in the blood and that the volume of blood remains constant. a. Write an initial value problem that models the mass of the drug in the blood, for $$\displaystyle{t}\ \geq\ {0}$$. Solve the initial value problem, and graph both the mass of the drug and the concentration of the drug. c. What is the steady-state mass of the drug in the blood? d. After how many minutes does the drug mass reach 90% of its steady-state level? 1)What is the position vector r(t) as a function of angle $$\displaystyle\theta{\left({t}\right)}$$. For later remember that $$\displaystyle\theta{\left({t}\right)}$$ is itself a function of time.
Give your answer in terms of $$\displaystyle{R},\theta{\left({t}\right)}$$, and unit vectors x and y corresponding to the coordinate system in thefigure. 2)For uniform circular motion, find $$\displaystyle\theta{\left({t}\right)}$$ at an arbitrary time t.
Give your answer in terms of $$\displaystyle\omega$$ and t.
3)Find r, a position vector at time.
Give your answer in terms of R and unit vectors x and/or y.
4)Determine an expression for the positionvector of a particle that starts on the positive y axis at (i.e., at ,(x_{0},y_{0})=(0,R)) and subsequently moves with constant $$\displaystyle\omega$$.
Express your answer in terms of R, \omega ,t ,and unit vectors x and Determine the algebraic modeling a.
One type of Iodine disintegrates continuously at a constant rate of $$\displaystyle{8.6}\%$$ per day.
Suppose the original amount, $$\displaystyle{P}_{{0}}$$, is 10 grams, and let t be measured in days.
Because the Iodine is decaying continuously at a constant rate, we use the model $$\displaystyle{P}={P}_{{0}}{e}^{k}{t}$$ for the decay equation, where k is the rate of continuous decay.
Using the given information, write the decay equation for this type of Iodine.
b.
Use your equation to determine the half-life ofthis type of Fodine, That is, find ‘out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs. A transverse sine wave with an amplitude of 2.50 mm and a wavelength of 1.80 m travels from left to right along a long, horizontal stretched string with a speed of 36.0 m/s. Take the origin at the left end of the undisturbed string. At time t = 0 the left end of the string has its maximum upward displacement. What is y (t) for a particle 1.35 m to the right of the origin? Round all numeric coefficients to exactly three significant figures. Two basketball players are essentially equal in all respects. In particular, by jumping they can raise their centers of mass the same vertical distance, H. The first player,Arabella, wishes to shoot over the second player, Boris, and forthis she needs to be as high above Boris as possible. Arabella Jumps at time t=0, and Boris jumps later, at time $$\displaystyle{t}_{{R}}$$(his reaction time). Assume that Arabella has not yet reached her maximum height when Boris jumps.
Part A.) Find the vertical displacement $$\displaystyle{D}{\left({t}\right)}={h}_{{A}}{\left({t}\right)}-{h}_{{B}}{\left({t}\right)}$$, as a function of time for the interval $$\displaystyle{0}{<}{t}{<}{t}_{{R}}$$, where $$\displaystyle{h}_{{A}}{\left({t}\right)}$$ is the height of the raised hands of Arabella, while $$\displaystyle{h}_{{B}}{\left({t}\right)}$$ is the height of the raised hands of Boris. (Express thevertical displacement in terms of H,g,and t.)
Part B.) Find the vertical displacement D(t) between the raised hands of the two players for the time period after Boris has jumped ($$\displaystyle{t}{>}{t}_{{R}}$$) but before Arabella has landed. (Express youranswer in terms of t,$$\displaystyle{t}_{{R}}$$, g,and H)
Part C.) What advice would you give Arabella To minimize the chance of her shot being blocked? A bird flies in the xy-plane with a position vector given by $$\displaystyle\vec{{{r}}}={\left(\alpha{t}-\beta{t}^{{3}}\right)}\hat{{{i}}}+\gamma{t}^{{2}}\hat{{{j}}}$$, with $$\displaystyle\alpha={2.4}\ \frac{{m}}{{s}},\beta={1.6}\ \frac{{m}}{{s}^{{3}}}$$ and $$\displaystyle\gamma={4.0}\ \frac{{m}}{{s}^{{2}}}$$. The positive y-direction is vertically upward. At the bird is at the origin.
Calculate the velocity vector of the bird as a function of time.
Calculate the acceleration vector of the bird as a function oftime.
What is the bird's altitude(y-coordinate) as it flies over x=0 for the first time after ? Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: $$(1) {1,i,-i}. (2) {-1+4i,-1-4i}. (3) {-1}. (4)$$ The empty diagram.
(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by $$w(t) = u(t)e^{-\frac{t}{2}} \sin(\frac{3t}{2})$$
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid? 