In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to the ratio 4∶3). Similarly, the ratio of lemons to oranges is 6∶8 (or 3∶4) and the ratio of oranges to the total amount of fruit is 8∶14 (or 4∶7).
The numbers in a ratio may be quantities of any kind, such as counts of people or objects, or such as measurements of lengths, weights, time, etc. In most contexts, both numbers are restricted to be positive.
A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a∶b", or by giving just the value of their quotient a/b. Equal quotients correspond to equal ratios.
Consequently, a ratio may be considered as an ordered pair of numbers, a fraction with the first number in the numerator and the second in the denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers. When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.
1. A ball of mass m and radius R is both sliding and spinning on a horizontal surface so
that its rotational kinetic energy equals its translational kinetic energy.What is the ratio of the ball’s center-of-mass speed to the speed due to rotation only of a point on the ball’s surface? The moment...
Homework Statement
Determine whether the series converges or diverges.
\sum (\sqrt {k} - \sqrt {k - 1})^k
Homework Equations
The Attempt at a Solution
(a_k)^\frac{1}{k} = \sqrt{k} - \sqrt{k - 1}
What do I do here..?
= \frac{1}{\sqrt{k} + \sqrt{k-1}} \to 0 ?
Suppose we've got the setup as shown in the figure (see attachment).
The idea is that the motor transfers its speed and force (rotary) to the actuator force and speed (linear) via some gears and a spindle.
Here:
R = radius [m]
J = inertia [kg m^2]
n = rotary to linear transmission [---]...
Hello friends,
Homework Statement
lim sup |\frac{a_{n+1}}{a_{n}}|
I know that this operation tests for convergence, but I don't understand how it's related to lim_{n -> \infty}|\frac{a_{n+1}}{a_{n}}|
I understand that both will give the same result, but how is the latter different? lim...
Hi, I need to calculate \Gamma(7/3)/\Gamma(1/3)
is there some trick to doing this type of problem? I'm finding it really hard to do from first principles.
Any help would be great, thanks!
Homework Statement
Two resistors of resistances R1 and R2 , with R2>R1 , are connected to a voltage source with voltage V0 . When the resistors are connected in series, the current is Is . When the resistors are connected in parallel, the current Ip from the source is equal to 10Is
Let r be...
Homework Statement
The problem is to find the inverse laplace of \frac{s^2-a^2}{(s^2+a^2)^2}
I am supposed to use the residue definition of inverse laplace (given below)
The poles of F(s) are at ai and at -ai and they are both double poles.
Homework Equations
f(t) =...
Homework Statement
Two resistors R1 and R2, where R1 > R2, are connected to a voltage source with voltage V. When the resistors are connected in series, the current is Is. When the resistors are connected in parallel, the current from the source is Ip = 8.0Is. What is the ratio R2/R1?
I have...
Let ∑ak be a series with positive terms.
Ratio test:
Suppose ak+1/ak -> c.
If c<1, then ∑ak converges.
If c>1, then ∑ak diverges.
If c=1, the test is inconclusive.
What if ak+1/ak diverges (i.e. ak+1/ak->∞)? Do we count this as falling into the case c>1? Can we say whether ∑ak converges...
Homework Statement
Hint: Disregard any dissipative effects of the
atmosphere of the planet.
A projectile is launched from the surface of a planet (mass M, radius R)
with a launch speed equal to 61 percent of the escape speed for that
planet. The projectile will rise to a maximum height...
Homework Statement
Prove: If the limit inf as k goes to infinity of abs(ak+1 / ak) > 1 then the sum from 1 to infinity of ak diverges
Homework Equations
The Attempt at a Solution
So far I have this:
Suppose lim inf abs(ak+1/ak) >1
then, there exists an r such that lim inf...
Homework Statement
Prove: If the limit inf as k goes to infinity of abs(ak+1 / ak) > 1 then the sum from 1 to infinity of ak diverges
Homework Equations
The Attempt at a Solution
So far I have this:
Suppose lim inf abs(ak+1/ak) >1
then, there exists an r such that lim inf...
Hi, my task is to calculate the ratio (gamma) between the specific heats of a gas (Cp and Cv). The only information I have is a table of data for the pressures and volumes of the gas at different temperatures. I don't know if its monatomic, diatomic etc. (it's a later task to determine this)...
what is the maximum "compression ratio"
lease visit this link
http://www.starrotor.com/Engines
what is the maximum "compression ratio", so to speak, achievable by compressors? i don't think it is very high.
