1. can you explain the concepts that apply 2. identify easily confused concepts 3.how to apply into real life. 4. solve these problems---**Question 1:**
**Question Stem:**
Planets X and Y orbit the same star. The average distance between planet X and the star is five times greater than the average distance between planet Y and the star. What is the ratio orbital period of planet X / orbital period of planet Y?
**Options:**
A. $\sqrt{5^3}$
B. $\sqrt{5}$
C. $\sqrt{5^2}$
D. $\sqrt{5^3}$
**Handwritten Calculations:**
$(T_X/T_Y)^2 = (R_X/R_Y)^3$
$(T_X/T_Y)^2 = (5R_Y/R_Y)^3$
$(T_X/T_Y)^2 = 5^3$
$(T_X/T_Y)^2 = 125$
$V = 2\pi r / T$
$T = 2\pi r / V$
$T_X/T_Y = \sqrt{125}$
$T_X/T_Y = 5\sqrt{5}$
**Question 2:**
**Question Stem:**
Kepler's Third law relates the orbital period $T$ of a planet about its sun to its orbital radius $r$. The mass of the Sun is $M$. What is a correct algebraic form of the law?
**Options:**
A. $T = \frac{2\pi r^{1.5}}{(GM)^{0.5}}$
B. $T = \frac{2\pi r^{1.5}}{GM}$
C. $T = \frac{4\pi r^{0.67}}{(GM)^2}$
D. $T = \frac{4\pi r^{0.67}}{GM}$
**Handwritten Notes/Calculations:**
omit
$T^2 \propto r^3$
$T^2 = \frac{4\pi^2}{GM} r^3$
$T = \sqrt{\frac{4\pi^2 r^3}{GM}}$
$T = \frac{2\pi \sqrt{r^3}}{\sqrt{GM}}$
$T = \frac{2\pi r^{3/2}}{(GM)^{1/2}}$
$T = \frac{2\pi r^{1.5}}{(GM)^{0.5}}$
**Question 3:**
**Question Stem:**
A charged rod is brought near an initially neutral metal sphere without touching it. When the sphere is grounded (earthed), there is an electric current for a short time from the sphere to the ground. The ground connection is then removed. What are the charge on the rod and the charge induced on the sphere when the connection is removed?
**Chart/Diagram Description:**
Type: Schematic diagram illustrating electrostatic induction and grounding.
Elements:
- A charged rod (labeled "charged rod"). It is depicted as a cylindrical shape.
- A metal sphere (labeled "metal sphere"). It is depicted as a circle.
- A line connecting the bottom of the metal sphere to a standard ground symbol.
- An arrow pointing downwards from the sphere towards the ground symbol, labeled "direction of conventional current".
**Table (Options):**
| | Charge on the rod | Charge induced on the sphere |
|----|-------------------|------------------------------|
| A. | negative | negative |
| B. | negative | positive |
| C. | positive | negative |
| D. | positive | positive |
**Handwritten Notes:**
1. Initial State: A neutral metal sphere has an equal number of positive and negative charges distributed uniformly. A charged rod is brought near it.
2. Induction (Rod brought near): The image shows that when the sphere is grounded, there is an electric current from the sphere to the ground, which is the direction of conventional current. Conventional current is defined as the direction of flow of positive charge.
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Kepler's Laws describe how planets move around stars. The first law states that planets orbit in ellipses. The second law describes how planets move faster when closer to the star. The third law, which we'll focus on, relates orbital period to orbital distance: the square of the period is proportional to the cube of the orbital radius.
Let's solve Question 1 step by step. We're given that Planet X is five times farther from the star than Planet Y. We need to find the ratio of their orbital periods. Using Kepler's Third Law, the square of the period ratio equals the cube of the distance ratio. Since the distance ratio is 5, we get the period ratio squared equals 5 cubed. Taking the square root gives us the final answer: the square root of 5 cubed.
For Question 2, we need to derive Kepler's Third Law algebraically. We start by equating gravitational force with centripetal force. The gravitational force is GMm over r squared, and the centripetal force is mv squared over r. We substitute the orbital velocity formula v equals 2 pi r over T. After algebraic manipulation, we get T squared equals 4 pi squared r cubed over GM. Taking the square root gives us the final formula: T equals 2 pi r to the 1.5 power over GM to the 0.5 power.
For Question 3, we analyze electrostatic induction. When a charged rod approaches a neutral metal sphere, it causes charge separation. The diagram shows conventional current flowing from sphere to ground, meaning positive charges are flowing away from the sphere. This happens when a negatively charged rod repels electrons in the sphere toward the ground. When grounded, these electrons flow away, leaving the sphere with a net positive charge. Therefore, the rod is negative and the induced charge on the sphere is positive.
These physics concepts have many real-world applications. Kepler's laws are essential for satellite navigation and space missions. Electrostatic induction is used in pollution control and high-voltage generators. Common confusion points include mixing up Kepler's different laws and the direction of conventional versus electron current. Remember that orbital period squared is proportional to radius cubed, and that electrostatic induction separates charges without physical contact. Our final answers are: Question 1 is A, Question 2 is A, and Question 3 is B.