A mass m on a length of string ? is displaced a small angle ?max and undergoes simple harmonic motion indefinitely (as a simple pendulum). 1) If t = 0 is the instant when the pendulum attains its maximum velocity, which equation below describes the angular displacement as a function of time? ? = ?max sin(?t) ? = ?max sin(?t - ?/4) ? = ?max cos(?t) ? = ?max cos(?t - ?/4) Submit 2) Which of the following changes would DECREASE the period of oscillation? decreasing the mass decreasing the length decreasing the amplitude of oscillation moving to a planet where the acceleration due to gravity is less Submit
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The figure shows a pendulum with length $L$ and the angle $\theta$ from the vertical to the pendulum. It can be shown that $\theta,$ as a function of time, satisfies the nonlinear differential equation $$\frac{d^{2} \theta}{d t^{2}}+\frac{g}{L} \sin \theta=0$$ where $g$ is the acceleration due to gravity. For small values of $\theta$ we can use the linear approximation $\sin \theta \approx \theta$ and then the differential equation becomes linear. (a) Find the equation of motion of a pendulum with length I m if $\theta$ is initially 0.2 rad and the initial angular velocity is $d \theta / d t=1$ rad/s. (b) What is the maximum angle from the vertical? (c) What is the period of the pendulum (that is, the time to complete one back-and-forth swing)? (d) When will the pendulum first be vertical? (e) What is the angular velocity when the pendulum is vertical?
Second-Order Differential Equations
Applications of Second Order Differential Equations
(II) A physical pendulum consists of an 85 -cm-long, 240 -g-mass, uniform wooden rod hung from a nail near one end (Fig, 38 ). The motion is damped because of friction in the pivot; the damping force is approximately proportional to $d \theta / d t .$ The rod is set in oscillation by displacing it $15^{\circ}$ from its equilibrium position and releasing it. After 8.0 $\mathrm{s}$ the amplitude of the oscillation has been reduced to $5.5^{\circ} .$ If the angular displacement can be written as $\theta=A e^{-\gamma t} \cos \omega^{\prime} t,$ find $(a) \gamma,(b)$ the approximate period of the motion, and $(c)$ how long it takes for the amplitude to be reduced to $\frac{1}{2}$ of its original value.
In Section 5 of "oscillations," the oscillation of a simple pendulum (Fig. 46$)$ is viewed as linear motion along the arc length $x$ and analyzed via $F=m a$ . Alternatively, the pendulum's movement can be regarded as rotational motion about its point of support and analyzed using $\tau=I \alpha$ . Carry out this alternative analysis and show that $\theta(t)=\theta_{\max } \cos \left(\sqrt{\frac{g}{\ell}} t+\phi\right)$ where $\quad \theta(t)$ is the angular displacement of the pendulum from the vertical at time $t,$ as long as its maximum value is less than about $15^{\circ}$ .
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