Answer:
A)6.15 cm to the left of the lens
Explanation:
We can solve the problem by using the lens equation:
![\frac{1}{q}=\frac{1}{f}-\frac{1}{p}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bq%7D%3D%5Cfrac%7B1%7D%7Bf%7D-%5Cfrac%7B1%7D%7Bp%7D)
where
q is the distance of the image from the lens
f is the focal length
p is the distance of the object from the lens
In this problem, we have
(the focal length is negative for a diverging lens)
is the distance of the object from the lens
Solvign the equation for q, we find
![\frac{1}{q}=\frac{1}{-16.0 cm}-\frac{1}{10.0 cm}=-0.163 cm^{-1}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bq%7D%3D%5Cfrac%7B1%7D%7B-16.0%20cm%7D-%5Cfrac%7B1%7D%7B10.0%20cm%7D%3D-0.163%20cm%5E%7B-1%7D)
![q=\frac{1}{-0.163 cm^{-1}}=-6.15 cm](https://tex.z-dn.net/?f=q%3D%5Cfrac%7B1%7D%7B-0.163%20cm%5E%7B-1%7D%7D%3D-6.15%20cm)
And the sign (negative) means the image is on the left of the lens, because it is a virtual image, so the correct answer is
A)6.15 cm to the left of the lens
To solve this problem it is necessary to apply the kinematic equations of angular motion.
Torque from the rotational movement is defined as
![\tau = I\alpha](https://tex.z-dn.net/?f=%5Ctau%20%3D%20I%5Calpha)
where
I = Moment of inertia
For a disk
Angular acceleration
The angular acceleration at the same time can be defined as function of angular velocity and angular displacement (Without considering time) through the expression:
![2 \alpha \theta = \omega_f^2-\omega_i^2](https://tex.z-dn.net/?f=2%20%5Calpha%20%5Ctheta%20%3D%20%5Comega_f%5E2-%5Comega_i%5E2)
Where
Final and Initial Angular velocity
Angular acceleration
Angular displacement
Our values are given as
![\omega_i = 0 rad/s](https://tex.z-dn.net/?f=%5Comega_i%20%3D%200%20rad%2Fs)
![\omega_f = 450rev/min (\frac{1min}{60s})(\frac{2\pi rad}{1rev})](https://tex.z-dn.net/?f=%5Comega_f%20%3D%20450rev%2Fmin%20%28%5Cfrac%7B1min%7D%7B60s%7D%29%28%5Cfrac%7B2%5Cpi%20rad%7D%7B1rev%7D%29)
![\omega_f = 47.12rad/s](https://tex.z-dn.net/?f=%5Comega_f%20%3D%2047.12rad%2Fs)
![\theta = 3 rev (\frac{2\pi rad}{1rev}) \rightarrow 6\pi rad](https://tex.z-dn.net/?f=%5Ctheta%20%3D%203%20rev%20%28%5Cfrac%7B2%5Cpi%20rad%7D%7B1rev%7D%29%20%5Crightarrow%206%5Cpi%20rad)
![r = 7cm = 7*10^{-2}m](https://tex.z-dn.net/?f=r%20%3D%207cm%20%3D%207%2A10%5E%7B-2%7Dm)
![m = 17g = 17*10^{-3}kg](https://tex.z-dn.net/?f=m%20%3D%2017g%20%3D%2017%2A10%5E%7B-3%7Dkg)
Using the expression of angular acceleration we can find the to then find the torque, that is,
![2\alpha\theta=\omega_f^2-\omega_i^2](https://tex.z-dn.net/?f=2%5Calpha%5Ctheta%3D%5Comega_f%5E2-%5Comega_i%5E2)
![\alpha=\frac{\omega_f^2-\omega_i^2}{2\theta}](https://tex.z-dn.net/?f=%5Calpha%3D%5Cfrac%7B%5Comega_f%5E2-%5Comega_i%5E2%7D%7B2%5Ctheta%7D)
![\alpha = \frac{47.12^2-0^2}{2*6\pi}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7B47.12%5E2-0%5E2%7D%7B2%2A6%5Cpi%7D)
![\alpha = 58.89rad/s^2](https://tex.z-dn.net/?f=%5Calpha%20%3D%2058.89rad%2Fs%5E2)
With the expression of the acceleration found it is now necessary to replace it on the torque equation and the respective moment of inertia for the disk, so
![\tau = I\alpha](https://tex.z-dn.net/?f=%5Ctau%20%3D%20I%5Calpha)
![\tau = (\frac{1}{2}mr^2)\alpha](https://tex.z-dn.net/?f=%5Ctau%20%3D%20%28%5Cfrac%7B1%7D%7B2%7Dmr%5E2%29%5Calpha)
![\tau = (\frac{1}{2}(17*10^{-3})(7*10^{-2})^2)(58.89)](https://tex.z-dn.net/?f=%5Ctau%20%3D%20%28%5Cfrac%7B1%7D%7B2%7D%2817%2A10%5E%7B-3%7D%29%287%2A10%5E%7B-2%7D%29%5E2%29%2858.89%29)
