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

The ratio of \(F_{T}\) to acceleration is negative when we account for the torque balance between \(F_{T}\) and friction during rolling without slipping. \(F_{T}\) does all the work in our system, equivalent to the rate of increase in the system's translational and rotational kinetic energies. When an external electric field is applied to a semiconductor, it's like the friction in our system and does the work of driving the charges, resulting in an overall increase in system energy, akin to increase in kinetic energies in our mechanical system.

Step-by-step solution

Proving the ratio of pulling force to acceleration is negative

Understand that the tension \(F_{T}\) tends to rotate the object and produce a torque \(T_{\text{rot}}\). This torque is given by \(T_{\text{rot}} = F_{T} x R\). If the object is rolling without slipping, create a pseudo force factoring in friction which works in the opposite direction to \(F_{T}\). This gives a torque due to friction \(T_{\text{fric}} = f x r\). The torques must balance for the object to roll without slipping, hence, \(F_{T} x R = f x r\). Considering f = µN (where µ is the coefficient of friction and N is the Normal force), and N = mg (since it's rolling without slipping), we substitute these into the equation to find the ratio \(F_{T}/a\) is negative as \(F_{T}\) and \(a\) would be in opposite directions.

Verification of work done by \(F_{T}\)

Remember that power is work done per unit time and the work done \(W\) is given by the force \(F_{T}\) times the distance covered (which is equal to the speed \(v\) times time). Hence power \(P = F_{T} x v\). On the other hand, the total kinetic energy is the sum of translational and rotational kinetic energies. The increase in kinetic energy is the work done by \(F_{T}\), hence proving that \(F_{T} x v = (1/2 \ mv^2 + 1/2 \ Iω^2)/t\), where \(ω = v/r\) is the angular velocity, and I is the moment of inertia.

Discussing the behavior of semiconductors

The electric field in semiconductors is akin to the frictional force in the system. In part (a), the electric field helps charges to move and in part (b), it’s like the external load that does all the work of driving the charges resulting in an increase in overall system energy, akin to increase in kinetic energies we saw in our mechanical system.

Key concepts

Understanding Rotational Motion

Rotational motion is the movement of a body around a fixed axis or point. Just like linear motion, which involves the change in an object's position, rotational motion is concerned with the change in orientation. For objects rolling without slipping, consider a scenario with two elements: translational motion (straight-line movement) and rotational motion (spinning). In physics, an object undergoing rotational motion has an angular velocity \(\omega\), which is the rate at which it rotates around the axis. During rolling without slipping, the distance covered by the object (its arc length) is equivalent to the linear distance traveled by a point on the edge of the object. This setup creates a special relationship where the linear speed \(v\) of the object is equal to \(\omega r\), where \(r\) is the radius of the wheel.

The exercise demonstrates this principle by focusing on an object being pulled by a string which results in a rotational motion about its axis. Inherent to this motion are concepts of torque and energy conservation, which are vital to understanding the mechanics of rotating systems in various applications, such as the movement of wheels on a vehicle or the spinning of a turbine blade.

The Role of Torque in Rotational Motion

Torque is the measure of the force causing an object to rotate around an axis, and is of critical importance in the analysis of rotational motion. It depends not just on the force magnitude but also on the distance from the force application to the axis of rotation. The formula for torque \(\tau\) is given by \(\tau = F \times r\), where \(F\) is the force applied and \(r\) is the radius or distance from the axis. It's the rotational equivalent of force in linear motion.

In the exercise, a string exerts a force \(F_{\mathrm{T}}\) which results in torque \(\tau_{\text{rot}} = F_{\mathrm{T}} \times R\). This torque must be countered by the friction between the object and the surface it rolls on, which provides a torque in the opposite direction, ensuring the object rolls without slipping. The balance between these torques is essential in maintaining a stable rotational motion, analogous to equilibrium in linear motion. This balance also determines the acceleration of the object, revealing the negative ratio of pulling force to acceleration.

Kinetic Energy in Rotational Motion

Kinetic energy is the energy that an object possesses due to its motion. For a rolling object, there are two types: translational kinetic energy and rotational kinetic energy. Translational kinetic energy is given by \(\frac{1}{2}mv^2\), where \(m\) is the mass of the object and \(v\) is its velocity. Rotational kinetic energy, on the other hand, is given by \(\frac{1}{2}I\omega^2\), where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

The exercise involves the work-energy principle, where the work done by the pulling force \(F_{\mathrm{T}}\) translates to an increase in both types of kinetic energy as the object rolls. Equating the power delivered by the tension force to the rate of increase in kinetic energy reinforces this principle. It emphasizes the conservation of energy, with the work done by torque transformed into kinetic energy, illustrating how the rotational motion transfers energy from one form to another without loss.

Semiconductor Behavior Under External Electric Field

Semiconductors exhibit specific behaviors when an external electric field is applied. In semiconductors, charge carriers (electrons and holes) move in response to the electric field, which is similar to how the frictional force acts in the rolling motion scenario from our exercise. The external field does work on these charge carriers, increasing the energy within the semiconductor material.

Comparing the physics of rolling objects with semiconductor behavior offers an insightful analogy. Part (a) of the exercise, where the ratio of force to acceleration is analyzed, can be related to how the mobility of charge carriers is influenced by the electric field. In part (b), akin to how the string's tension does all the work in a mechanical system, the electric field is responsible for doing the work on charge carriers, increasing the semiconductor's energy. This parallel helps clarify the dynamics of semiconductor behavior in a vivid and mechanical context.