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7.2E: Right Triangle Trigonometry (Exercises) - Mathematics


For the following exercises, use side lengths to evaluate.

17. (cos frac{pi}{4})

18. (cot frac{pi}{3})

19. ( an frac{pi}{6})

20. (cos left(frac{pi}{2} ight)=sin left(longrightarrow^{circ} ight))

21. (csc left(18^{circ} ight)=sec left(longrightarrow^{circ} ight))

For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.

22. (cos B=frac{3}{5}, a=6)

23. ( an A=frac{5}{9}, b=6)

For the following exercises, use Figure 1 to evaluate each trigonometric function.

Figure 1

24. (sin A)

25. ( an B)

For the following exercises, solve for the unknown sides of the given triangle.

26.

27.

28. A 15-ft ladder leans against a building so that the angle between the ground and the ladder is (70^{circ} .) How high does the ladder reach up the side of the building? Find the answer to four decimal places.

29. The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. Find the answer to four decimal places.


Representation of the Reynolds stress tensor through quadrant analysis for a near-neutral atmospheric surface layer flow

To disseminate the role of the eddy motions on the anisotropic states of the Reynolds stress tensor ( (varvec) ), we devise a novel methodology based on the quadrant analysis, where the distributions of the invariants of (varvec) are studied on a (u^) – (w^) quadrant plane, with (u^) and (w^) being the turbulent fluctuations in the streamwise and vertical velocities. We apply this methodology to a near-neutral atmospheric surface layer (ASL) flow, derived from a field experiment dataset having multi-level turbulence measurements. The results show that in a near-neutral ASL flow, the anisotropic states of (varvec) are determined by the distribution of the streamwise and cross-stream velocity variances ( (sigma _^2) and (sigma _^2) ) on the (u^) – (w^) quadrant plane. By studying the contour maps of the invariants of (varvec) on the (u^) – (w^) quadrant plane, we discover three distinct zones with elliptical boundaries in the (u^) – (w^) plane, across which the anisotropic states of the eddy motions evolve. We find that the eddy motions which occur

Inside the inner elliptical zone display rod-like anisotropy being determined by (sigma _^2) ,

Within the annular zone between the inner and outer ellipses display pancake-like anisotropy being determined by both (sigma _^2) and (sigma _^2) ,

Outside the outer elliptical zone display rod-like anisotropy being determined by (sigma _^2) .

We also notice that the distinction between these three zones in the (u^) – (w^) plane is prominent at the lowest measurement level, but becomes progressively indistinguishable as the measurement height increases.

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DVMs for Polyatomic Molecules

We consider a single species with polyatomic molecules, in the sense that each particle has an associated internal energy [12, 19, 29]. The total number of possible internal energies is assumed to be finite, i.e., there are s different internal energies (E^<1>,ldots ,E^) that can be associated with the particles, which either can be considered as that there only is a finite number of different (internal) energy states or that one (in some way) have modelled a continuous internal energy variable by discretizing it, cf. [21]. We model the Boltzmann equation by discretizing the continuous velocity variable, i.e., we assume that the velocity only can take a finite number of different vector values. In order to do the discretization, we fix a set of velocity vectors (V_=< mathbf _<1>^,ldots ,mathbf _<>>^> subset mathbb ^) (in applications (d=2,,3) ) for each of the internal energies (E^.) Now there are (n=n_<1>+cdots +n_) different pairs, being composed of a velocity vector and an internal energy, contained in the set

Note that the same velocity can appear many times in the set, while the pairs are unique. An important question is how to choose the sets of velocities, in such a way that there will be no extra, so called spurious, collision invariants, in plus to the physical ones. Our main concern is how the velocity sets can be constructed, such that the resulting DVM will be normal, i.e., without spurious collision invariants. Explicit implementations will not be considered here, but we refer to e.g., [21, 28] for some interesting examples. It would also be of interest to see if the results in [20] could be extended to include also (mixtures and/or) polyatomic molecules.

