Chapter 8: Q13P (page 436)

The exact equation of motion of a simple pendulum is $d^{2} θ / d t^{2} = - ω^{2} s i n θ$ where $ω^{2} = g / l$ . By method (c) above, integrate this equation once to find $d θ / d t$ if $d θ / d t = 0$ when $θ = 90 °$ . Write a formula for $t (θ)$ as an integral. See Problem 5.34.

Short Answer

Expert verified

The solution of the given differential equation is $t (θ) = \int \sqrt{\frac{l}{2 g}} \cdot \sqrt{s e c θ} d θ + c$

Step by step solution

Given information

The given equation.

$\frac{d^{2} θ}{d t^{2}} = - ω^{2} s i n θ$

Differential equation

Definition: A differential equation is an equation that relates one or more unknown functions and their derivatives.

Solve for differential equation

Equation is $\frac{d^{2} θ}{d t^{2}} = - ω^{2} s i n θ$

Multiply both side of the equation by $d θ / d t$

Solve further,

It is given that $\frac{d θ}{d t} = 0$ at $θ = 90 °$ .

So,

$0 = 2 ω^{2} c o s 90^{°} + 2 c 0 = 0 + 2 c c = 0$

Thus,

Integrate above equation

Integrate both sides,

Therefore, the solution of the given differential equation is $t (θ) = \int \sqrt{\frac{l}{g}} \cdot \sqrt{s e c θ} d θ + c$

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The exact equation of motion of a simple pendulum is $d^{2} θ / d t^{2} = - ω^{2} s i n θ$ where $ω^{2} = g / l$ . By method (c) above, integrate this equation once to find $d θ / d t$ if $d θ / d t = 0$ when $θ = 90 °$ . Write a formula for $t (θ)$ as an integral. See Problem 5.34.

Short Answer

Step by step solution

Given information

Differential equation

Solve for differential equation

Integrate above equation

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The exact equation of motion of a simple pendulum isd2θ/dt2=-ω2sin⁡θwhereω2=g/l. By method (c) above, integrate this equation once to finddθ/dtifdθ/dt=0whenθ=90°. Write a formula fort(θ)as an integral. See Problem 5.34.

Short Answer

Step by step solution

Given information

Differential equation

Solve for differential equation

Integrate above equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Engineering Physics

Scientific Method Physics

Fields in Physics

Translational Dynamics

Waves Physics

Solid State Physics

Study anywhere. Anytime. Across all devices.

The exact equation of motion of a simple pendulum is $d^{2} θ / d t^{2} = - ω^{2} s i n θ$ where $ω^{2} = g / l$ . By method (c) above, integrate this equation once to find $d θ / d t$ if $d θ / d t = 0$ when $θ = 90 °$ . Write a formula for $t (θ)$ as an integral. See Problem 5.34.