PHYSICS 15A Midterm: PHYS15A Harvard 04 Fall Midterm1

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turquoisegnu128

15 Feb 2019

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0 (b) /2 (b) (c) (c) (a) 1 dx. 4 (d) e2 1 (d) ln( /2) (e) ln e4 1 (f) 3. 46 (e) e (f) 1. 14 (d) (e) ln 2 (f) 2. 3 (c) 32/51 (d) /8 (e) e/9 (f) 0. 28. 1 (x 1)3 dx. (e) ln 2 + 1. 3 (f) divergent: find the arclength of the part of the curve y = 2. 3 (x 4)3/2 between the points (7, 2 3) and (12, (a) 19. What is the surface area (which contains both the inside and the outside surfaces) of such a glass? (a) (cid:16) 8 2 4 (d) (cid:16) 5 5 1. + 2(cid:17) (b) (cid:16) 8 2 4 (e) (cid:16) 5 5 1. 6 (c) (cid:16) 8 2 4 (f) (cid:16) 5 5 1. 1: what is the x-coordinate of the centroid of the region bounded by the graph of y = 3 x, the line x = 8, and the x-axis? (a) 8 (b)

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I NEED THIS ANSWERED BEFORE MIDNIGHT PST! I WILL RATELIFESAVER!

A very thin glass rod 4 meters long is rubbed all over with a silkcloth. It gains a uniformly distributed charge 1.6X10^â6 C. Twosmall spherical rubber balloons of radius 1.2 cm are rubbed allover with wool. They each gain a uniformly distributed charge of-8X10^â8 C. The balloons are near the midpoint of the glass rod,with their centers 3 cm from the rod. The balloons are 2 cm apart(4.4 cm between centers).

(a) Find the net electric field at the location marked by the X,0.6 cm to the right of the center of the left balloon. Things tothink about: Which objects make nonzero contributions to E field atthis location? What are the correct distances from sources to theobservation location? What is the direction of E field due to therod? (The drawing is not to scale; read the problem statementcarefully. Assume the +x axis is to the right, the +y axis is up,and the +z axis is out.)

E = ________ N/C

(b) What approximations did you make, if any? Which apply(1-4)?
1. Neglect polarization of balloons.
2. Use approximate formula for electric field of a chargedspherical shell.
3. Neglect polarization of rod.
4. Assume distance to observation location is small compared tolength of rod.

(c) Next a proton is placed at that same location (marked by thex). What is the force acting on the proton?

F = ________ N

Note: in parts (a) and (c) there should be an arrow over the"E" and the "F" but the custom equation function would not workproperly.

Consider the electric field given by ~E =(2xi-3j) N/C, where x isin units of meters.
Now suppose we have a rectangular volume, a distance c from the y -z plane, with two edges length a and the third length b. Theoutgoing flux through the right surface
is:
1. right = -2 a b (a + c) .
2. right = -2 (a + c) .
3. right = -[2 (a + c) - 3] a b .
4. right = 2 (a + c) .
5. right = -3 a b .
6. right = [2 (a + c) - 3] a b .
7. right = 2 a b .
8. right = 2 a b (a + c) .
9. right = -2 a b .
10. right = 3 a b .

Figure 1 illustrates a Keplerian orbit, with Cartesiancoordinates (x,y) and polar coordinates (r,Ï).

Parametric equations for a general Keplerian orbit are r(Ï) = a(1 â e cosÏ)

t(Ï) = *(T)/2Ï+ ( Ïâ e sinÏ)

tan(Ï/2) = *(1+e)/(1âe)+1/2 tan(Ï/2)

Here Ï is an independent variable. The solution is periodic in Ïwith period âÏ = 2Ï. Also, a, e, T are constants that determine theparameters of the orbit.

Figure 2 shows the relation between Ï and the spatialcoordinates. In the figure, the coordinates (Î¾,Î·) are defined by Î¾= x + ae and Î· = y a /b,

withb=a*1âe2 ]1/2.

(A) What kind of curve is the orbit?

(D) Determine Î¾(Ï) and Î·(Ï).

(E) Express r as a function Ï .

(F) The angular momentum is L = m r2dÏ/dt. Determine L in termsof {a,e,T} Hence verify that L is a constant of the motion.

(G) The energy isE = [1/2] m ( dr/dt )2+ [1/2] m r2(dÏ/dt)2â GMm/r.Determine E in terms of {a,e,T}.

(H) E must be a constant of the motion. Hence determine T and Efrom the equation that you obtained in (F).

(I) Write T in terms of the spatial orbit parameters and theforce constant GM.

(J) Now express L and E in terms of a, e, GM.

(K) In Figure 1, determine the coordinates of the points P, A,and R

(L) In Figure 1, determine the time t at the point R.

PHYSICS 15A Midterm: PHYS15A Harvard 04 Fall Midterm1

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