The general DVM, or the discrete Boltzmann equation, for a single species with polyatomic molecules reads

where (f_=f_( mathbf ,,t) =f( mathbf ,,t,,mathbf _,,E_) ) for (i=1,ldots ,n,) and (f=f( mathbf ,,t,,mathbf ,,E) ) represents the microscopic density of particles with internal energy E and velocity (mathbf ) at time (tin mathbb _<+>) and position (mathbf in mathbb ^.)

For a function (h=h(mathbf ,,E)) (possibly depending on more variables than (mathbf ) and E), we identify h with its restrictions to the pairs (( mathbf _,,E_) in mathscr ,) i.e.,

The collision operators (Q_( f,,f) ) in (1) are given by

where we assume that the collision coefficients satisfy the symmetry relations

with equality unless we have conservation of momentum and total energy

Furthermore, we assume that (Gamma _^=0) if

This assumption was not considered in [7, 8]. However, even if particles with the same velocity, by obvious reasons, will never collide in reality (neither we assume molecules to clog together during a collision), such reactions can still appear as combinations of collisions involving other velocities (belonging to the considered velocity sets, or not). Moreover, they can also be chosen as a represent for collisions of molecules with velocities close to that “same” velocity (however, they might still be rare enough to neglect). Therefore, in principle, it would be possible to allow also these collisions as in [7, 8], even if we do not choose to do it here. Anyway, these reactions might still appear implicitly as combinations of allowed collisions.

To obtain the symmetry relations (3) we may need to scale the distribution functions first (cf. [19])

After a possible scaling (6) of the distribution functions, the symmetry relations (3) are a natural assumption, assuming a convenient reciprocity relation, see, e.g., [25, p. 9]. We like to point out that the symmetry relations (3) are fulfilled for the proposed models in [21] (with (g_<1>^=( E_<1>^) ^,) where (delta ) is the number of internal degrees of freedom).

A collision is obtained by the exchange of velocities and internal energies

and can occur if and only if (Gamma _^ e 0.) Geometrically, the collision obtained by (7) is, as in the case of single species, represented by a rectangle in (mathbb ^,) if (< E_,,E_> =< E_,,E_>, ) with the corners in ( < mathbf _,,mathbf _,,mathbf _,,mathbf _>, ) where (mathbf _) and (mathbf _) (and therefore, also (mathbf _) and (mathbf _) ) are diagonal corners. In general, the collision obtained by (7) is geometrically represented by a parallelogram in (mathbb ^) [16], with the corners in ( < mathbf _,,mathbf _,,mathbf _,,mathbf _>, ) where (mathbf _) and (mathbf _) (and therefore, also (mathbf _) and (mathbf _) ) are diagonal corners, such that

Note that the parallelogram is allowed to be degenerate, in the sense that it has no hight (or area), i.e., all corners of the parallelogram lie on the same line.

A function (phi =phi (mathbf ),) is a (global) collision invariant if and only if

for all indices such that (Gamma _^ e 0.) We note that in general a collision invariant can also depend on time and position. However, that dependence is of no importance for our studies here and therefore, we will below assume that all collision invariants are global, i.e., that they are independent of time and position. The vector space of collision invariants contains the trivial collision invariants (or physical collision invariants)

where (v^=(v_<1>^,ldots ,v_^)) (here we denote (mathbf _=( v_^<1>,ldots ,v_^) ) ), (vert mathbf vert ^<2>=(vert mathbf _<1>vert ^<2>,ldots ,vert mathbf _vert ^<2>),) and (E=E(mathbf )=(E_<1>,ldots ,E_).)

In the continuous case, these are the only collision invariants. However, for DVMs there can also be extra, so called spurious, collision invariants. DVMs without spurious collision invariants are called normal, if the (d+2) collision invariants (8) are linearly independent. A DVM such that the collision invariants (8) are linearly dependent are called degenerate, and otherwise non-degenerate. Methods of construction of normal DVMs for monatomic single species and mixtures can be found in e.g., [10, 15, 16]. We again stress that the collision invariants include, and for normal models are restricted to

for some constant (a,,cin mathbb ,) (mathbf in mathbb ^.) Our main interest in this work is the construction of such normal models.