Homework Statement
The Helmholtz coil current = 2.45 A, the magnetic field = 1.91 x 10^(-3) T, the accelerating voltage = 295 V, and the beam radius = .03m.
a) Calculate the e/m ratio of an electron. Show that e/m = 2V / r²B²
b) During the initial use of this experiment, physicists also...
Hi All,
This is a similar post to one before, but that didn't offer much more than i know already.
a = width of cell - depth of cell
c= height of cell
c/2 = height of central atom
So i have various formula for a few isosceles lengths, namely a/2*cos30 for the base line of the vertical...
Homework Statement
Two astronauts are floating in space. Astronaut A's mass is 1.89 times greater as that of astronaut B. IF astronaut A pushes astronaut B, what is the ration of their accelerations, Ab/Aa.
Homework Equations
F=ma
MaAa=MbAb (M is the mass, A is acceleration)...
Homework Statement
This is more of a conceptual question. I understand that the heat capacity ratio is basically the ratio between the heat capacity at constant pressure to the heat capacity at constant volume.
I was asked to calculate the ratio of heat capacities for Nitrogen at two speeds...
Aspect ratio of cylinder floating in water?
Hi,
Basically the question is:
What is the aspect ratio (h/d) for a cylinder of specific gravity 0.65 above which it will float vertically? (Note use GM = I/V - BG)
So i found a solution that is simliar and can help solve the problem. (please...
please could you help me understand this is very simple terms.
1.0 x 10-2 = 1:100
(as in ten to the power of minus 2)
i do not understand what i need to do to get it into a ratio.
Hi everybody:
I am trying to solve this question:
[PLAIN]http://imageshugger.com/images/wviidazp3mf06ej32.jpg
I have made the following:
u=y
v=xy
Calculated the Jacobian, which is |-y|=y
then as I know that is unnormalized I need to normalized first using a double integral as a...
Just out of curiosity : Consider a transformer with winding ratio of 1 : 10.
What happens when AC 120 v is applied on primary and on secondary we apply DC voltage of 120 v from a battery source is series or in parallel? What will be the output in both cases?
If the phase angle for a block-spring system in SHM is φ and the block's position is given by x = xm cos(ωt+φ), what is the ratio of the kinetic energy to the potential energy at time t = 0? State your answers in terms of the given variables.
x =xm Cos(wt + phi)
v = -w(xm)sin(wt +phi)...
Homework Statement
If the cross-section of the barrel of a pump is 5 square centimeters, the stroke of the piston is 20 centimeters, and the volume of the receiver, in which the air is drawn from, is 1 liter, calculate the ratio of the pressure of the air in the receiver to the original...
Hi,
I am doing a project on communications engineering. I've done some background reading on CDMA and spread spectrum technologies such as Direct Sequence and Frequency Hopping. My project supervisor has asked me to find out the equation for the signal to interference ratio of CDMA systems...
I was stuck on a problem in my Diff Eq book and I asked my teacher for some guidance. This was the problem:
Let M dx + N dy = 0 be a homogeneous diff eq. Using the substitution y = r sin(t), x = r cos(t), show that this equation is a separable equation of r and t.
Now it was obvious that...
I'm a bit confused about power output and gear ratios. I think it's best to explain by going by my example (imperical units, sorry Americas & Brits):
Mass: 1300kg
max. power: 128 kW @ 5500 rpm
max. torque: 245 Nm
Gear ratios (from a slightly different car type, but should be similar):
1st...
Homework Statement
Determine whether the series below is convergent or not:
\sum 7*\frac{n!}{n^{n-10}}
n=8 and the series goes to infinity
(Sorry, I couldn't get the formatting correct.)
Homework Equations
n/aThe Attempt at a Solution
Well, originally I thought the series was divergent...
Homework Statement
I have often heard that given say, ds/dt=k, that it is not entirely rigorous to say that ds=kdt. Why is that? If I view the derivative as nothing more than a ratio of the differentials, that seems perfectly reasonable. Also I see this done all the time, with acknowledgment...
Hello everybody,
I am having some troubles with this exercise. I have attached the text and the solutions to my post. My problem is, after I have coupled the angular momenta and used the CK coefficients like in (|\pi ^{+}p>=|1,1>|1/2, 1/2>=|3/2,3/2>) (pag.432) I don't understand what...
Homework Statement
Two tubes carry the same incompressible fluid with viscosity 1.5 Pl. They have lengths L1 = 6 and L2 = 22 m and diameters d1 = 1.2 and d2 = 4.5 cm. What is the ratio of their flow rates F1/F2?
