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Chapter 10: Problem 79

A string wrapped around a hub of radius \(R\) pulls with force \(F_{\mathrm{T}}\) on an object that rolls without slipping along horizontal rails on "wheels" of radius \(r

Short Answer

Expert verified
In the described scenario, it was demonstrated that the ratio of force due to the string (wrapped around the hub) to the object's acceleration is negative. This force does all the work leading to an increase in both the translational and rotational kinetic energies of the object. An analogy can be drawn to behavior of a semiconductor in an external electric field, the field acts like \(F_{T}\), causing electrons (analogous to the rolling object) to move and create current (kinetic energy).

Step by step solution

01

Provide context for the scenario

Suppose the object is rolling without slipping. Then, the linear acceleration, \(a\), of the object's COM (center of mass) is related to its angular acceleration, \(\alpha\), by the equation \(a = \alpha \cdot r\). Moreover, the torque acting on the object about the COM due to \(F_{T}\) is \(F_{T} \cdot R\).
02

Applying Newton's Second Law

We now apply Newton's Second Law in its linear and rotational forms. For linear motion, \(F_{T} - F_{friction} = m \cdot a\), and for rotational motion, \( F_{T} \cdot R = I \cdot \alpha\). Substituting for \(a\), we get \(F_{T} - F_{friction} = m \cdot \alpha \cdot r\) and \( F_{T} \cdot R = I \cdot \alpha\). These equations can be solved to show that the ratio of \(F_{T}\) to \(a\) is negative.
03

Calculating power and kinetic energy.

To demonstrate the principle of work and power, calculate the power delivered by \(F_{T}\), which equals \(F_{T} \cdot v\); here \(v = \omega \cdot R\) is the speed at which the string moves. The rate of change of kinetic energy of the object can be shown to equal this power value, demonstrating that \(F_{T}\) does all the work in this system.
04

Drawing an analogy with semiconductors

The behaviours described in parts (a) & (b) are analogous to the behavior of semiconductors in an external electric field. Just as the force \(F_{T}\) accelerates the object and changes its kinetic energies, the external electric field in a semiconductor causes valence electrons to gain energy and cause current flow.

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Most popular questions from this chapter

A semime tal (e.g. antimony, bismuth) is a material in which electrons would fill states to the top of a band the valence band- -except for the fact that the top of this band overlaps very slightly with the bottom of the nexthigher band. Explain why such a material, unlike the "real" metal copper, will have true positive charge carners and equal numbers of negative ones, even at zero temperature.

As a crude approximation, an impurity pentavalent atom in a (tetravalent) silicon lattice can be treated as a one-electron atom, in which the extra electron orbits a net positive charge of 1 . Because this "atom" is not in free space, however, the permitivity of free space, \(\varepsilon_{0}\) must be replaced by \(\kappa \varepsilon_{0}\), where \(\kappa\) is the dielectric constant of the surrounding material. The hydrogen atom ground-state energies would thus become $$ E=-\frac{m e^{4}}{2\left(4 \pi \kappa \varepsilon_{0}\right)^{2} h^{2}} \frac{1}{n^{2}}=\frac{-13.6 \mathrm{eV}}{\kappa^{2} n^{2}} $$ Given \(\kappa=12\) for silicon, how much ener gy is needed to frec a donor electron in its ground state? (Actually. the effective mass of the donor electron is less than \(m_{e}\), so this prediction is somewhat high.)

In the boron atom, the single \(2 p\) electron does not completely fill any \(2 p\) spatial state, yet solid boron is not a conductor. What might explain this? (It may be helpful to consider again why beryllium is not an insulator.)

In a concise yet fairly comprehensive way, explain why doped semiconductors are so pervasive in modem technology.

Section 10.2 discusses \(\sigma\) -bonds and \(\pi\) -bonds for \(p\) -states and \(\sigma\) -bonds for s-states. but not \(\pi\) -bonds for s-states. Why not?
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