The stationary points are Maxwellian distributions (or just Maxwellians) of the form

where (for normal models) (phi ) is given by Eq. (9).

Note also that under the assumptions above we will have an H-theorem as usual, cf. [8].

Supernormal DVMs for Polyatomic Molecules

For DVMs for polyatomic molecules, one can, as in the case of DVMs for mixtures cf. [10, 16] have different kinds of normality. Similarly as in the case of mixtures in [10], we introduce different kinds or levels of normality. We start with the usual definition of normality.

Definition 1

with internal energies (< E^<1>,ldots ,E^> ,) is called normal if it is non-degenerate and has exactly (d+2) linearly independent collision invariants.

Note that for normal DVMs, the (d+2) linearly independent collision invariants in Definition 1 will be linear combinations of the (d+2) trivial collision invariants (8).

A drawback with Definition 1 is that if we look separately on the restriction to a specific energy level, the reduced model does not have to be normal. Therefore, we extend the definition above.

Definition 2

A DVM (11), with internal energies (< E^<1>,ldots ,E^>, ) is called semi-supernormal if it is normal and the restriction to each velocity set (mathrm _,) (1le ile s,) is a normal DVM.

However, still if we consider subsets of energy levels, the restrictions to those energy levels do not have to be normal. Therefore, we make a further extension.

Definition 3

A DVM (11), with internal energies (< E^<1>,ldots ,E^>, ) is called supernormal if the restriction to each collection

Depending on what we are interested to study we can be satisfied with different levels of normality, where normal is the lowest level and supernormal the highest one (including the other ones).

As we construct semi-supernormal DVMs we can be helped by the following theorem.

Theorem 1

A DVM (11), with internal energies ( < E^<1>,ldots ,E^>, ) is semi-supernormal if, for each (2le jle s) there exists (1le i<jle s,) such that the restriction to the pair (< < mathrm _,,E^> ,, < mathrm _,,E^> > ) is a supernormal DVM.

Proof

The restriction to each velocity set (mathrm _= < mathbf _<1>^,ldots ,mathbf _<>>^> ,) (1le ile s,) is normal by the supernormality of (< < mathrm _,,E^> ,,< mathrm _,,E^> > .) Hence, the collision invariants will be of the form (phi =( phi ^<1>,ldots ,phi ^), ) where (phi _^=a_+mmathbf ^cdot mathbf _^+c_( mvert mathbf _^vert ^<2>+2E^) ) for (1le jle n_) and (1le ile s.)

Denote (a_<1>=a,) (mathbf ^>mathbf <=b>,) and (c_<1>=c.) Assume that (a_=a_=cdots =a_<1>=a) , (mathbf ^mathbf <=b>^ mathbf <=cdots =b>^>mathbf <=b>,) and (c_=c_=cdots =c_<1>=c) for some (2le jle s.) Then there exists (1le ile j-1,) such that the restriction to the pair (<< mathrm _,,E^> ,,< mathrm _,,E^> > ) is normal and therefore ( a_=a_=a,) (mathbf ^mathbf <=b>^mathbf <=b>,) and ( c_=c_=c.) Hence, the collision invariants will be of the form (phi =( phi ^<1>,ldots ,phi ^<>>), ) where (phi _^=a+m mathbf _^+c( mvert mathbf _^vert ^<2>+2E^) ) for (1le jle n_) and (1le ile s.) (square )

For constructing supernormal DVMs (or checking if existing DVMs are supernormal), the following theorem can be useful.

Theorem 2

A DVM (11), with internal energies (< E_<1>,ldots ,E_>, ) is supernormal if and only if the restriction to each pair

of velocity sets is a supernormal DVM.