Homework Equations
Poiseuille's law: 8nLI/(pi*r^4)
while n is viscosity
L...
Homework Statement
Calculate the ratio of the kinetic energy to the potential energy of a simple harmonic oscillator when its displacement is half its amplitude.
Homework Equations
KE=1/2mv2 = 1/2kA2sin2(wt)
U=1/2kx2 = 1/2kA2cos2(wt)
KEmax=1/2kA2
Umax=1/2KA2
The Attempt at a...
electron is deflected in a circular path
electron speed = 2.7 x 10^8 m/s
diameter = 3cm
magnetic field = 0.24T
1. work out charge to mass ratio - 7.5 x 10^10
2. the textbook value is 1.76 x 10^11 - comment on this value
The answer is given as electron traveling close to speed of...
So I've read that phi (1.6) has been found at the quantum level, and from what I've been told this is a major breakthrough. But I fail to see the relevance. I was wondering if anyone felt that this is a major breakthrough, and could explain this to me?
edit: posting link to article...
Here is a problem involving ratio , and the answer plus work as given by my textbook. I get a little lost at the underlines part. I don't understabnd how we arrived at a ratio of 24:9. Can anyone help me out?
Mr Smith starts a business with $50,000.
3 months later, Mr Jones joins the...
Why do we take the "probability of an event" ratio as favorable / possible outcome
If A is an event, then probability that A will occur is given by
P(A)= no. of favorable outcomes / total no. of possible outcomes
Is this just a definition, or is there some special significance in taking...
Hi,
Could anyone explain to me why it is that cosmic radiation supplies slightly more positive than negative muons? I know that negative muons disappear a little faster than the positive ones as they can decay or be captured, but I don't know if this would have any impact on what I'm asking...
Calculate the heat capacity ratio for a diatomic ideal gas.
----
Using the equipartition theorem, I calculate:
Cp / Cv = (Cv + R) / Cv = (7/2 R + R) / (7/2 R) = (9/2 R) / (7/2 R) = 9/7 ~ 1.286
According to the equipartition theorem, I assign each vibrational degree of freedom 1 R and...
All the LCDM models with variations (see http://lambda.gsfc.nasa.gov/product/map/current/parameters.cfm ) have similar ratios of dark matter: baryonic matter density of very close to 5:1.
Is this ratio determined from WMAP data such as angular size of a peak, or is it a not very tightly...
v = \epsilontrans/ \epsilonaxial
where
ν is the resulting Poisson's ratio,
\epsilontrans is transverse strain
\epsilonaxial is axial strain
and Poissions Ratio have range and its not more then 0.5 and not less then -1?
so my question is how its value can turn "-1"?
As...
Homework Statement
let f: R->R be a continuous function
Suppose k>=1 is an integer such that
lim f(x)/x^k = lim f(x)/x^k = 0
x->inf x->-inf
set g(x)= x^k + f(x)
g: R->R
Prove that
(i) if k is odd, then g is surjective
(ii) if k is even, then there is...
I'm building pairs of coils with standard core heights and diameters and linking each pair with a yoke. The magnets all run at 6V DC with windings of 24 AWG magnet wire in 6/8/10 layers.
Is there a ratio of coil core mass to yoke mass for maximum efficiency? (shown below)
Homework Statement
I'm supposed to determine e/m using Zeeman effect.
B=0.45 T
Difference between two spectral lines is also known (0.649x10^10 Hz)Homework Equations
Normal or anomalous Zeeman effect? How you know it?
I'm guessing normal, so
\Delta E = m_{l} \frac{e \hbar}{2m_{e}}B
\Delta E =...
1. Uranium has two naturally occurring isotopes. 238U has a natural abundance of 99.3% and 235U has an abundance of 0.7%. It is the rarer 235U that is needed for nuclear reactors. The isotopes are separated by forming uranium hexafluoride, which is a gas, then allowing it to diffuse through a...
Homework Statement
Harry and Sally are sitting on opposite sides of a circus tent when and elephant trumpets a loud blast. If Harry experiences a sound intensity level of 65dB and Sally only 55dB, what is the ratio of the distance between Sally and the elephant to the distance between the...
Homework Statement
Work out the branching ratio of (W --> e-Ve)
The book uses fermi's golden rule to do it, but I am wondering if my different solution is also valid?
Homework Equations
Γ = 1/τ and ΔEΔT = hbar
The Attempt at a Solution
If ΔEΔT = hbar, then i can write ΔT =...