1. Homework Statement
Show that ##\lim_{x \to a} f(x) = L## if and only if ##\lim_{x \to 0} f(x+a) = L##
2. Homework Equations

3. The Attempt at a Solution
For the forward direction (ie ##1 \Rightarrow 2##), I tried to first assume that 1. holds true (ie ##\forall \epsilon>0, \exists...
1. Homework Statement
Consider two circular, parallel, coaxial loops which are almost in contact. They are separated by 1.20mm, have 13.6cm as radius each. Both loops carry a current of I = 127A but in opposite directions. Find the magnetic force that the top loop exerts on the bottom loop...
Thanks, that's helpful. I understand that there are less waves in the vacuum medium, but how does this relate to the path length? Does the "extra number of wavelengths" mean path length? And as well, I still don't understand why having the glass would deflect the light rays in that way (both in...
1. Homework Statement
A double slit experiment is set up using a heliumneon laser (wavelength 633 nm). Suppose we add a small piece of glass (n = 1.50) over one of the slits. Then, the central point on the screen is occupied by what had been the m = 10 dark fringe. Determine the thickness t...
1. Homework Statement
Two laser beams, one of wavelength 633 nm and intensity I1 = 4.00x106 W/m2 with a circular beam profile D1 = 1.33 mm in diameter, and one of wavelength 442 nm and intensity I2 = 4.00x106 W/m2 with the same circular beam profile are fired, crossing at an angle of θ = 45°...
Thanks for everything thus far. Apparently there's been an error with the question grading, so I'm pretty sure our answer we have come up to is the correct one. Thanks for everything.
My bad again sorry. The computer system is telling me that I have made a mistake (I'm not sure where though). In the meantime, is my reasoning for the flux current correct? Thanks for your time and patience.
1. Homework Statement
Let d = 0.048m, L = 0.15m, r = 0.10m in the following diagram. Assume that the current I1 = 80.0A and I2 = 40.0A. Find the net force on the rectangle of wire and the direction it points, and state the direction of the emf if the current I1 is increasing in the direction...
It would be N = 1000 turns as we divide the total length by the spacing. I used this and i stilll get a wrong anser for a. (Using it as the total length of wire).
1. Homework Statement
A solenoid has length L = 2.0m, radius R = 28.0cm, field strength B = 2.7T. The field is maintained by superconducting wires, with cross section 2mm x 2mm, wound with a 2mm spacing between windings and 20 layers of winding. Use the result that the current is I = 215A.
a...
1. Homework Statement
A household appliance consumes 1800W of power from a 120V wall plug. Calculate the power consumed due to a wire of length L = 22.0m and diameter D1 = 1.628mm (with resistivity p = 1.680 x 108Ωm). If a house spends $1400 per year in electricity bills, how much money can...
1. Homework Statement
Consider a line of length L=0.950 m and uniform positive charge per unit length λ=2.9 μC. Point A is a distance x=0.23 m from the center of the line, while B is the same distance from the line but a distance y=0.28 m farther along the line; point C is a distance z=0.050 m...
1. Homework Statement
An object of mass m1 of 4.00kg is tied to an object of mass m2 of 3.00kg with a string of length 0.5m. The combination is then swung in a verticular circular path on a second string of length 4.00m. The two strings are always colinear (they are on the same line). At the...
"Extreme situations" in physics problems
I don't know if this question is pertinent for this discussion forum. I was making some study notes, and i was trying to recollect statements that are always "extreme" in physics problems (such as mechanics). Until now, i have found two of them: Lose...
So i found the phase of the second pendulum is \displaystyle\frac{\pi}{2}rad. Then they would not have the same phase after 0.1 seconds, and they would not have the same initial phase. After a time T, they would have reverted to their initial angular velocities (so i'm not sure what they mean...
The centripetal force would be:
N=F_{c}=\displaystyle\frac{mv^{2}}{r}
And then v would be the equation we found, v=v_{0}μNt. So we would replace it in this equation? However, I find a second degree equation (with the mass I can't get rid of). What is the way to solve it?
Thank you...
So then δ=\displaystyle\frac{\pi}{2} or is it 0? Then shouldn't the second pendulum as well have an initial phase of zero as it starts from an angle 0? Or do we have to consider the initial velocity as well? I have that:
0.1=Aω\sin(wt+\phi) (then this would mean that the initial phase is not...
1. Homework Statement
Two identical simple pendulums of length 0.25m and period T, are mounted sidebyside. One is released from rest with an initial angular displacement of \displaystyle\frac{\pi}{12}rad, and the other is started with an initial angular velocity of 0.1 rad/s at θ=0. They...
So this "total" acceleration is the one that changes the velocity of the block? (so it would be different comparing it to centripetal velocity). Isn't the equation that relates tangential acceleration to speed:
v=v_{o}+at?
So, wouldn't we just plug in the acceleration value which is the...
1. Homework Statement
A small block of mass m is sliding around the inside of an Lshaped track of radius r. The bottom of the track is frictionless; the coefficient of kinetic friction and the wall of the track is μk. The block's speed is v0 at t0=0. Find an expression for the block's speed...
I have tried differentiating but to no avail. The answers are here if anyone can find out what it is about:
Extrema positions: (9.28x10^{6}m, 7.51x10^{8}m)
The thing is that supposedly we aren't required to use calculus for all of the exercises. I feel that b. as well requires some calc in order to optimize the function (and wouldn't it be negative everywhere, and have the extrema at the end points?
Edit: Wouldn't the function that gives the...
1. Homework Statement
Consider the earth's radius to be 6.40x103km, its mass to be 6.00x1024kg, the Moon's mass to be 7.36x1022kg, and the Moon's radius to be 1.74x103km. The average value for the EarthMoon distance is 3.84x105kg. Neglect friction and rotation.
a. Sketch the potential...
So now i did your method and found that the "tangent" (direction vector) was:
2i + 4j + 3k
And the gradient was:
4i + 8j + 6k
Then i argued, as these two vectors are parallel (scalar multiples of each other), we should have that the direction vector is parallel to the normal vector...
1. Homework Statement
Given that the surface x^{6}y^{5}+y^{4}z^{5}+z^{9}x^{7}+4xyz=7 has the equation z = f(x, y) in a neighborhood of the point (1, 1, 1) with f(x,y) differentiable, find:
\displaystyle\frac{\partial^{2} f}{\partial x^{2}}(1,1) = ?
2. Homework Equations
3. The Attempt at...