Proof

The theorem follows directly from the definition of supernormal DVMs and Theorem 1. (square )

Algorithms for Construction of Semi-supernormal and Supernormal DVMs for Polyatomic Molecules

We will below use the concept of “linearly independent” collisions [10]. Intuitively, a set of collisions is linearly dependent if one of them can be obtained by a combination of (some of) the other collisions (including corresponding reverse collisions), and correspondingly linearly independent if this is not the case. More formally, each collision can be represented by an n-dimensional vector with 0, (-1,) and 1 as the only coordinates, see, e.g., [16, 18], in the way that collision (7) is represented by a vector (with non-zero elements at the positions ( i,,j,,k,) and l)

We then say that a set of collisions is linearly independent if and only if the set of the corresponding vectors is linearly independent. Furthermore, all vectors for possible collisions, i.e., collisions such that the collision coefficient (Gamma _^) (3) is nonzero, (Gamma _^ e 0,) can be put as rows in a matrix of collisions (Lambda ) (in fact, it is enough that the vectors for all linearly independent collisions are included) [16, 18]. Then the kernel of (Lambda ) will be equal to the vector space of collisions invariants [16, 18], and hence, the number of collision invariants is given by

Therefore, for a normal DVM we need to have [16, 18]

Note that by the conditions on the collision coefficient (Gamma _^) (3), we will always have the trivial collision invariants (8), and hence,

We will now present a possible strategy for constructing (semi-)supernormal DVMs for polyatomic molecules.

Algorithm for construction of semi-supernormal DVMs for polyatomic molecules

Choose a set of velocities (mathrm _<1>) such that it corresponds to a normal DVM for a monatomic species. Here, and in all the steps below, the set should be chosen in such a way, that we can obtain normal models for any mass ratio and/or energy levels we intend to consider otherwise we might also be able to extend the set(s) later, as we realize that it is needed.

Iteration step. For (j=2,ldots ,s) :

Choose a set of velocities (mathrm _) [in a similar way as in step (1)] such that, there is (1le i<j,) such that

Remark 1

If we do not allow any collisions between the two levels of internal energies, we will have exactly (2d+4) linearly independent collision invariants, since, if (n_) and (n_) are the number of velocities in (mathrm _) and (mathrm _,) respectively, the rank of the matrix of collisions (Lambda ) will be

By Definition 1, we would like to have (d+2) linearly independent collision invariants. Hence, we need to have (d+2) linearly independent collisions between the two levels of internal energies. Hence, if, as in the case of all the (explicitly and implicitly) constructed DVMs in Sect. 3 (but, also for many others), there are (d+1) linearly independent elastic collisions, we will only need to find one “basal” inelastic collision, if we have a maximal number of linearly independent elastic collisions. In fact, then by inequality (13) there will be at most one linearly independent inelastic collision with respect to the elastic collisions. This will not at all mean that we do not have more inelastic collisions, since a certain number of other inelastic collisions can be obtained by combining the basal one with elastic collisions (collisions represented by linear combinations of the vectors of collisions of the basal inelastic collision and elastic collisions). However, the obtained inelastic collisions will be linearly dependent with the elastic collisions and the basal inelastic one.

Note that in the spirit of Remark 1 we will have satisfactorily many inelastic collisions as long as we have satisfactorily many elastic collisions.

Algorithm for construction of supernormal DVMs for polyatomic molecules

Choose a set of velocities (mathrm _<1>) such that it corresponds to a normal DVM for a monatomic species. As above, here, and in all the steps below, the set should be chosen in such a way, that we can obtain normal models for any mass ratio and/or energy levels we intend to consider otherwise we might also be able to extend the set(s) later, as we realize that it is needed.

Iteration step. For (j=2,ldots ,s) :

Choose a set of velocities (mathrm _) [in a similar way as in step (1)] such that

is a normal DVM for each (1le i<j.) Also here, Remark 1 is applicable, in all cases.


March Meeting 2018 – Days 2 to 4!

The March Meeting is always so exciting — there is so much information here!

Graphene origami and micron-sized laser controlled robots at Marc Miskin’s talk on Tuesday morning. SO COOL!

On Tuesday morning, I went to an outstanding session on Atomic Origami. There is some truly amazing work out there with people designing shapes of graphene (mostly) that fold up on their own into boxes or flowers. Post-doc Marc Miskin gave a really inspiring talk, including showing a little piece of graphene that folds itself up into a triangle only 15 microns on a side — reversibly and repeatedly! Harvard professor David Nelson (who gave a colloquium at Wooster just a few years ago) also gave a wonderfully dynamic talk on criticality and crumpling of paper.

