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3 years ago

11

What forms do the equations of motion for an idealized string take if either (a) the linear density varies with position, or (b)

the string hangs vertically supported only at the upper end?

1 answer:

stepan [7]3 years ago

7 0

Answer:

a) the varying densities are to be considered with respect to position

Step-by-step explanation:

the complete answer is found in the attachment

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15. If x=a Sin2t (I+Cos2t) and y=b Cos 2t (1-Cos2t) then find<br>dy/dx at =22/7*4<br>

Sav [38]

By the chain rule,

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}$

It looks like we're given

$\begin{cases}x=a\sin(2t)(1+\cos(2t))\\y=b\cos(2t)(1-\cos(2t))\end{cases}$

where <em>a</em> and <em>b</em> are presumably constant.

Recall that

$\cos^2t=\dfrac{1+\cos(2t)}2$

$\sin^2t=\dfrac{1-\cos(2t)}2$

so that

$\begin{cases}x=2a\sin(2t)\cos^2t\\y=2b\cos(2t)\sin^2t\end{cases}$

Then we have

$\dfrac{\mathrm dx}{\mathrm dt}=4a\cos(2t)\cos^2t-4a\sin(2t)\cos t\sin t$

$\dfrac{\mathrm dy}{\mathrm dt}=-4b\sin(2t)\sin^2t+4b\cos(2t)\sin t\cos t$

$\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{4b\cos(2t)\sin t\cos t-4b\sin(2t)\sin^2}{4a\cos(2t)\cos^2t-4a\sin(2t)\cos t\sin t}$

$\implies\boxed{\dfrac{\mathrm dy}{\mathrm dx}=\dfrac ba\tan t}$

where the last reduction follows from dividing through everything by $\cos(2t)\cos^2t$ and simplifying.

I'm not sure at which point you're supposed to evaluate the derivative (22/7*4, as in 88/7? or something else?), so I'll leave that to you.

8 0

3 years ago

What is the slope of this line?<br> y = -2x + 6

KengaRu [80]

Answer:

<h2>slope = -2</h2>

Step-by-step explanation:

The slope-intercept form of an equation of a line:

<em>y = mx + b</em>

<em>m</em><em> - slope</em>

<em>b</em><em> - y-intercept</em>

<em />

We have the equation:

<em>y = -2x + 6</em>

therefore

<em>m = -2</em>

<em>b = 6</em>

8 0

3 years ago

Bob rents bicycles in a local park. The rental fee is described as follows: A $3 flat fee plus $4 per hour. Suppose you want to

Marizza181 [45]

Answer: la respuesta es 24

Step-by-step explanation:

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2 years ago

PLEASE HELP I NEED THIS QUICK IM SO STUCK.<br>

madreJ [45]

The answer is A. You can check by seeing that for 2 lbs $5 were paid, which means one lb is 2.50

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2 years ago

2032 base four expanded notation

BabaBlast [244]

Assuming 2032 is itself a number in base 4, you have

$2032_4=2\times4^3+3\times4^1+2\times4^0$

Or, if you mean to ask about converting 2032 into base 4, you have

$\dfrac{2032}4=508$ with remainder 0, which means the "ones" digit is 0.
$\dfrac{508}4=127$ with remainder 0, which means the "tens" digit is also 0.
$\dfrac{127}4=31$ with remainder 3, so the "hundreds" digit is 3.
$\dfrac{31}4=7$ with remainder 3, so the "thousands" digit is also 3.
$\dfrac74=1$ with remainder 3, so the next digit is also 3.
$\dfrac14=0$ with remainder 1, so the next (and final) digit is 1.

So, $2032_{10}=133300_4$ .

4 0

3 years ago