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    Definition of a limit of a function confusion

    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...
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    Force between two current-carrying loops

    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...
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    Path length difference and Diffraction

    1. Homework Statement A double slit experiment is set up using a helium-neon 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...
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    Scattering of two laser beams

    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°...
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    Force on a rectangular loop

    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...
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    Energy and force in a solenoid with multiple wirings

    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...
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    Power usage in wires

    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 10-8Ωm). If a house spends $1400 per year in electricity bills, how much money can...
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    Potential and electric fields

    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...
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    Tensions and centripetal acceleration

    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...
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    Extreme situations in physics problems

    "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...
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    Pendulums and SHM

    1. Homework Statement Two identical simple pendulums of length 0.25m and period T, are mounted side-by-side. 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...
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    2D Dynamics Problem

    1. Homework Statement A small block of mass m is sliding around the inside of an L-shaped 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...
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    Gravitational potential energy between Earth and Moon

    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 Earth-Moon distance is 3.84x105kg. Neglect friction and rotation. a. Sketch the potential...
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    Complicated implicit multivariable differentiation problem

    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...
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    Proving that a curve intersects a surface at a right angle

    1. Homework Statement a) Show that the curve determined by: x=2(t^{3}+2)/3, y=2t^{2}, z=3t-2 intersects the surface: x^{2}+2y^{2}+3z^{2}=15 at a right angle at the point (2, 2, 1) b) Verify that the curve x^{2}-y^{2}+z^{2}=1, xy+xz=2 is tangent to the surface xyz-x^{2}-6y+6y=0 at the...
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    Center of mass and angular momentum

    1. Homework Statement On a frictionless table, a glob of clay of mass 0.240kg strikes a bar of mass 1.560kg perpendicularly at a point 0.390m from the center of the bar and sticks to it. a) If the bar is 1.260m long and the clay is moving at 9.300m/s before striking the bar, what is the...
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    Angular momentum and pulleys

    1. Homework Statement In the graph shown in the picture, an expression for the acceleration of the pulleys is obtained. 2. Homework Equations 3. The Attempt at a Solution The thing i don't understand is, how do we find the angular momentum for the system? In class, I was told that...
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    Eigenspaces and geometric reasoning

    1. Homework Statement Let T be the reflection about the line 6x + 1y = 0 in the euclidean plane. Find the standard matrix A of T. Then, write down one of the eigenvalues and its corresponding eigenspace (in the form span {[ ]}). Then, find the other eigenvalue of A and its corresponding...
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    Torque and angular acceleration

    1. Homework Statement A massless beam is supported only at one point, called the pivot point, as shown in the diagram. A block with mass m1 sits at the left end of the beam, a distance L1 from the pivot point. A block with mass m2 sits at the right end of the beam, a distance L2 from the pivot...
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    Matrix transformations and effects on the unit square

    I was looking over my notes today, and I realized that there was a point that isn't pretty clear. If we have the image under T (being T a matrix transformation induced by a matrix A) of the unit square, then its area should be abs(det(A)). Why is this though? I was looking at the proof and I...
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    Transforming differentials and motion

    Hello, I currently have a problem with interpreting how this statement was interpreted: We have a rate of change which is dv/dt, and in the given notes, they transformed the expression dv/dt into dv/dx dx/dt (using the chain rule). Then, the whole expression simply turns into v dv/dx (as...
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    Differential equations incongruecy

    1. Homework Statement I am going to copy-paste this text that my friend made (because we both have the same doubt and we don't know to work around it. This is a long post, so warning): "I'm currently unsure of how these two problems work. I've tried working at them in different ways but i...
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    Momentum in a space shuttle and thrust

    1. Homework Statement The space shuttle, with an initial mass M = 2.41 x 106kg, is launched from the surface of the earth with an initial net acceleration a = 26.1 m/s^2 . The rate of fuel consumption is R = 6.90 x 103kg/s. The shuttle reaches outer space with a velocity v0= 4632m/s, and a...
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    Giving the equation of a line

    1. Homework Statement Find the scalar equations of the line passing through p(6, 0, 3), intersecting the line (x y z) = (1 2 -3) + t(1 -2 0) and perpendicular to it 2. Homework Equations - 3. The Attempt at a Solution I don't know where to start with the problem, I tried using...
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    Radioactive samples and differential equations

    1. Homework Statement A certain nuclear plant produces radioactive waste in the form of strontium-90 at the constant rate of 500kg. The waste decays exponentially with a half-life of 28 years. How much of this radioactive waste from the nuclear plant will be present after the following...
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    Negative work and energy

    How can we define negative work to be? I can't get my head around negative work or energy. However, I'm thinking that it must be work that is done opposite in the direction of motion of a particle or body (say, friction). Could someone please help me understand this concept?
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    Differential equations and law of cooling

    1. Homework Statement A homicide victim is found to have a temperature of 31°C at the stroke of midnight. At 1:00AM his temperature dropped to 29°C. Assuming that the temperature of the room stays at 20°C, when did the murder take place? 2. Homework Equations - 3. The Attempt at a...
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    FO Differential equations and account balance

    1. Homework Statement a. Assume that yo dollars are deposited into an account paying r percent compounded continuously. If withdrawals are at an annual rate of 200t dollars (assume these are continuous), find the amount in the account after T years. b. Consider the special case if r = 10% and...
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    L'Hôpital's rule and weird expression

    1. Homework Statement Find, using l'Hôpital's rule: lim x-> ∞ ((tan(ax)-atanx)/(sin(ax)-asinx)) Where a is a non constant greater than ± 1. 2. Homework Equations - 3. The Attempt at a Solution I can't work out anything from this point. I don't know how to change or factorize the...
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    Understanding constant velocity, inertia and rest

    I've come to the conclusion that i don't understand at all the concept behind inertia. How can we know when an object stays at rest or at constant velocity? (take for example a girl in a rope. She would tighten her grip on the rope until the tension balances the force of gravity (assume rope is...
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