<span>d.rotating counterclockwise and slowing down
This is a matter of understanding the notation and conventions of angular rotations. Positive rotations are counter clockwise and negative rotations are clockwise. An easy way to remember this is the "right hand rule". Make a closed fist with your right hand and have the thumb sticking outwards. If you orient your thumb such that it's pointing in the direction of the positive value along the axis, your fingers will be curled in the positive rotational direction. So in the described scenario, the sphere is rotating in the positive direction (counter clockwise) and decelerating due to the negative angular acceleration. That immediately indicates that options "a", "b", and "e" are wrong since they mention the sphere going clockwise at the beginning. Of the two remaining options "c" and "d", we can discard option "c" since it has the rotation speeding up, and that leaves us with option "d" where the sphere is rotating counter clockwise and slowing down.</span>
Answer:
magnitude: 21.6; direction: 33.7 degrees
Explanation:
When we multiply a vector by a scalar, we have to multiply each component of the vector by the scalar number. In this case, we have
vector: (-3,-2)
Scalar: -6
so the vector multiplied by the scalar will have components
![(-3\cdot (-6), -2 \cdot (-6))=(18,12)](https://tex.z-dn.net/?f=%28-3%5Ccdot%20%28-6%29%2C%20-2%20%5Ccdot%20%28-6%29%29%3D%2818%2C12%29)
The magnitude is given by Pythagorean's theorem:
![m=\sqrt{18^2+12^2}=21.6](https://tex.z-dn.net/?f=m%3D%5Csqrt%7B18%5E2%2B12%5E2%7D%3D21.6)
and the direction is given by the arctan of the ratio between the y-component and the x-component:
![\theta = tan^{-1} (\frac{12}{18})=33.7^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%20%3D%20tan%5E%7B-1%7D%20%28%5Cfrac%7B12%7D%7B18%7D%29%3D33.7%5E%7B%5Ccirc%7D)
If he’s walking at a constant velocity there is no acceleration.
If Juan used a Celsius thermometer, it would tell him the Celsius temperature.
If he added 273 to that number, he'd have the "absolute" or Kelvin temperature.
The radius of a nucleus of hydrogen is approximately
![r_{n1}=1\cdot 10^{-15}m](https://tex.z-dn.net/?f=r_%7Bn1%7D%3D1%5Ccdot%2010%5E%7B-15%7Dm)
, while we can use the Borh radius as the distance of an electron from the nucleus in a hydrogen atom:
![r_{e1}=5.3 \cdot 10^{-11}m](https://tex.z-dn.net/?f=r_%7Be1%7D%3D5.3%20%5Ccdot%2010%5E%7B-11%7Dm)
The radius of a dime is approximately
![r_{n2} = 9\cdot 10^{-3}m](https://tex.z-dn.net/?f=r_%7Bn2%7D%20%3D%209%5Ccdot%2010%5E%7B-3%7Dm)
: if we assume that the radius of the nucleus is exactly this value, then we can find how far is the electron by using the proportion
![r_{n1}:r_{e1}=r_{n2}:r_{e2}](https://tex.z-dn.net/?f=r_%7Bn1%7D%3Ar_%7Be1%7D%3Dr_%7Bn2%7D%3Ar_%7Be2%7D)
from which we find
![r_{e2}= \frac{r_{e1} r_{n2}}{r_{n1}}= \frac{(5.3 \cdot 10^{-11}m)(9\cdot 10^{-3}m)}{1 \cdot 10^{-15}m}=477 m](https://tex.z-dn.net/?f=r_%7Be2%7D%3D%20%5Cfrac%7Br_%7Be1%7D%20r_%7Bn2%7D%7D%7Br_%7Bn1%7D%7D%3D%20%5Cfrac%7B%285.3%20%5Ccdot%2010%5E%7B-11%7Dm%29%289%5Ccdot%2010%5E%7B-3%7Dm%29%7D%7B1%20%5Ccdot%2010%5E%7B-15%7Dm%7D%3D477%20m%20%20)
So, if the nucleus had the size of a dime, we would find the electron approximately 500 meters away.