Physics by Rohit: February 2013

Thursday, February 28, 2013

Analyzing orbit through differential equation (courtesy: wikipedia)

Students,

We discussed in class on how to express equation of conic sections (circle, ellipse, parabola, hyperbola) in polar coordinates. Now we can write differential equation for motion of planet around sun. Solving this differential equation we get equation of general conic section. Depending on value of constants we will end up with equation of circle/ellipse/parabola/hyperbola.

Note: All these derivations are only for reference and are not part of board or IIT-JEE syllabus. But since some of you have frequently raised queries on how we arrive at trajectory so this discussion is just to answer your query.

To analyze the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the center of force. In such coordinates the radial and transverse components of the acceleration are, respectively:

$a_r = \ddot{r}-r\dot{\theta }^2 \,$

and

$a_\theta =\frac{1}{r}\frac{d}{dt}\left( r^2 \dot \theta \right).$

Since the force is entirely radial, and since acceleration is proportional to force, it follows that the transverse acceleration is zero. As a result,

$a_\theta = 0. \,$

After integrating, we have

$r^2 \dot \theta = \text{constant} \,$

which is actually the theoretical proof of Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, h, is the angular momentum per unit mass. It then follows that

$\dot\theta =\frac{h}{r^2} = hu^2 \,$

where we have introduced the auxiliary variable

$u = { 1 \over r }.$

The radial force ƒ(r) per unit mass is the radial acceleration a_r defined above. Solving the above differential equation with respect to time^[9](See also Binet equation) yields:

$\frac{d^2u}{d\theta^2} + u = -\frac{f(1 / u)}{h^2u^2}.$

In the case of gravity, Newton's law of universal gravitation states that the force is proportional to the inverse square of the distance:

$f(1/u) = a_r = { -GM \over r^2 } = -GM u^2$

where G is the constant of universal gravitation, m is the mass of the orbiting body (planet) - note that m is absent from the equation since it cancels out, and M is the mass of the central body (the Sun). Substituting into the prior equation, we have

$\frac{d^2u}{d\theta^2} + u = \frac{ GM }{h^2}.$

So for the gravitational force — or, more generally, for any inverse square force law — the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is:

$u(\theta) = \frac{ GM }{h^2} + A \cos(\theta-\theta_0)$

where A and θ₀ are arbitrary constants.
The equation of the orbit described by the particle is thus:

$r = \frac{1}{u} = \frac{ h^2 / GM }{1 + e \cos (\theta - \theta_0)},$

where e is:

$e \equiv \frac{h^2A}{G M}.$

In general, this can be recognized as the equation of a conic section in polar coordinates (r, θ). We can make a further connection with the classic description of conic section with:

$\frac{h^2}{GM} = a(1-e^2).$

If parameter e is smaller than one, e is the eccentricity and a the semi-major axis of an ellipse.
(courtesy: wikipedia)

Monday, February 25, 2013

Result Phase Test-3 (CBSE Pattern Test) (Pinnacle 2012-14)

NAME	PHY(30)	CHEM(30)	MATHS(30)	TOTAL (90)
TRISHNA N NAIR	27.5	27	28.5	83
SAMEER KUMAR SINGH	29	26	27.5	82.5
ADITYA GOEL	25	28	28.5	81.5
APURV GUPTA	27.5	22	29.5	79
ARTH SHAH	28	27	24	79
GARGI SANDEEP VIPRA	29	28	22	79
ANAND V LANGALIA	25.5	26	26.5	78
KARAN MAKHIJA	24	25	27	76
R C SAI CHARAN	28.5	23	24	75.5
ADITYA PISHARODY	28	23	24	75
AKASH NAIR	20	28	26.5	74.5
AADITYA KAUL	24.5	19	29	72.5
AYUSH BHARDWAJ	23	21	26.5	70.5
V PRASHANT	23	28	17	68
JAYANT AGRAWAL	26	24	15.5	65.5
SANKAB SHARMA	24.5	18	22	64.5
C.V. RAMAKRISHNA	20	18	25.5	63.5
JAY P DAVE	21.5	24	15	60.5
ANAND PATEL	17	19	22	58
ABHISHEK JAIN	17.5	17	22.5	57
ROHAN POONIWALA	18	14	24	56
ISHAQ ASWAFI	21	17	14.5	52.5
SHUBHAM TIWARI	19	19	14	52
ARANYA CHAKRAVORTY	15	12	24	51
RAJAN SHARMA	22	20	9	51
HARDEE DAVE	22	6	22	50
ANUJ AGRAWAL	14.5	16	14	44.5
ANISHA GHOSH	13.5	9	17.5	40
JAY RAKESH SHAH	6.5	22	9.5	38
BEDANGA SAIKIA	11.5	15	10.5	37
AARSH A PATWA	13	9	14	36
NIRMIT SHARMA	12	11	13	36
TANAY TUSHAR SHAH	8	18	8	34
RISHABH AGRAWAL	12.5	12	9	33.5
MEET PANCHAL	7.5	11	11	29.5
PARTH PATEL	7.5	14	11	32.5
RISHIRAJ DAS	9.5	8	12	29.5
SOUMENDU GHOSH	7	10	10	27
SURAJ KUMAR PARIDA	14.5	7	3.5	25
ADITI SINGH	9.5	7	6.5	23
B.KAILASHH	7.5	4	7	18.5
ADITYA CHAUDHARY	12	3	2	17
HARSH AMBASTHA	4	6	7	17
MOHIT RAJ	6	0	9	15

Sunday, February 10, 2013

Summer Vacations for Pinnnacle (2012-14)

This is to inform all parents and students of Pinnacle 2012-14 Batch about summer vacations for coming academic session.

