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  GR9277 #93
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GREPhysics.NET Official Solution    Alternate Solutions
\prob{93}
9277_93

THe figure above shows a small mass connected to a string, which is attached to a vertical post. If the mass is released when the string is horizontal as shown, the magnitude of the total acceleration of the mass as a function of the angle $\theta$ is


  1. $g\sin\theta$
  2. $2g\cos\theta$
  3. $2g\cos\theta$
  4. $g\sqrt{3\cos^2\theta+1}$
  5. $g\sqrt{3\sin^2\theta+1}$
Mechanics}Boundary Condition

Getting low on time, one should begin scoring points based more of testing strategy than sound rigorous physics. At the initial release point, the acceleration is due to gravity and the tension is 0 (no centripetal acceleration). The only choice that gives a(\theta=0) = g is choice (E).
See below for user comments and alternate solutions! See below for user comments and alternate solutions!
Alternate Solutions
jburkart
2007-11-02 00:23:15		A real way to do the problem is as follows. Kinetic energy is T=mR^2\dot{\theta}^2/2, where R is the length of the string; potential is U=-mgR\sin\theta. Setting the Lagrangian to L=T-U, Lagrange's equation is \frac{\partial L}{\partial q}=\frac{\partial}{\partial t}\frac{\partial L}{\partial\dot{q}}. In our case q=\theta and we find \ddot{\theta}=g\cos\theta/R, which then yields a_\parallel=\ddot{\theta}/R=g\cos\theta. But this is only the acceleration along the direction of motion; there is also a perpendicular centripetal acceleration given by v^2/R. To find v we can invoke conservation of energy: mv^2/2=mgR\sin\theta, so v^2=2gR\sin\theta, so the centripetal acceleration is a_{\perp}=v^2/R=2g\sin\theta.

Taking the norm of the the two acceleration components, we get a=g\sqrt{\cos^2\theta+4\sin^2\theta}=g\sqrt{1+3\sin^2\theta}, answer (E).

A common mistake would be to omit the calculation of the centripetal acceleration, which would yield an incorrect answer of g\cos\theta, which ETS kindly does not give as an option.		Alternate Solution - Unverified
Comments
elzoido238
2008-11-06 11:02:47		Answer choice C should be 2g sin\theta.NEC
Poop Loops
2008-10-25 20:22:52		Oh Jesus, people, get all this nonsense away. Calculations on the GRE? That's suicide!

Here's how you think of it:

When \theta is 0, the mass wants to go straight down, so a = g.

That's it. Only E survives that.

But, if you had a different choice of answers, you can also think that acceleration is a maximum at \theta = \frac{\pi}{2}, because it also has centripetal acceleration, which adds to the gravitational force at that point. This obviously corresponds to sin, so there you go.		NEC
ayabepaula
2008-10-22 00:36:22		time= zero

{a_{total}}=g

U=mgh
F=mg=m{a_{total}}

time

t={t_{bottom}}

{a_{total}}=2g

mgh=m\frac{v^{2}}{2}

{v^{2}}=2gh

{a_{total}}=2g=\frac{v^{2}}{h}

F=m.{a_{total}}={F_{cp}}=m{a_{cp}}=m(2g)		NEC
wangjj0120
2008-10-08 12:42:19		I am confused, why can't I use
mgsin(\theta)=T= force on string, so a_{perp}=0, and mgcos(\theta)=m(l\ddot{\theta})=ma_{para}, where l=length of string. From this, a_{para}=gcos(\theta)is correct, but a_{perp} is wrong. Obviously the concept is incorrect here, but why?

Help
jburkart
2007-11-02 00:23:15		A real way to do the problem is as follows. Kinetic energy is T=mR^2\dot{\theta}^2/2, where R is the length of the string; potential is U=-mgR\sin\theta. Setting the Lagrangian to L=T-U, Lagrange's equation is \frac{\partial L}{\partial q}=\frac{\partial}{\partial t}\frac{\partial L}{\partial\dot{q}}. In our case q=\theta and we find \ddot{\theta}=g\cos\theta/R, which then yields a_\parallel=\ddot{\theta}/R=g\cos\theta. But this is only the acceleration along the direction of motion; there is also a perpendicular centripetal acceleration given by v^2/R. To find v we can invoke conservation of energy: mv^2/2=mgR\sin\theta, so v^2=2gR\sin\theta, so the centripetal acceleration is a_{\perp}=v^2/R=2g\sin\theta.

Taking the norm of the the two acceleration components, we get a=g\sqrt{\cos^2\theta+4\sin^2\theta}=g\sqrt{1+3\sin^2\theta}, answer (E).

A common mistake would be to omit the calculation of the centripetal acceleration, which would yield an incorrect answer of g\cos\theta, which ETS kindly does not give as an option.		Alternate Solution - Unverified
kb
2007-10-28 22:49:32		is E is correct, can someone explain why
a(pi/2) = sqrt(3+1)g = 2g?
kb
2007-10-29 16:52:27		 sorry, typo

"if E is correct, can someone explain why a(pi/2) = sqrt(3+1)g = 2g?"

and should go under "help!"
jburkart
2007-11-02 00:25:23		 Because at that point the mass is experiencing a lot of g's! Try swinging in a swing sometime--you feel a lot of acceleration at the bottom.
evanb
2008-06-26 19:26:17		 At the top, the energy is mgh, and at the bottom it is .5mv^2.

This leaves us with v^2 = 2 g h, and a = v^2 / r, but r = h, so a = 2g.
NEC
rmyers
2006-12-01 13:29:01		The first four words should be edited out of this solution.
lambda
2007-10-21 22:13:29		 I agree. This test can't be finished in the time allowed if one tries to apply "rigorous physics"!
NEC
Andresito
2006-03-30 12:05:12		Thank you for the advice. :)
ayabepaula
2008-10-22 00:03:57		 1)http://dev.physicslab.org/img/db0bf4f1-2f9b-428f-81a6-536588e8a7ce.gif


2)http://www.physics-lab.net/applets/simple-pendulum




U=mgh=\frac{mv^{2}}{2}
then
{v}^{2}=2gh

Gravitational acceleration:
********radial direction***************
{a_{cp}} = \frac{v^2}{Radius=h}
{a_{cp}}=\frac{v^2}{h}
then
{a_{cp}}= \frac{v^2}{h}=-2{g_{radial}}(negative sign -->{a_{cp}} and {g_{radial} --> opposite direction.
{a_{cp}}= \frac{v^2}{h}=-2{g_{radial}}( r-direction)
{a_{cp}}=-2g(cos(90-\theta))=2g(sin90)(sin\theta)
{a_{cp}}=-2g(sin\theta)
{a_{cp}}^{2}=4{g^2}sin{^2}\theta

Gravitational acceleration
********tangential direction********
{a_{tang}}={g_{tang}} (t-direction)

{a_{tang}}={g}cos\theta

{a_{tang}^{2}}={g^{2}}{{\cos^{2}{\theta}}

********total acceleration*********
{a_{total}}=\sqrt{{{a_{cp}}^{2}}+ {a_{tang}}^{2}}
***********************************************
using
cos^2\theta+sin^{2}\theta=1

{a_{total}}=\sqrt{{g^{2}}\cos^{2}{\theta}+ 4{g^2}sin{^2}\theta}

{a_{total}}=\sqrt{{g^{2}}(1-{\sin^{2}{\theta})+ 4{g^2}sin{^2}\theta}

********
therefore:

The magnitude of the total acceleration of the mass as a function of the angle \theta is:
********
{a_{total}}=g\sqrt{3sin^{2}\theta+1}

********
Paula Ayabe Pereira
NEC

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