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Answers by Kirill Z.

a) In the no-slipping case, energy conservation tells us:   mgh=mv2/2+Iω2/2; here ω=v/R, R is the radius of the marble, I=(2/5)mR2; Plugging in all the data into the first equation yields:   mgh=(7/10)mv2; From this we obtain:   h=(7/10)(v2/g); h=0...

Use energy conservation. Potential energy transforms into rotational energy.   mgh=Iω2/2; Here I is the moment of inertia with respect to the axis passing through the point at the rim, where the rope leaves the disk. If you draw a picture, you will realize that the disk at every instant...

The field on the axis is directed perpendicular to the disk and is given by:   E=∫035 2π*ρ*x/(x2+y2)3/2*y*dx; Here ρ=7.90*10-3 C/m2, charge density, x is the distance from the axis to a point on the disc, y is the distance from the disc to the point on the axis where...

dot product (answer)

Since (a+βb) and (a-βb) are perpendicular, it means that (a+βb)•(a-βb)=0 or   a•a-β2*(b•b)=|a|2-β2*|b|2=0   Substituting values for |a| and |b| into the equation, we obtain:   9-25β2=0 or β=±3/5

-2(2m+1)^2-17=11 (answer)

-2(2m+1)2-17=11 ⇔ -2(2m+1)2=28 ⇔ (2m+1)2=-14 ⇔ 2m+1=±i√14, where i=√(-1);    Finally,    m=-½±(i/2)√14

I would like to add to answers something for you to think about. Why the sky is not violet? It is true that the shorter the wavelength, the stronger the scattering is. Then the sky shall be violet, since the violet light has shorter wavelength than the blue light. So why not?

Factor -54. It is -9*6 or 9*(-6). Then check the sum of factors, it shall come to -3, opposite to the coefficient in front of x. -9+6=-3, so roots are x1=-9 and x2=6. The factoring is given by:   (x-x1)(x-x2), in your case,  (x+9)(x-6)

I would try to see which interval/intervals the roots may lie into. First, I would see what I get if I plug in x=0, I will get -2; If I plug in -1, I will get zero, which means x=-1 is the root (lucky me!). It turns out that x=1 is also the root. Thus you can now divide your polynomial by (x-1)*(x+1)=x2-1...

Physics Problem (answer)

1) v=v0+at, so since v0=0 (officer starts from rest), a=2 m/s2, you obtain:   40=2t, so t=20 seconds.   2) In this case it is better to imagine things from the driver's perspective. In its reference frame, the officer is at -100 m at t=0 with the velocity of -24 m/s...

You have 6 possibilities for each, since there are two dice, you end up with 62=36 possible combinations. Note that similarly colored dice will give you 15 less combinations (because dice would be indistinguishable)

Reduce the expression by combining two ratios in numerator together. The whole expression then becomes:                1 - -------------    2(2+x)   this expression is defined ar x=0 and its value is -¼