Food trucks and extremely long, slow lines in the sun. Good food though.

Emma Brinton 󈧖 gave her poster in the afternoon, and got good interest from the crowd.

Meanwhile Justine Walker 󈧖 and I went to a session on the life and legacy of Millie Dresselhaus. Millie was an absolutely outstanding person and physicist, and the first woman to do so many things. I knew of Millie, of course, but learned so much more about her. I also got to catch up with my Ph.D. advisor Laurie McNeil, who was chairing this session.

Spot the Wooster physicists in the crowd waiting for the graphene superconductivity talk. Hint: look for Michelle’s backpack.

Day 3, Wednesday, started off with a buzz of excitement around an invited talk about graphene and a new discovery of superconductivity. This was actually really interesting since I learned a good bit about graphene and carbon compounds in general the day before at the Millie Dresselhaus session, since she was a tremendous pioneer with carbon (and is known as the Queen of Carbon). I don’t know if everyone was already excited about the talk, but the APS sent out an email basically saying “this talk is so cool we’re projecting it in the cafe”, so then of course everyone came. We had seats at a table, but moved to the balcony when too many people stood in front of us. It was interesting, but I’m not sure if it’s quite the Woodstock of physics event we were hoping for.

Andrew Blaikie 󈧑 quizzes Chase Fuller 󈧗 about BZ waves.

Our remaining four students presented their posters and did very well, and had fun gathering up free toys from the vendors in the exhibit hall.

Time for a group photo, and a couple more sessions before a big group dinner. This is always a highlight of the trip (as long as the scheduling works) and this time we got to include alumnus Andrew Blaikie 󈧑 who is finishing grad school at the University of Oregon.

Wooster physics out for dinner, with bonus alumnus Andrew Blaikie 󈧑!

Whew! So much more that I could say, but frankly I’m exhausted. On Thursday, Avi Vajpeyi 󈧖 presented his bead pile simulation, and I presented the latest experimental bead pile results later in the same session.

Me, taking credit for Gabe’s excellent work

As I said at the start, the March Meeting is exhilarating, but the counterpoint is that it is exhausting. We are starting to hit brain overload here, but fortunately things are winding down. The weather has been lovely, so here’s a final image from the atrium at the convention center. I notice reflections all around, and I liked the blue sky with the reflections in the floor.

Sunlight and reflection in the atrium


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2 OBSERVATION AND DATA REDUCTION

The analysis is based on photometry from the Kepler space telescope (Borucki et al. 2010 Gilliland et al. 2010 Jenkins et al. 2010a, 2010b Koch et al. 2010). The data set is 775 d long, observed in six quarters (Q1–Q6) at long cadence (LC time resolution of 29.4 min) and three quarters (Q7–Q9) at short cadence (SC time resolution of 58.9 s). Since HD 181068 is a ∼7 mag star, it is heavily saturated, resulting in charge bleeding. Therefore, the SC observations were obtained using a custom made aperture mask. This was uploaded directly to the spacecraft lookup table and shaped precisely to match the shape of the target on the detector including the bleeding area.

2.1 Measuring the times of minima

The 2.1-yr-long observations cover ∼885 orbital cycles of the close pair and 17 revolutions of the wide system. Approximately 10 per cent of the eclipses of the close binary (hereafter we refer to them as shallow minima) occur during the eclipse events of the wide system (hereafter deep minima) and cannot be observed. Additionally, a few hundred events escaped observation due to data gaps. In all, 1177 of the 1770 shallow minima were analysed. The analysis of these minima was quite a complex task. As shown by Derekas et al. ( 2011), the red giant component shows oscillations on a time-scale similar to the half of the orbital period of the short-period binary. In addition, there are long-term variations, discussed in Section 4, which slightly distort the shape of the shallow minima, as shown in Fig. 1. This distortion has a significant effect on the measurement of the exact times of minima.

Example of a primary (upper panel) and a secondary (lower panel) shallow minimum to illustrate the states before (black triangles) and after (red crosses) the detrending of the minima. The dashed line is the fit used for the detrending (see Section 2.1).

