Chapter 8: Q5P (page 394)

Find the position x of a particle at time t if its acceleration is $\frac{d^{2} x}{d t^{2}} = A t ω s i n$ .

Short Answer

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Answer

The position x of a particle at time t is $x ω = - A ω t^{- 2} \sin + c t + d$ when acceleration is $\frac{d^{2} x}{d t^{2}} = A t ω \sin$ .

Step by step solution

Given information

In this question, the acceleration is given as $\frac{d^{2} x}{d t^{2}} = A t ω \sin$

Meaning of differential equation

In mathematics, an equation with only one independent variable and one or more of its derivatives with respect to the variable is referred to as an ordinary differential equation or ODE. In other words, the ODE is a relation with one independent variable x, a real dependent variable y, and several derivatives $y', y ", . . . ., y^{n}$ in relation to x.

Find the position of a particle

The given acceleration is $\frac{d^{2} x}{d t^{2}} = A t ω \sin$ .

Integrate the acceleration two times.

$\frac{d x}{d t} = \int A t ω \sin (d t) = - A ω^{- 1} \cos ω t + c$

The second integration will be

$d x = (- A ω t^{- 1} \cos + d t) \int d x = \int - A t ω d t^{- 1} \cos d t + \int x ω = - A ω t^{- 2} \sin + c t + d$

Therefore, the position of a particle at time t is $x ω = - A ω t^{- 2} \sin + c t + d$ .

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Find the position x of a particle at time t if its acceleration is $\frac{d^{2} x}{d t^{2}} = A t ω s i n$ .

Short Answer

Step by step solution

Given information

Meaning of differential equation

Find the position of a particle

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Step by step solution

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Meaning of differential equation

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Find the position x of a particle at time t if its acceleration is $\frac{d^{2} x}{d t^{2}} = A t ω s i n$ .