FOF EXercises 81-82, consider a 1-kg object oscillating at the end of a horizontal spring. The horizontal position x(t) of the object is given by Vo x(1) =sin(wr) + xocos (wt) www -3-2 -1 0 1 2 3 where v is the initial velocity, xo is the initial position, and o is the number of back-and-forth cycles that the object makes per unit time t. 81. At time t=0 sec, the object is moved 3 ft to the left of the equilibrium position and then given a velocity of 4 ft/sec to the right (Vo =4 ft/sec). %3D a. If the object completes 1 cycle in 1 sec (@ = 1), write a model of the form %3D x(t) Vo sin(wt) + xocos(wr) to represent the horizontal motion of the spring. b. Use the reduction formula to write the function in part (a) in the form x() =k sin (t+ a). Round a to 2 decimal places. c. What is the maximum displacement of the object from its equilibrium position?
FOF EXercises 81-82, consider a 1-kg object oscillating at the end of a horizontal spring. The horizontal position x(t) of the object is given by Vo x(1) =sin(wr) + xocos (wt) www -3-2 -1 0 1 2 3 where v is the initial velocity, xo is the initial position, and o is the number of back-and-forth cycles that the object makes per unit time t. 81. At time t=0 sec, the object is moved 3 ft to the left of the equilibrium position and then given a velocity of 4 ft/sec to the right (Vo =4 ft/sec). %3D a. If the object completes 1 cycle in 1 sec (@ = 1), write a model of the form %3D x(t) Vo sin(wt) + xocos(wr) to represent the horizontal motion of the spring. b. Use the reduction formula to write the function in part (a) in the form x() =k sin (t+ a). Round a to 2 decimal places. c. What is the maximum displacement of the object from its equilibrium position?
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![For Exercises 81-82, consider a 1-lb object oscillating at the end of a horizontal spring. The horizontal position \(x(t)\) of the object is given by:
\[
x(t) = \frac{v_0}{\omega} \sin(\omega t) + x_0 \cos(\omega t)
\]
where \(v_0\) is the initial velocity, \(x_0\) is the initial position, and \(\omega\) is the number of back-and-forth cycles that the object makes per unit time.
81. At time \(t = 0\) sec, the object is moved 3 ft to the left of the equilibrium position and then given a velocity of 4 ft/sec to the right \((v_0 = 4 \text{ ft/sec})\).
a. If the object completes 1 cycle in 1 sec \((\omega = 1)\), write a model of the form \(x(t) = \frac{v_0}{\omega} \sin(\omega t) + x_0 \cos(\omega t)\) to represent the horizontal motion of the spring.
b. Use the reduction formula to write the function in part (a) in the form \(x(t) = k \sin(t + \alpha)\). Round \(\alpha\) to 2 decimal places.
c. What is the maximum displacement of the object from its equilibrium position?
**Diagram Explanation:**
The diagram shows a spring-mass system with a scale indicating positions from -3 to 3. The object is initially positioned at -3, indicating it is 3 ft to the left of the equilibrium position. The spring is depicted in a compressed state.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabbe649f-9542-4750-a644-5a898c154717%2Fbc4a0b0c-aeb9-4db2-8ffa-c88e577a2b53%2Fhfmvmw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For Exercises 81-82, consider a 1-lb object oscillating at the end of a horizontal spring. The horizontal position \(x(t)\) of the object is given by:
\[
x(t) = \frac{v_0}{\omega} \sin(\omega t) + x_0 \cos(\omega t)
\]
where \(v_0\) is the initial velocity, \(x_0\) is the initial position, and \(\omega\) is the number of back-and-forth cycles that the object makes per unit time.
81. At time \(t = 0\) sec, the object is moved 3 ft to the left of the equilibrium position and then given a velocity of 4 ft/sec to the right \((v_0 = 4 \text{ ft/sec})\).
a. If the object completes 1 cycle in 1 sec \((\omega = 1)\), write a model of the form \(x(t) = \frac{v_0}{\omega} \sin(\omega t) + x_0 \cos(\omega t)\) to represent the horizontal motion of the spring.
b. Use the reduction formula to write the function in part (a) in the form \(x(t) = k \sin(t + \alpha)\). Round \(\alpha\) to 2 decimal places.
c. What is the maximum displacement of the object from its equilibrium position?
**Diagram Explanation:**
The diagram shows a spring-mass system with a scale indicating positions from -3 to 3. The object is initially positioned at -3, indicating it is 3 ft to the left of the equilibrium position. The spring is depicted in a compressed state.
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