Example of a primary (upper panel) and a secondary (lower panel) shallow minimum to illustrate the states before (black triangles) and after (red crosses) the detrending of the minima. The dashed line is the fit used for the detrending (see Section 2.1).

To correct for these distortions, we applied the following method in determining the times of minima. We took the ±0.225 d interval around each minimum and fitted low-order (4–6) polynomials outside the eclipses. Then we corrected each subset, which resulted in a detrended light curve. Finally, to determine the times of minima, we fitted low-order (5 and 6) polynomials to the lowest parts of the minima.

We also analysed the available deep minima. Out of the 34 events we were able to determine times of minima in 28 cases. (One of these events was omitted from the final analysis, due to its large deviation from the general trend of the data, which might be caused by its incomplete sampling.) To determine these times of minima, first we removed the effects of the intrinsic brightness variations from the light curves, and then fitted each outer transit and occultation event individually with our newly developed simultaneous light-curve solution code. Both the code and the complete light-curve analysis are described in Section 4.

The determined times of minima are listed in Tables 1 and 2 for the close and the wide pairs, respectively.

A sample of times of minima for the close pair (the whole table is published only electronically see the Supporting Information).

BJD . σ . Type . BJD . σ . Type . BJD . σ . Type . BJD . σ . Type .
245 4963.8399 0.0010 II 245 4994.6312 0.0010 II 245 5101.5010 0.0010 II 245 5132.2946 0.0010 II
245 4964.2926 0.0010 I 245 4995.0838 0.0010 I 245 5101.9551 0.0010 I 245 5132.7470 0.0010 I
245 4965.1967 0.0010 I 245 4995.5359 0.0010 II 245 5102.4099 0.0010 II 245 5133.1999 0.0010 II
245 4965.6478 0.0010 II 245 4995.9891 0.0010 I 245 5102.8605 0.0010 I 245 5133.6511 0.0010 I
245 4966.1021 0.0010 I 245 4996.4429 0.0010 II 245 5103.3130 0.0010 II 245 5134.1046 0.0010 II
BJD . σ . Type . BJD . σ . Type . BJD . σ . Type . BJD . σ . Type .
245 4963.8399 0.0010 II 245 4994.6312 0.0010 II 245 5101.5010 0.0010 II 245 5132.2946 0.0010 II
245 4964.2926 0.0010 I 245 4995.0838 0.0010 I 245 5101.9551 0.0010 I 245 5132.7470 0.0010 I
245 4965.1967 0.0010 I 245 4995.5359 0.0010 II 245 5102.4099 0.0010 II 245 5133.1999 0.0010 II
245 4965.6478 0.0010 II 245 4995.9891 0.0010 I 245 5102.8605 0.0010 I 245 5133.6511 0.0010 I
245 4966.1021 0.0010 I 245 4996.4429 0.0010 II 245 5103.3130 0.0010 II 245 5134.1046 0.0010 II

A sample of times of minima for the close pair (the whole table is published only electronically see the Supporting Information).

BJD . σ . Type . BJD . σ . Type . BJD . σ . Type . BJD . σ . Type .
245 4963.8399 0.0010 II 245 4994.6312 0.0010 II 245 5101.5010 0.0010 II 245 5132.2946 0.0010 II
245 4964.2926 0.0010 I 245 4995.0838 0.0010 I 245 5101.9551 0.0010 I 245 5132.7470 0.0010 I
245 4965.1967 0.0010 I 245 4995.5359 0.0010 II 245 5102.4099 0.0010 II 245 5133.1999 0.0010 II
245 4965.6478 0.0010 II 245 4995.9891 0.0010 I 245 5102.8605 0.0010 I 245 5133.6511 0.0010 I
245 4966.1021 0.0010 I 245 4996.4429 0.0010 II 245 5103.3130 0.0010 II 245 5134.1046 0.0010 II
BJD . σ . Type . BJD . σ . Type . BJD . σ . Type . BJD . σ . Type .
245 4963.8399 0.0010 II 245 4994.6312 0.0010 II 245 5101.5010 0.0010 II 245 5132.2946 0.0010 II
245 4964.2926 0.0010 I 245 4995.0838 0.0010 I 245 5101.9551 0.0010 I 245 5132.7470 0.0010 I
245 4965.1967 0.0010 I 245 4995.5359 0.0010 II 245 5102.4099 0.0010 II 245 5133.1999 0.0010 II
245 4965.6478 0.0010 II 245 4995.9891 0.0010 I 245 5102.8605 0.0010 I 245 5133.6511 0.0010 I
245 4966.1021 0.0010 I 245 4996.4429 0.0010 II 245 5103.3130 0.0010 II 245 5134.1046 0.0010 II

