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Chapter 8: Q. 22 (page 550)

An object of mass m (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion.

(a) Develop a model that relates the distance d of the object from its rest position after t seconds.

(b) Graph the equation found in part (a) for 5 oscillations using a graphing utility.

Given values:m=20,a=15,b=0.75,T=6

Short Answer

Expert verified

a) The equation of motion can be given as

d=15e-0.01875tcos(1.047t)

b) The graph can be given as

Step by step solution

01

Given information

We are givenm=20,a=15,b=0.75,T=6

02

Part (a) Step 1: Find angular velocity

We get,

蝇=2蟺T蝇=2蟺6蝇=蟺3

03

Part (a) Step 2: Find the equation

We get,

d(t)=ae-bt2mcos(蝇2-b24m2t)d(t)=15e-0.75t2(20)cos((蟺3)2-0.7524(20)2t)d(t)=15e-0.01875tcos((蟺29)-0.56251600t)d(t)=15e-0.01875tcos(1.047t)

04

Part (b) Step 1: Plot the graph

We get,

05

Conclusion

a) The equation of motion isd=15e-0.01875tcos(1.047t)

b) The graph can be given as

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