Homework Help is Here – Start Your Trial Now!
Math
Calculus
EXAMPLE 3 An object at the end of a vertical spring is stretched 8 cm beyond its rest position and released at time t = 0. (Note the downward direction is positive in the figure.) Its position at time t is s= f(t) = 8 cos(t) Find the velocity and acceleration at time t and use them to analyze the motion of the object. SOLUTION The velocity and acceleration are ds V = dt d(8 cos(t)) = 8- dt 8d(cos(t)) = dt dv a = dt 으(-8 sin(t)) = -8유(sin(t)) dt The object oscillates from the lowest point (s = 8 cm) to the highest point (s = -8 cm). The period of the oscillation is , the period of cos(t). The speed is Iv| = which is greatest when Isin(t)| = that is, when cos(t) = J. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin(t) that is, at the high and low points. The acceleration a = = 0 when s = 0. It has greatest magnitude at the high and low points. See the graphs to the left.

EXAMPLE 3 An object at the end of a vertical spring is stretched 8 cm beyond its rest position and released at time t = 0. (Note the downward direction is positive in the figure.) Its position at time t is s= f(t) = 8 cos(t) Find the velocity and acceleration at time t and use them to analyze the motion of the object. SOLUTION The velocity and acceleration are ds V = dt d(8 cos(t)) = 8- dt 8d(cos(t)) = dt dv a = dt 으(-8 sin(t)) = -8유(sin(t)) dt The object oscillates from the lowest point (s = 8 cm) to the highest point (s = -8 cm). The period of the oscillation is , the period of cos(t). The speed is Iv| = which is greatest when Isin(t)| = that is, when cos(t) = J. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin(t) that is, at the high and low points. The acceleration a = = 0 when s = 0. It has greatest magnitude at the high and low points. See the graphs to the left.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
See similar textbooks
icon
Related questions
Question
-0- b = 8 y 8. 6 4 -2 -4 -6 -8 2. 2.
Transcribed Image Text:-0- b = 8 y 8. 6 4 -2 -4 -6 -8 2. 2.
EXAMPLE 3 An object at the end of a vertical spring is stretched 8 cm beyond its rest position and released at time t = 0. (Note the downward direction is positive in the figure.) Its position at time t is s= f(t) = 8 cos(t) Find the velocity and acceleration at time t and use them to analyze the motion of the object. SOLUTION The velocity and acceleration are ds V = dt d(8 cos(t)) = 8- dt 8d(cos(t)) = dt dv a = dt 으(-8 sin(t)) = -8유(sin(t)) dt The object oscillates from the lowest point (s = 8 cm) to the highest point (s = -8 cm). The period of the oscillation is , the period of cos(t). The speed is Iv| = which is greatest when Isin(t)| = that is, when cos(t) = J. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin(t) that is, at the high and low points. The acceleration a = = 0 when s = 0. It has greatest magnitude at the high and low points. See the graphs to the left.
Transcribed Image Text:EXAMPLE 3 An object at the end of a vertical spring is stretched 8 cm beyond its rest position and released at time t = 0. (Note the downward direction is positive in the figure.) Its position at time t is s= f(t) = 8 cos(t) Find the velocity and acceleration at time t and use them to analyze the motion of the object. SOLUTION The velocity and acceleration are ds V = dt d(8 cos(t)) = 8- dt 8d(cos(t)) = dt dv a = dt 으(-8 sin(t)) = -8유(sin(t)) dt The object oscillates from the lowest point (s = 8 cm) to the highest point (s = -8 cm). The period of the oscillation is , the period of cos(t). The speed is Iv| = which is greatest when Isin(t)| = that is, when cos(t) = J. So the object moves fastest as it passes through its equilibrium position (s = 0). Its speed is 0 when sin(t) that is, at the high and low points. The acceleration a = = 0 when s = 0. It has greatest magnitude at the high and low points. See the graphs to the left.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Harmonic Motion
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS