Explanation:
Hints:
Use the formula below to find the period.
![f = \frac{1}{t}](https://tex.z-dn.net/?f=f%20%3D%20%20%5Cfrac%7B1%7D%7Bt%7D%20)
Where f is the frequency in Hertz, and T is the time period in seconds.
To calculate the wavelength
use:
frequency=speed×wavelength
use the speed of sound. Make wavelength the subject of the formula and solve
Answer:
<em>0.97c</em>
<em></em>
Explanation:
From the relativistic equation for length contraction, we have
= ![l_{0}\sqrt{1 - \beta }](https://tex.z-dn.net/?f=l_%7B0%7D%5Csqrt%7B1%20-%20%5Cbeta%20%7D)
where
is the final length of the object
is the original length of the object before contraction
β = ![v^{2} /c^2](https://tex.z-dn.net/?f=v%5E%7B2%7D%20%2Fc%5E2)
where v is the speed of the object
c is the speed of light in free space = 3 x 10^8 m/s
The equation can be re-written as
/
= ![\sqrt{1 - \beta }](https://tex.z-dn.net/?f=%5Csqrt%7B1%20-%20%5Cbeta%20%7D)
For the length to contract to one-fourth of the proper length, then
/
= 1/4
substituting into the equation, we'll have
1/4 = ![\sqrt{1 - \beta }](https://tex.z-dn.net/?f=%5Csqrt%7B1%20-%20%5Cbeta%20%7D)
substituting for β, we'll have
1/4 = ![\sqrt{1 - v^2/c^2 }](https://tex.z-dn.net/?f=%5Csqrt%7B1%20-%20v%5E2%2Fc%5E2%20%7D)
squaring both side of the equation, we'll have
1/16 = 1 - ![v^2/c^2](https://tex.z-dn.net/?f=v%5E2%2Fc%5E2)
= 1 - 1/16
= 15/16
square root both sides of the equation, we have
v/c = 0.968
v = <em>0.97c</em>
Answer:
Part a)
![KE = 77.95 J](https://tex.z-dn.net/?f=KE%20%3D%2077.95%20J)
Part b)
![L = 3.16 m](https://tex.z-dn.net/?f=L%20%3D%203.16%20m)
Part c)
distance L is independent of the mass of the sphere
Explanation:
Part a)
As we know that rotational kinetic energy of the sphere is given as
![KE = \frac{1}{2}I\omega_2 + \frac{1}{2}mv^2](https://tex.z-dn.net/?f=KE%20%3D%20%5Cfrac%7B1%7D%7B2%7DI%5Comega_2%20%2B%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
so we will have
![KE = \frac{1}{2}(\frac{2}{5}mR^2)(\frac{v}{R})^2 + \frac{1}{2}mv^2](https://tex.z-dn.net/?f=KE%20%3D%20%5Cfrac%7B1%7D%7B2%7D%28%5Cfrac%7B2%7D%7B5%7DmR%5E2%29%28%5Cfrac%7Bv%7D%7BR%7D%29%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
so we will have
![KE = \frac{1}{5} mv^2 + \frac{1}{2}mv^2](https://tex.z-dn.net/?f=KE%20%3D%20%5Cfrac%7B1%7D%7B5%7D%20mv%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2)
![KE = \frac{7}{10} mv^2](https://tex.z-dn.net/?f=KE%20%3D%20%5Cfrac%7B7%7D%7B10%7D%20mv%5E2)
![KE = \frac{7}{10}(\frac{42}{9.81})(5.10^2)](https://tex.z-dn.net/?f=KE%20%3D%20%5Cfrac%7B7%7D%7B10%7D%28%5Cfrac%7B42%7D%7B9.81%7D%29%285.10%5E2%29)
![KE = 77.95 J](https://tex.z-dn.net/?f=KE%20%3D%2077.95%20J)
Part b)
By mechanical energy conservation law we know that
Work done against gravity = initial kinetic energy of the sphere
So we will have
![mgLsin\theta = KE](https://tex.z-dn.net/?f=mgLsin%5Ctheta%20%3D%20KE)
![\frac{42}{9.81}(9.81)L sin36 = 77.95](https://tex.z-dn.net/?f=%5Cfrac%7B42%7D%7B9.81%7D%289.81%29L%20sin36%20%3D%2077.95)
![L = 3.16 m](https://tex.z-dn.net/?f=L%20%3D%203.16%20m)
Part c)
by equation of energy conservation we know that
![\frac{7}{10}mv^2 = mgL sin\theta](https://tex.z-dn.net/?f=%5Cfrac%7B7%7D%7B10%7Dmv%5E2%20%3D%20mgL%20sin%5Ctheta)
so here we can see that distance L is independent of the mass of the sphere