Question

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

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

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 \(F_{T} \cdot R\).

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.

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.

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.