The cellular motor driving the flagellum in \(E\). coli (see Problem 47 ) exerts a typical torque of \(400 \mathrm{pN}\) -nm on the flagellum. If this torque results from a force applied tangentially to the outside of the 12 -nm-radius flagellum, what's the magnitude of that force?

Understanding the concept of torque is fundamental in the realm of physics, especially when dealing with rotational systems. Torque, often symbolized as \( \tau \) or \( T \), measures the tendency of a force to rotate an object around an axis. It is a vector quantity, meaning it has both a magnitude and a direction. The calculation of torque is straightforward yet profound:
\[ T = F \times r \times \sin(\theta) \]
Where \( F \) is the force applied, \( r \) is the distance from the axis of rotation to the point where the force is applied, and \( \theta \) is the angle between the force vector and the lever arm. In many cases, including the cell's flagellum problem, the force is applied tangentially, or perpendicular, to the lever arm, making \( \sin(\theta) = 1 \) and simplifying the torque formula to \( T = F \times r \).
When solving for torque, it's essential to use consistent units. In this context, the torque was given in picoNewtons-nanometers (pN-nm), requiring that the force also be in picoNewtons (pN) and the distance in nanometers (nm). By rearranging the torque equation, we can calculate the force exerted by the cellular motor:

Force from Torque

To find the force, we simply divide the torque by the radius, assuming the force is applied perpendicularly to the radius:
\[ F = \frac{T}{r} \]
This relationship illustrates a key principle in physics: the interdependence of force, distance, and torque. A smaller force can produce the same torque if it is applied at a larger distance from the rotation axis, highlighting the importance of leverage in mechanical systems.

The mechanical force, in physics, is essentially the push or pull upon an object resulting from the object's interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects that can cause the objects to be pushed or pulled in particular directions. Force is a vector quantity—inclusive of both magnitude and direction—and is measured in the SI unit of newtons (N). In the microscopic world of bacteria flagella, forces are often expressed in picoNewtons (pN), which are a trillionth of a newton.
In the aforementioned physics problem, the force exerted by the cellular motor is tangential to the flagellum's surface—a situation frequently encountered in physics problems involving rotating bodies or angular motion. The direction of this force is significant, as it determines the direction of the torque, which subsequently dictates the rotational motion of the object.

The Role of Force in Torque

It's important to note that the larger the force or the greater the distance from the rotation axis (radius), the larger the torque. This kind of relationship makes it possible to fine-tune mechanical systems by adjusting one or both of these variables—like a wrench being used to loosen a nut, where the length of the wrench amplifies the force exerted by a person's hands.

Angular motion is motion in a circular path and is pertinent in the study of objects that rotate about an axis. This type of motion is pervasive in our daily lives, from the spinning of tires on a car to the rotation of celestial bodies. The basic parameters that describe angular motion include angular velocity, which tells us how fast an object spins or rotates, and angular acceleration, which describes how quickly the angular velocity changes with time.
Understanding torque is essential when studying angular motion because torque affects angular acceleration in much the same way that force affects linear acceleration in Newton's second law of motion. Just as force causes an object to accelerate linearly, torque causes an object to gain angular acceleration.

Connecting Torque and Angular Motion

In the problem related to the E. coli flagellum, the torque applied by the motor induces angular motion, causing the flagellum to spin. The relationship of torque to angular motion is governed by the equation:\[ \tau = I \times \alpha \], where \( \tau \) is the torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular acceleration. This equation is analogous to Newton's second law (\( F=ma \)), where instead of mass we have moment of inertia, and instead of linear acceleration, we have angular acceleration.
Analyzing angular motion through these principles enables us to predict and control the behavior of rotational systems, providing insights necessary for technological advancements, from microscopic biological motors to large-scale mechanical engines.