12th May, 2013 to 19th May, 2013 (both days inclusive)

Wednesday, February 6, 2013

Phase Test -3 Result (Pinnacle 2012-14)

(PHASE - 3)	PAPER -1				PAPER -2				Aggregate Performance
Name of the Student	CHEM 80	MATH 80	PHY 80	Total 240	CHEM 80	MATH 80	PHY 80	Total 240	Tot: 480	%	RPI
Aditya Goel	27	64	45	136	24	48	58	130	266	56	4000-4500
Arth Shah	43	43	48	134	39	31	55	125	259	54	4500-5000
Trishna N Nair	28	49	46	123	23	50	50	123	246	52	5000-5500
Akash Nair	42	3	44	89	41	27	54	122	211	44	NOT SELECTED
KARAN MAKHIJA	34	20	39	93	43	23	44	110	203	43	NOT SELECTED
Aditya Pisharody	39	17	47	103	25	18	46	89	192	40	NOT SELECTED
Jayant Agarwal	26	16	37	79	16	39	46	101	180	38	NOT SELECTED
Jay P Dave	36	30	13	79	27	30	44	101	180	38	11000-11500
Aarsh A Patwa	25	24	40	89	7	38	42	87	176	37	NOT SELECTED
Ayush Bhardwaj	39	16	33	88	17	29	36	82	170	36	NOT SELECTED
Abhishek Jain	30	21	31	82	29	32	26	87	169	36	12000-12500
Sameer Kumar Singh	37	-1	42	78	22	18	49	89	167	35	NOT SELECTED
Anisha Ghosh	34	29	30	93	23	17	32	72	165	35	NOT SELECTED
Shubham Sanjay Tiwari	19	22	37	78	17	18	49	84	162	34	NOT SELECTED
Gargi Sandeep Vipra	18	33	31	82	19	19	37	75	157	33	NOT SELECTED
V Prashant	24	7	37	68	30	25	33	88	156	33	NOT SELECTED
Rajan Sharma	26	20	33	79	11	31	33	75	154	33	NOT SELECTED
R C Sai Charan	26	14	29	69	31	14	39	84	153	32	NOT SELECTED
ROHAN POONIWALA	15	26	27	68	13	34	37	84	152	32	NOT SELECTED
C. V. Ramakrishna	13	18	33	64	7	20	51	78	142	30	NOT SELECTED
Anand V Langalia	11	5	38	54	20	22	43	85	139	29	NOT SELECTED
Aaditya Kaul	22	17	30	69	20	6	37	63	132	28	NOT SELECTED
Aranya Chakravorty	26	27	22	75	4	8	38	50	125	27	NOT SELECTED
Sankab Sharma	23	12	27	62	13	9	40	62	124	26	NOT SELECTED
Rishabh Agrawal	16	6	22	44	17	28	34	79	123	26	NOT SELECTED
Ishaq Aswafi	12	8	31	51	12	10	49	71	122	26	NOT SELECTED
Anand Patel	25	7	22	54	15	11	39	65	119	25	NOT SELECTED
Bedanga Saikia	21	27	23	71	7	10	17	34	105	22	NOT SELECTED
Mohit Raj	20	8	20	48	15	16	23	54	102	22	NOT SELECTED
Jay Rakesh Shah	31	11	9	51	5	15	30	50	101	22	NOT SELECTED
Rishiraj Das	30	6	17	53	7	9	21	37	90	19	NOT SELECTED
Hardee S Dave	16	9	7	32	12	17	25	54	86	18	NOT SELECTED
Tanay Tushar Shah	19	6	2	27	16	22	19	57	84	18	NOT SELECTED
SURAJ KUMAR PARIDA	18	3	6	27	24	2	23	49	76	16	NOT SELECTED
SOUMENDU GHOSH	15	3	15	33	11	15	13	39	72	15	NOT SELECTED
Panchal Meet	18	13	3	34	4	2	29	35	69	15	NOT SELECTED
Anuj Agrawal	10	7	8	25	9	3	29	41	66	14	NOT SELECTED
Parth Patel	17	8	13	38	2	4	19	25	63	14	NOT SELECTED
Nirmit Sharma	6	0	11	17	8	10	27	45	62	13	NOT SELECTED
Aditi Singh	8	4	8	20	2	13	17	32	52	11	NOT SELECTED
Harsh Ambastha	9	19	4	32	-1	5	15	19	51	11	NOT SELECTED
Aditya Chaudhary	1	4	0	5	3	11	17	31	36	8	NOT SELECTED
B.Kailashh	4	-2	-5	-3	6	10	19	35	32	7	NOT SELECTED
Apurv Gupta			0	0	0	0	0	0	0	0	ABSENT

Subject	Max. Marks	Difficulty Level	Cut Off Marks
Physics	160	2	54.4
Chemistry	160	3	49.6
Maths	160	3	49.6
Total	480	2.666666667

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