Suppose a pendulum with length LL (meters) has angle θθ (radians) from the vertical. It can be shown that θθ as a function of time satisfies the differential equation: d^2θ/dt^2 + g/Lsinθ = 0 where g=9.8/sec is the acceleration due to gravity. For small values of θ we can use the approximation sin(θ)∼θ, and with that substitution, the differential equation becomes linear. A. Determine the equation of motion of a pendulum with length 0.5 meters and initial angle 0.2 radians and initial angular velocity dθ/dt 0.4 radians/sec. B. At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer) seconds
Suppose a pendulum with length LL (meters) has angle θθ (radians) from the vertical. It can be shown that θθ as a function of time satisfies the differential equation:
where g=9.8/sec is the acceleration due to gravity. For small values of θ we can use the approximation sin(θ)∼θ, and with that substitution, the differential equation becomes linear.
A. Determine the equation of motion of a pendulum with length 0.5 meters and initial angle 0.2 radians and initial angular velocity dθ/dt 0.4 radians/sec.
B. At what time does the pendulum first reach its maximum angle from vertical? (You may want to use an inverse trig function in your answer)
seconds
C. What is the maximum angle (in radians) from vertical?
D. How long after reaching its maximum angle until the pendulum reaches maximum deflection in the other direction? (Hint: where is the next critical point?)
seconds
E. What is the period of the pendulum, that is the time for one swing back and forth?
seconds
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