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19.4: Resistors
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/19%3A_Electric_Current/19.04%3A_Resistors
To good approximation, one can model the two ends of the conductor as parallel plates, so that the magnitude of the electric field throughout the conductor is constant in magnitude and direction and h...To good approximation, one can model the two ends of the conductor as parallel plates, so that the magnitude of the electric field throughout the conductor is constant in magnitude and direction and has strength given by: $\begin{aligned} E=\frac{\Delta V}{L}\end{aligned}$ Combining this with Ohm’s Law, we have: $\begin{aligned} j&=\sigma E\\ \therefore j&=\sigma\frac{\Delta V}{L}\\\end{aligned}$ Since the current density is a microscopic quantity, we can replace it with the current, $I$ ,…
3.1: Motion with Constant Speed
https://phys.libretexts.org/Courses/Berea_College/Introductory_Physics%3A_Berea_College/03%3A_Describing_Motion_in_One_Dimension/3.01%3A_Motion_with_Constant_Speed
Since the position as a function of time for the ball plotted in Figure $\PageIndex{1}$ is linear, we can summarize our description of the motion using a function, $x(t)$ , instead of having to tab...Since the position as a function of time for the ball plotted in Figure $\PageIndex{1}$ is linear, we can summarize our description of the motion using a function, $x(t)$ , instead of having to tabulate the values as we did in Table 3.1.1. The velocity, $v_{x}$ , is simply the difference in position, $∆x$ , between any two points divided by the amount of time, $∆t$ , that it took the object to move between those to points (“rise over run” for the graph of $x(t)$ ):
26.1: Functions of Real Numbers
https://phys.libretexts.org/Courses/Berea_College/Introductory_Physics%3A_Berea_College/26%3A_Calculus/26.01%3A_Functions_of_Real_Numbers
Unfortunately, it becomes difficult to visualize functions of more than 2 variables, although one can usually look at projections of those functions to try and visualize some of the features (for exam...Unfortunately, it becomes difficult to visualize functions of more than 2 variables, although one can usually look at projections of those functions to try and visualize some of the features (for example, contour maps are 2D projections of 3D surfaces, as shown in the $xy$ plane of Figure A1.1.2).
23.9: Sample problems and solutions
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/23%3A_Electromagnetic_Induction/23.09%3A_Sample_problems_and_solutions
As the airplane moves as illustrated (towards the left, in an upwards magnetic field), the electrons in the wing of the airplane will be pushed into the page. Eventually, the electric field from the e...As the airplane moves as illustrated (towards the left, in an upwards magnetic field), the electrons in the wing of the airplane will be pushed into the page. Eventually, the electric field from the electrons will prevent further electrons from accumulating at that side of the wing, and there will be a constant (Hall) voltage, $\Delta V$ , across the wing tips.
9.7: Sample problems and solutions
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/09%3A_Gravity/9.07%3A_Sample_problems_and_solutions
To obtain the “altitude”, $h$ , namely the distance from the surface of the Earth to the satellite, we must subtract the radius of the Earth, $R_{⊕} = 6.371 × 10^{6}\text{ m}$ from this distance: T...To obtain the “altitude”, $h$ , namely the distance from the surface of the Earth to the satellite, we must subtract the radius of the Earth, $R_{⊕} = 6.371 × 10^{6}\text{ m}$ from this distance: The initial mechanical energy of the satellite, $E_{A}$ , is given by its gravitational potential energy (it has no kinetic energy at the surface of the Earth when at the North Pole - on the equator, it would have kinetic energy due to the Earth’s rotation):
19.9: Sample problems and solutions
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/19%3A_Electric_Current/19.09%3A_Sample_problems_and_solutions
The cross-sectional area of one ring is given by: $\begin{aligned} dA = 2\pi r dr\end{aligned}$ so that the current through one ring is given by: \[\begin{aligned} dI = j(r) dA = 2\pi a r^2 dr\end{a...The cross-sectional area of one ring is given by: $\begin{aligned} dA = 2\pi r dr\end{aligned}$ so that the current through one ring is given by: $\begin{aligned} dI = j(r) dA = 2\pi a r^2 dr\end{aligned}$ The current through the whole wire is then found by summing the currents through each ring: $\begin{aligned} I=\int dI = \int_0^R 2\pi a r^2 dr=\frac{2}{3}\pi aR^3\end{aligned}$
7.6: Sample Problems and Solutions
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/07%3A_Work_and_energy/7.06%3A_Sample_Problems_and_Solutions
The net work is thus the sum of the work done by the force gravity, $W_g$ , and the work done by the force of friction, $W_f$ , over the displacement corresponding to the length of the ramp: Thus th...The net work is thus the sum of the work done by the force gravity, $W_g$ , and the work done by the force of friction, $W_f$ , over the displacement corresponding to the length of the ramp: Thus the net work done by the force of friction is the difference in kinetic energies between the final landing point and the beginning of the ramp, because friction is the only force that did a net amount of (negative) work over the whole trajectory (gravity did no net work over the whole trajectory).
12.6: Sample problems and solutions
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/12%3A_Rotational_Energy_and_Momentum/12.06%3A_Sample_problems_and_solutions
where $TR_{1}$ is the magnitude of the torque from the force of tension, since the tension is perpendicular to the vector $\vec r$ between the center of mass and the point where the tension is exe...where $TR_{1}$ is the magnitude of the torque from the force of tension, since the tension is perpendicular to the vector $\vec r$ between the center of mass and the point where the tension is exerted. We can calculate the particle’s angular momentum just before the collision, so that $\vec r$ is the vector from the center of the circle to the point where the particle collides (with magnitude $R$ , and perpendicular to $\vec v$ ).
10.6: Sample problems and solutions
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/10%3A_Linear_Momentum_and_the_Center_of_Mass/10.06%3A_Sample_problems_and_solutions
The two are related by: $\begin{aligned} ds = Rd\theta\end{aligned}$ The mass element, $dm$ , can then be expressed in terms of the mass per unit length of the wire and the length, $Rd\theta$ , of...The two are related by: $\begin{aligned} ds = Rd\theta\end{aligned}$ The mass element, $dm$ , can then be expressed in terms of the mass per unit length of the wire and the length, $Rd\theta$ , of the mass element: $\begin{aligned} dm = \lambda ds = \lambda Rd\theta\end{aligned}$ We also need to express the $y$ position of the mass element using $\theta$ : $\begin{aligned} y = R\sin\theta\end{aligned}$ Now that we have expressed $dm$ and $y$ in terms of $\theta$ , we can determi…
3.5: Summary
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/03%3A_Describing_Motion_in_One_Dimension/3.05%3A_Summary
If an object has a position $x^A(t)$ in a given inertial frame of reference, $x$ , that is moving with a velocity $v^{'B}$ compared to a different inertial frame of reference, $x^{'}$ , then the...If an object has a position $x^A(t)$ in a given inertial frame of reference, $x$ , that is moving with a velocity $v^{'B}$ compared to a different inertial frame of reference, $x^{'}$ , then the position of the object in the $x^{'}$ frame of reference is given by $x'(t)=v^{'B}+x^{A}(t)$ .
1: The Scientific Method and Physics
https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/01%3A_The_Scientific_Method_and_Physics
This textbook will introduce the theories from Classical Physics, which were mostly established and tested between the seventeenth and nineteenth centuries. We will take it as given that readers of th...This textbook will introduce the theories from Classical Physics, which were mostly established and tested between the seventeenth and nineteenth centuries. We will take it as given that readers of this textbook are not likely to perform experiments that challenge those well-established theories. The main challenge will be, given a theory, to define a model that describes a particular situation, and then to test that model.

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