In the process of mitosis (cell division), two motor proteins pull on a spindle pole, each with a 7.3 -pN force. The two force vectors make a \(65^{\circ}\) angle. What's the magnitude of the force the two motor proteins exert on the spindle pole?

In physics, understanding the concept of the vector sum of forces is crucial when dealing with multiple forces acting at different angles. A force vector represents both the magnitude and the direction of the force. When two or more forces are applied to an object, they often do not act in the same direction, and therefore their impact cannot be assessed by simply adding their magnitudes.

Consider two forces represented by vectors; their combined effect—known as the resultant force—doesn't necessarily equal the arithmetic sum of their magnitudes. Instead, we need to add these vectors together taking into account their directions. When the forces are aligned, the sum is straightforward: you simply add up the magnitudes. However, when forces are applied at angles to each other, one has to consider vector components or vector addition rules. To find the vector sum of forces, we can use methods such as the parallelogram law, graphical representations, or analytical methods, which involve breaking down each vector into perpendicular components and then combining these to find the sum.

To precisely calculate the vector sum when the forces have an angle between them, the law of cosines is a mathematical treasure. It's a generalization of the Pythagorean theorem and works for any triangle, not just right-angled ones. The law of cosines states that for any triangle with sides 'a', 'b', and 'c', and the angle opposite to 'c' being \( \theta \), the relationship between the sides and the angle is given by:
\[ c^2 = a^2 + b^2 - 2ab\cos(\theta) \]

Where \( c \) is the length of the side opposite the angle \( \theta \), while \( a \) and \( b \) are the lengths of the other two sides. This formula lets us solve for an unknown side when we know the other two sides and the included angle. In the context of vector forces, \( c \) would represent the magnitude of the resultant force, while \( a \) and \( b \) would be the magnitudes of the individual forces, and \( \theta \) the angle between them. This is exactly how we approach the force vectors exerted by motor proteins during cell division.

Motor proteins are fascinating molecular machines within cells that use energy to move and generate forces, essential in numerous cellular processes, including the mitosis mentioned in the exercise. They often function in teams to produce concerted action, like the two motor proteins pulling on a spindle pole. These proteins can convert chemical energy directly into mechanical work.

During mitosis, motor proteins play a key role in manipulating the structure known as the spindle apparatus. This structure is responsible for distributing chromosomes into the two daughter cells. The force they produce is measured in piconewtons (pN), and as illustrated in the exercise, understanding how these tiny forces combine using vector addition and the law of cosines can give us insights into the biomechanical events of cellular division.