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...
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...
Since perimeter is the sum of all side lengths, it scales just as the sides themselves. So two perimeters scale as 2:3. If 56 cm is the perimeter of a small triangle, the perimeter of the large one is (3/2)*56=3*28=84 cm.
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 ⇔ -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)
For direct variation y=kx, where k is the number. If y=5 for x=10, then k=y/x=5/10=1/2. So we have: y=1/2*x; For y=15 x=2*y=2*15=30 Answer:30
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...
Centrifugal force is given by: F=mω2R, where m is the mass of the hand, R is the radius of the circle the hand makes, ω is the angular frequency in rad/s. Mass of one hand is 1.25% of the body mass, that is 50*0.0125=0.625 kg; radius equals to 0.8 m (half the distance between...
a=dv/dt; Let us differentiate v(x): dv/dt=dv/dx*dx/dt=dv/dx*v, since dx/dt is velocity itself. I used a chain rule here, as you may notice. dv/dx is given by: dv/dx=2x+1; Thus we obtain: a=dv/dt=(2x+1)*v(x)=(2x+1)*(x2+x)=2x3+3x2+x Now...
Use work-energy theorem. It states that the change in kinetic energy equals to the work done on the object. Work done on the object when moved from x=1 to x=3 is: W=∫13(x3-3x)dx=[x4/4-3/2*x2]|13=81/4-27/2-1/4+3/2=8 J. Since the body was at rest at...
Since box is not moving, the forces are balanced. There are three forces: gravity, normal force, and horizontal force F. 1) Let x-axis be parallel to the incline. In this case, component of force F along this axis is F*cos(60). Component of the gravity force along the same axis is...
You have two places and any letter can be at any place, thus, you have N=42=16 codes. AA AB AC AD BB BA BC BD CA CB CC CD DA DB DC DD
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...
The number of ways to pick up two balls from 8+3=11 balls is given by: N=C112=11*10/2=55 The number of outcomes that those two balls are white is given by: Nw=C82=8*7/2=28 The probability is the ratio of two, that is, ...
Commutative property of addition.
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