Times of minima for the wide system.

BJD . Cycle number a . BJD . Cycle number a .
245 4977.0831 −11.5 245 5363.5693 −3.0
245 5022.5375 −10.5 245 5386.3163 −2.5
245 5045.2970 −10.0 245 5409.0662 −2.0
245 5068.0335 −9.5 245 5431.7818 −1.5
245 5113.5169 −8.5 245 5454.5345 −1.0
245 5136.2170 −8.0 245 5477.2681 −0.5
245 5158.9550 −7.5 245 5499.9950 0.0
245 5204.4405 −6.5 245 5545.4559 1.0
245 5227.1669 −6.0 245 5590.9390 2.0
245 5249.9048 −5.5 245 5613.6734 2.5
245 5272.6355 −5.0 245 5659.1425 3.5
245 5295.3893 −4.5 245 5681.8955 4.0
245 5318.1113 −4.0 245 5704.6063 4.5
245 5340.8384 −3.5 245 5727.3559 5.0
BJD . Cycle number a . BJD . Cycle number a .
245 4977.0831 −11.5 245 5363.5693 −3.0
245 5022.5375 −10.5 245 5386.3163 −2.5
245 5045.2970 −10.0 245 5409.0662 −2.0
245 5068.0335 −9.5 245 5431.7818 −1.5
245 5113.5169 −8.5 245 5454.5345 −1.0
245 5136.2170 −8.0 245 5477.2681 −0.5
245 5158.9550 −7.5 245 5499.9950 0.0
245 5204.4405 −6.5 245 5545.4559 1.0
245 5227.1669 −6.0 245 5590.9390 2.0
245 5249.9048 −5.5 245 5613.6734 2.5
245 5272.6355 −5.0 245 5659.1425 3.5
245 5295.3893 −4.5 245 5681.8955 4.0
245 5318.1113 −4.0 245 5704.6063 4.5
245 5340.8384 −3.5 245 5727.3559 5.0

a Half-integer values refer to secondary minima.

Times of minima for the wide system.

BJD . Cycle number a . BJD . Cycle number a .
245 4977.0831 −11.5 245 5363.5693 −3.0
245 5022.5375 −10.5 245 5386.3163 −2.5
245 5045.2970 −10.0 245 5409.0662 −2.0
245 5068.0335 −9.5 245 5431.7818 −1.5
245 5113.5169 −8.5 245 5454.5345 −1.0
245 5136.2170 −8.0 245 5477.2681 −0.5
245 5158.9550 −7.5 245 5499.9950 0.0
245 5204.4405 −6.5 245 5545.4559 1.0
245 5227.1669 −6.0 245 5590.9390 2.0
245 5249.9048 −5.5 245 5613.6734 2.5
245 5272.6355 −5.0 245 5659.1425 3.5
245 5295.3893 −4.5 245 5681.8955 4.0
245 5318.1113 −4.0 245 5704.6063 4.5
245 5340.8384 −3.5 245 5727.3559 5.0
BJD . Cycle number a . BJD . Cycle number a .
245 4977.0831 −11.5 245 5363.5693 −3.0
245 5022.5375 −10.5 245 5386.3163 −2.5
245 5045.2970 −10.0 245 5409.0662 −2.0
245 5068.0335 −9.5 245 5431.7818 −1.5
245 5113.5169 −8.5 245 5454.5345 −1.0
245 5136.2170 −8.0 245 5477.2681 −0.5
245 5158.9550 −7.5 245 5499.9950 0.0
245 5204.4405 −6.5 245 5545.4559 1.0
245 5227.1669 −6.0 245 5590.9390 2.0
245 5249.9048 −5.5 245 5613.6734 2.5
245 5272.6355 −5.0 245 5659.1425 3.5
245 5295.3893 −4.5 245 5681.8955 4.0
245 5318.1113 −4.0 245 5704.6063 4.5
245 5340.8384 −3.5 245 5727.3559 5.0

a Half-integer values refer to secondary minima.


