The period of a physical pendulum is measured first at sea level, where the pendulum completes 100 cycles in 1550 s, and then on a mountaintop, where it comepletes 100 cycles in 1551 s. Assuming that the change in period is caused only by the variation in g with altitude, find the height of the mountain.

Question

irishboy123 · Answer

go find the equation for the period of  pendulum and you will see T is proportional to 1/ root (g) or T = A / root (g) where A = const.  you can also use Newton's grav equation F = GMm/r^2, meaning that g is inverseley proportional to distance from earth's centre or g = B / r^2 where B = const.

combining these two gives you a very simple relationship between T and r.  remembering of course that r = distance from earth's centre and not height above sea level.

anonymous · Answer

this on is hard ye? I will have to find some relation between force of gravity and length of pendulum? @IrishBoy123

irishboy123 · Answer

yes, see above

anonymous · Answer

the equation for pendulum is$$T=2\pi*\sqrt{l/g}$$?

$$g=B/r^2. Solve.for.r=> r=\sqrt{B/g}$$

and then B is amplitude?

anonymous · Answer

or is B the period? 15.5 s?

anonymous · Answer

@IrishBoy123

irishboy123 · Answer

A and B are just constants that should cancel out when you do the arithmetic.....

anonymous · Answer

could you write the equations in equations?

anonymous · Answer

@IrishBoy123

irishboy123 · Answer

$$T = \frac{A}{\sqrt{g}}$$
$$g = \frac{B}{r^{2}}$$
$$T = \frac{A}{\sqrt{{\frac{B}{r^{2}}}}} = Cr,  where C =  constant $$

Thus, T is directly proportional to r, I think.  do you?

if so we cany say T1/T2 = r1/r2 and solve that way, assuming you agree....

anonymous · Answer

I did it like this..

$$T=2\pi \sqrt{l/g}$$

solve for l

$$l=g*T^2/4\pi^2$$

$$l_1=l_2$$
so

$$g_1*T^2_1/4\pi^2=g_2*T^2_2/4\pi^2$$

and then 4pi^2 can be equalised

so we have:

$$g_1*T^2_1=g_2*T^2_2$$

solve for g_2 since g1 is  at sea level.

$$g_2=9.82m/s^2*(15.5s)^2/(15.51s)^2 = g_2=9.807m/s^2$$

from there I used
$$F=G*Mm/r^2=m*g_2$$

solve for r:

$$r=\sqrt{G*M/g_2}$$

and the answear from that is minus the earth radius?

anonymous · Answer

@IrishBoy123

anonymous · Answer

Or can I put in g2 in the l equation? in that case I get that the length is 3.85 meters?

irishboy123 · Answer

your methodology looks good to me.  i just cut out some more of the variables and thus the middle step.  i get your answer to be 6374km.  and yes, you need to subtract 6371 from that, as that is the ave radius from Wiki (reliable?!?!).  meaning the mountain is 3 km high.

my method says 15.51/15.50 = r/6371
r = 6375.11
mountain = c 4 km.


both methods get you in the ballpark and demonstrate an understanding of the physics.  this is not an inclined plane problem where the answer is exactly 2.5 or π!!

anonymous · Answer

I get my answear to 6372 km :/ maybe its calculater wrong...

anonymous · Answer

so that the mountain is 1km high :S

irishboy123 · Answer

yeah, my method doen't need G or M but it is a bit at the mercy of the radius figure you use.  i get 6379 for you method when i use the constants in my physics book.  that is probably the best place to look for them.  wiki can be all over the place.

anonymous · Answer

ye true! But what constants for G and M do you have? 

G=6.67*10^-11?
M=5.97*10^24?

@IrishBoy123