Suppose a pendulum of length L meters makes an angle of radians with the vertical, as in the figure. It can be shown that as a function of time, satisfies the differential equation d²0 9 + dt2 L where g = 9.8 m/s² is the acceleration due to gravity. For near zero we can use the linear approximation sin(0) 0 to get a linear differential equation d²0 9 dt² L Use the linear differential equation to answer the following questions. 0(t) = sin 0 = 0, (a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.1 radians and initial angular velocity = 0.1 radians per second. de dt Period= + -0 = 0. seconds (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? radians
Suppose a pendulum of length L meters makes an angle of radians with the vertical, as in the figure. It can be shown that as a function of time, satisfies the differential equation d²0 9 + dt2 L where g = 9.8 m/s² is the acceleration due to gravity. For near zero we can use the linear approximation sin(0) 0 to get a linear differential equation d²0 9 dt² L Use the linear differential equation to answer the following questions. 0(t) = sin 0 = 0, (a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.1 radians and initial angular velocity = 0.1 radians per second. de dt Period= + -0 = 0. seconds (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? radians
Chapter6: Exponential And Logarithmic Functions
Section6.1: Exponential Functions
Problem 60SE: The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is...
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Transcribed Image Text:Suppose a pendulum of length L meters makes an angle of radians with the vertical, as in the
figure. It can be shown that as a function of time, 0 satisfies the differential equation
where g = 9.8 m/s² is the acceleration due to gravity. For near zero we can use the linear
approximation sin(0) ≈ 0 to get a linear differential equation
d²0
9
+ -sin = 0,
dt² L
0 (t)
d²0
dt²
Period =
9
Use the linear differential equation to answer the following questions.
-0 = 0.
(a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.1
radians and initial angular velocity = 0.1 radians per second.
de
dt
seconds
L
(b) What is the period of the pendulum? That is, what is the time for one swing back and forth?
radians
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