Solved papers for JEE Main & Advanced JEE Main Paper (Held On 10-Jan-2019 Evening)

question_answer2) A rigid massless rod of length 3l has two masses attached at each end as shown in the figure. The rod is pivoted at point P on the horizontal axis (see figure). When released from initial horizontal position, its instantaneous angular acceleration will be- [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer6) For the circuit shown below, the current through the Zener diode is- [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer11) Consider a Young's double slit experiment as shown in figure. What should be the slit separation d in terms of wavelength [lambda ] such that the first minima occurs directly in front of the slit[left( <_<1>> ight)]? [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer13) Charges -q and +q located at A and B. respectively, constitute an electric dipole. Distance [AB=2a], 0 is the mid-point of the dipole and OP is perpendicular to AB. A charge Q is placed at P where [OP=y] and [y,,>,,>,,2a]. The charge Q experiences an electrostatic force F. If Q is now moved along the equatorial line to P? such that [OP'=left( frac <3> ight)], the force on Q will be close to- [left( frac<3>>>2a ight)] [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer19) A particle starts from the origin at time [t=0] and moves along the positive x-axis. The graph of velocity with respect to time is shown in figure. What is the position of the particle at time[t=5s]? [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer24) The Wheatstone bridge shown in figure, here, gets balanced when the carbon resistor used as [<_<1>>] has the colour code (Orange, Red, Brown). he resistors [<_<2>> ext< >and ext< ><_<4>> ext< >are ext< >80Omega ,,and ext< >40Omega ] , respectively. Assuming that the colour code for the carbon resistors gives their accurate values, the colour code for the carbon resistor, used as Rs, would be- [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer25) The actual value of resistance R, shown in the figure is [30,Omega ]. This is measured in an experiment as shown using the standard formula [R=frac] where V and I are the readings of the voltmeter and ammeter, respectively. If the measured value of R is [5%] less, then the internal resistance of the voltmeter is- [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer29) Two identical spherical balls of mass M and radius R each are stuck on two ends of a rod of length 2R and mass M (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is: [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

question_answer39) The major product of the following reaction is [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

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question_answer48) The major product of the following reaction [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

question_answer49) The major product obtained in the following reaction is: [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

question_answer51) Which is the most suitable reagent for the following transformation? [JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer55) The correct match between item - I and item - II is:
Item - I (compound) Item - II (reagent)
(A) Lysine (P) 1-naphthol
(B) Furfural (Q) ninhydrin
(C) Benzylalcohol (R) [KMn<_<4>>]
(D) Styrene (S) Ceric ammonium nitate
[JEE Main Online Paper (Held On 10-Jan-2019 Evening] question_answer57) The major product of the following reaction is: [JEE Main Online Paper (Held On 10-Jan-2019 Evening]

question_answer60) 5.1 g [N<_<4>>SH] is introduced in 3.0 L evacuated flask at [327<>^circ C]. 30% of the solid [N<_<4>>SH] decomposed to [N<_<3>>] and [<_<2>>S] as gases. The [<_

>] of the reaction at [327<>^circ C] is ([R=0.082] L atm [mo<^<-1>> ext< ><^<->>^<1>], Molar mass of [S=32] g [mo<^<->>^<1>] molar mass of [N=14,g ext< >mo<^<-1>>]) [JEE Main Online Paper (Held On 10-Jan-2019 Evening]


A statue 5m high is standing on a base 8m high. Is an observer's eye is 2.5 m above the ground, how far should he stand from the base of the statue in order that the angle between his lines of sight to the top and bottom of the statue be a maximum.

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Chemistry

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