![\tau = 0.00245N\cdot m \approx 2.45*10^{-3}N\cdot m](https://tex.z-dn.net/?f=%5Ctau%20%3D%200.00245N%5Ccdot%20m%20%5Capprox%202.45%2A10%5E%7B-3%7DN%5Ccdot%20m%20)
Therefore the torque exerted on it is ![2.45*10^{-3}N\cdot m](https://tex.z-dn.net/?f=2.45%2A10%5E%7B-3%7DN%5Ccdot%20m)
Answer:
a
The hiker (you ) is 200 m below his/her(your) starting point
b
The resultant displacement in the north east direction is
![a = 6562.0 \ m](https://tex.z-dn.net/?f=a%20%20%3D%206562.0%20%5C%20%20m%20)
The resultant displacement in vertical direction (i.e the altitude change )
![b =6503.1 \ m](https://tex.z-dn.net/?f=b%20%3D6503.1%20%5C%20%20m%20)
Explanation:
From the question we are told that
The displacement in the morning is ![S_{morning} = (2200 \m , east) + (4000\ m\ north) + (100 \ m ,\ vertical)](https://tex.z-dn.net/?f=S_%7Bmorning%7D%20%3D%20%20%282200%20%5Cm%20%2C%20east%29%20%2B%20%284000%5C%20m%5C%20north%29%20%2B%20%28100%20%5C%20m%20%2C%5C%20vertical%29)
The displacement in the afternoon is ![S _{afternoon}= (1300\ m ,\ west) + (2500 \ m ,\ north) - (300\ m ,\ vertical)](https://tex.z-dn.net/?f=S%20_%7Bafternoon%7D%3D%20%281300%5C%20m%20%2C%5C%20west%29%20%2B%20%282500%20%5C%20m%20%2C%5C%20north%29%20-%20%28300%5C%20m%20%2C%5C%20vertical%29)
Generally the direction west is negative , the direction east is positive
the direction south is negative , the direction north is positive
resultant displacement is mathematically evaluated as
![(2200 \m , east) +( - 1300\ m ,\ west) = 900 \ m \ east](https://tex.z-dn.net/?f=%282200%20%5Cm%20%2C%20east%29%20%2B%28%20-%201300%5C%20m%20%2C%5C%20west%29%20%3D%20900%20%5C%20m%20%5C%20east)
![(4000\ m\ north) + (2500 \ m ,\ north) = 6500 \ m ,\ north](https://tex.z-dn.net/?f=%284000%5C%20m%5C%20north%29%20%20%2B%20%282500%20%5C%20m%20%2C%5C%20north%29%20%3D%206500%20%20%5C%20m%20%2C%5C%20north)
![(100 \ m ,\ vertical) - (300\ m ,\ vertical) = -200 \ m](https://tex.z-dn.net/?f=%28100%20%5C%20m%20%2C%5C%20vertical%29%20-%20%28300%5C%20m%20%2C%5C%20vertical%29%20%3D%20-200%20%5C%20m)
From the above calculation we see that at the end of the hiking the hiker (you) is 200 m below his/her(your) initial position
Generally from Pythagoras theorem , the resultant displacement in the north east direction is
![a = \sqrt{900^2 + 6500^2}](https://tex.z-dn.net/?f=a%20%20%3D%20%20%5Csqrt%7B900%5E2%20%2B%206500%5E2%7D)
=> ![a = 6562.0 \ m](https://tex.z-dn.net/?f=a%20%20%3D%206562.0%20%5C%20%20m%20)
Generally from Pythagoras theorem , the resultant displacement in vertical direction (i.e the altitude change )
![b = \sqrt{6500^2 +(-200)^2 }](https://tex.z-dn.net/?f=b%20%3D%20%5Csqrt%7B6500%5E2%20%2B%28-200%29%5E2%20%20%7D)
=> ![b =6503.1 \ m](https://tex.z-dn.net/?f=b%20%3D6503.1%20%5C%20%20m%20)
Answer:
Any scientific investigation in its simplest form involves collecting information, while a controlled experiment is a tightly controlled investigation that tests a hypothesis.
Explanation:
All the scientific investigations start with a particular question. The question decides what type of investigation should research take. A controlled experiment is an investigation which allows a particular variable to change and all other variable controlled to remain constant to see the effect on dependent variable to test a particular hypothesis while other types of investigations are used to increase the knowledge or information about the particular research subject.