# Chapter 1 Physics for Introductory Biology

## 1.1 Scalar and vector quantities

If a measure has magnitude only, it is a scalar quantity. If a measure has a magnitude and a direction, it is a vector quantity. A scalar more generally is a single number. A vector more generally is a column (or row) of numbers. A vector is called a vector because the column of numbers can be represented by an arrow from the origin to the point in space with coordinates equal to each number in the column. The magnitude is the length of the arrow. The direction is the direction of the arrow. An example of a vector quantity in Physics for Introductory Biology is the electrochemical “driving force” of an ion – the magnitude is the capacity to move the ions across the membrane and the direction is “into” or “out of” the cell.

## 1.2 Concepts of motion

displacement is the change of position of an object. This change includes a distance and a direction so it is a vector quantity. For example. If I’m measuring the flow of labeled water in xylem sap, I’ll measure the labelled water at two points in time. The displacement is the distance traveled in the direction of the flow.

Velocity is the change in displacement over time. So in our example the xylem sap moved 9.8 cm during the measured time period, which is 60 s, so the velocity .16 cm s$$^{-1}$$. Velocity is a vector quantity, meaning there is a magnitude component, which is also called speed, and a directional component. So xylem flowing up a tree at .16 cm s$$^{-1}$$ and phloem flowing down a tree at .16 cm s$$^{-1}$$ have different velocities but the same speed. This comparison seems absurd because all we care about is the speed in this comparison, which is the same for the xylem and phloem. But direction (and so velocity) does matter in many physiological comparisons. If you walk in a straight line at 8 miles per hour or you walk in a circle with a radius of 2 feet at 8 miles per hour than your velocity is not changing in the first but is in the second. Your inner ear senses this change in velocity and is the first step in you becoming dizzy.

Acceleration is the change in velocity over time. Your inner ear contains an organ that functions as an accelerometer. We cannot sense velocity - inside a plane I cannot tell if the plane is sitting on the tarmac or flying at 500 mph - but we can sense change in velocity (or acceleration). Acceleration also is a vector quantity. Importantly, in everyday language we use “accelerate” to mean “getting faster” and “decelerate” to mean “slowing down”, but in science “getting faster” is positive acceleration and “slowing down” is negative acceleration (that is, a negative number), and this is with respect to some direction.

Jerk is the change in acceleration over time. We won’t talk about jerk!

## 1.3 Concepts of density, mass, inertia, force, momentum

The density of an object is how much the space bound by the object is filled with matter (atoms). The closer the atoms or molecules in the object are to each other, the less “nothing” there is and the more dense the object. A box of air isn’t very dense because air is a collection of gas molecules that are relatively far apart with nothing in between. A rock is more dense than air because the atoms that make up the minerals in the rock are all bound closely together.

There are two ways to define mass. The material definition of mass is a measure of the total amount of matter in an object (where density is a per volume measure), so this is the density times the volume1, where $$\rho$$ (the greek letter rho) is density. The inertial definition is: mass is the property of an object that resists acceleration (this property is inertia). This way of thinking about mass blew my mind when I learned it, because it is much more useful in functional biology. To understand the inertial definition, we need to know what makes an object accelerate, which is a force.

A force is the something applied to an object that potentially causes the object to accelerate. A force isn’t necessary for an object to move. A force applied to an object slows it down or speeds it up. So blood moving through an artery is slowed down by friction (a type of force) and speeded up by the heart pressurizing the blood (another kind of force).

Newton’s second law states that force is the product of mass times acceleration2. We can re-arrange this to $$A=\frac{F}{M}$$. Given the same force applied to two objects, the more massive object (bigger $$M$$) will have a smaller acceleration. So this is the inertial definition of mass: mass is the property of an object that resists acceleration. This concept leads directly to the concept of momentum.

Momentum is the mass of an object times its velocity3, where $$\nu$$ is the greek letter nu. We usually think of momentum as we would inertia: an object with more momentum resists change in direction and/or speed more than an object with less momentum. But it’s really the mass (inertial) component of momentum that makes this so.

Finally, note that the change in momentum over time is $$\frac{\Delta M \nu}{T} = M \frac{\Delta \nu}{T} = MA = F$$! That is, force is the change in momentum over time.

## 1.4 Energy and power

Energy is a very elusive concept but here are a couple of notes. First, energy is never created or destroyed, it just changes from one form to another and this really is the story of much of science, including biology. One form of energy is called mechanical energy and it’s the energy of doing work where work is the energy necessary to displace an object over some distance. Work is equal to the force applied to the object times the distance the object moves4. A tree has to use energy to move xylem sap up its trunk and we say that it “does work on the xylem”. This work (mechanical energy) is the force applied to the xylem times the distance the xylem moves. The longer the distance the more work (given the same force). Another form of energy is kinetic energy which is the energy of a moving object, and is one-half of the product of the object’s mass and velocity squared5. When a cheetah runs, its hand and foot impact the ground with a certain amount of kinetic energy which suddenly goes to zero so this energy is transferred into the skeleton of the limbs. The cheetah will want a skeleton that can absorb and release this energy without permanently deforming or breaking the skeleton! Another form of energy is potential energy which is the capacity to do work. huh? A way of thinking about this is, potential energy isn’t doing anything but it can! When it does, the potential energy is converted to some other form of energy. Potential energy comes in several forms. It could be the energy of an elevated mass in a gravitational field. Gravity makes the mass fall, which transfers this potential energy to kinetic energy. Or it could be the electrochemical energy in an ion gradient, such as the H$$^+$$ gradient in a mitochondria. The potential energy of the H$$^+$$ gradient is transferred into the kinetic energy of the ATP synthase mechanism which is then transferred into the potential energy of the phosphate bond in ATP. Or it could be the elastic strain energy stored in the myosin head. We will talk a lot about this kind of energy transfer.

Power is the rate of working or the rate that energy is used, so is equal to the Work divided by the time spent doing the work6. It takes about the same amount of energy (work) for a cheetah to walk or to run a mile but the running cheetah expends this energy over a much shorter amount of time so running requires more Power. Note that since $$W=FD$$ then $$P=F\frac{D}{T} = F\nu$$, that is power is the product of force and velocity7. So high power activities are activities with high force at a high velocity or done over a short amount of time.

1. $$M=\rho V$$

2. $$F=MA$$

3. $$p = M\nu$$

4. $$W=FD$$

5. $$KE = \frac{1}{2}MV^2$$

6. $$P=\frac{W}{T}$$

7. $$P=F\nu$$