Answer:
a) [tex]a=33.73mm/s^{2}[/tex]
b) mg>N
c) [tex]\%_{change}=0.343\%[/tex]
d) [tex]a=24.07mm/s^{2}[/tex]
Explanation:
In order to solve part a) of the problem, we can start by drawing a free body diagram of the presented situation. (see attached picture).
In this case, we know the centripetal acceleration is given by the following formula:
[tex]a_{c}=\omega ^{2}r[/tex]
where:
[tex]\omega=\frac{2\pi}{T}[/tex]
we know the period of rotation of the earth is about 24 hours, so:
[tex]T=24hr*\frac{3600s}{1hr}=86400s[/tex]
so we can now find the angular speed:
[tex]\omega=\frac{2\pi}{86400s}[/tex]
[tex]\omega=72.72x10^{-6} rad/s^{2}[/tex]
So the centripetal acceleration will be:
[tex]a_{c} =(72.72x10^{-6} rad/s^{2})^{2}(6478x10^{3}m)[/tex]
which yields:
[tex]a_{c}=33.73mm/s^{2}[/tex]
b)
In order to answer part b, we must draw a free body diagram of us sitting on a chair. (See attached picture.)
So we can do a sum of forces in equilibrium:
[tex]\sum F=0[/tex]
so we get that:
[tex]N-mg+ma_{c} = 0[/tex]
and solve for the normal force:
[tex]N=mg-ma_{c}[/tex]
In this case, we can clearly see that:
[tex]mg>mg-ma_{c}[/tex]
therefore mg>N
This is because the centripetal acceleration is pulling us upwards, that will make the magnitude of the normal force smaller than the product of the mass times the acceleration of gravity.
c)
So let's calculate our weight and normal force:
Let's say we weight a total of 60kg, so:
[tex]mg=(60kg)(9.81m/s^{2})=588.6N[/tex]
and let's calculate the normal force:
[tex]N=m(g-a_{c})[/tex]
[tex]N=(60kg)(9.81m/s^{2}-33.73x10^{-3}m/s^{2})[/tex]
N=586.58N
so now we can calculate the percentage change:
[tex]\%_{change} = \frac{mg-N}{mg}x100\%[/tex]
so we get:
[tex]\%_{change} = \frac{588.6N-586.58N}{588.6N} x 100\%[/tex]
[tex]\%_{change}=0.343\%[/tex]
which is a really small change.
d) In order to find this acceleration, we need to start by calculating the radius of rotation at that point of earth. (See attached picture).
There, we can see that the radius can be found by using the cos function:
[tex]cos \theta = \frac{AS}{h}[/tex]
In this case:
[tex]cos \theta = \frac{r}{R_{E}}[/tex]
so we can solve for r, so we get:
[tex]r= R_{E}cos \theta[/tex]
in this case we'll use the average radius of earch which is 6,371 km, so we get:
[tex]r = (6371x10^{3}m)cos (44.4^{o})[/tex]
which yields:
r=4,551.91 km
and now we can calculate the acceleration at that point:
[tex]a=\omega ^{2}r[/tex]
[tex]a=(72.72x10^{-6} rad/s)^{2}(4,551.91x10^{3}m[/tex]
[tex]a=24.07 mm/s^{2}[/tex]
In this exercise we have to use the knowledge of mechanics to solve the magnitude of the acceleration and the acceleration of gravity, in this way we find that:
a)[tex]a=33.73 mm/s^2[/tex]
b)[tex]mg>N[/tex]
c) [tex]Change=0.343%[/tex]
d) [tex]a=24.07 mm/s^2[/tex]
Then calculating from the information given in the text;
a)In this case, we know the centripetal acceleration is given by the following formula:
[tex]a_c=w^2r\\w=\frac{2\pi}{T} \\T=24*\frac{3600}{60} = 86400 s[/tex]
so we can now find the angular speed:
[tex]w=\frac{2\pi}{86400} \\w=72.72*10^{-6} rad/s^2[/tex]
So the centripetal acceleration will be:
[tex]a_c=(72.72*10^{-6}rad/s^2)^2(6478*10^3m)\\a_c=33.73mm/s^2[/tex]
b)So we can do a sum of forces in equilibrium:
[tex]\sum F=o\\N-mg+ma_c=0\\N-mg+ma_c=0\\N=mg-ma_c\\mg>mg-ma_c\\mg>N[/tex]
This exist cause the centripetal increasing speed happen pulling united states of america upwards, that will create the size of the normal force tinier than the result or goods created of the bulk times the increasing speed of importance.
c) So let's calculate our weight and normal force:
[tex]mg=(60)(9.81)=588.6N\\N=m(g-a_c)\\N=(60)(9.81-33.73*10^{-3})\\N=586.58 N[/tex]
So now we can calculate the percentage change:
[tex]\%change= \frac{mg-N}{mg}*100\% \\=\frac{588.6-586.58}{588.6}*100\% \\=0.343\%[/tex]
d) There, we can see that the radius can be found by using the cos function:
[tex]cos\theta=\Delta S/h\\cos\theta=r/R_E\\r=R_E cos\theta\\r=(6371*10^3)cos(44.4)\\r=4,551.91 km[/tex]
And now we can calculate the acceleration at that point:
[tex]a=w^2r\\a=(72.72*10^{-6})^2(4,551.91*10^3)\\a=24.07 mm/s^2[/tex]
See more about acceleration at brainly.com/question/2437624
