PHY 271: Physics for Biology Comprehensive Study Notes

Introduction to Physics for Biology and Measurement

Physics is fundamentally a process of learning about our physical world by finding systematic ways to make sense of what we observe and measure. It is a science that deals with the study of matter in relation to energy, where matter is characterized by its ability to do work because of the energy it possesses. For a precise and accurate study of matter, measurement must be carried out, and units must be assigned to every quantity. Key quantities include length, time, mass, temperature, pressure, and electric current. Physics is based on these measurements, and the discipline is discovered by learning how to measure these quantities. Measurement tells us about the property of a particular thing and assigns a number to that property, such as how heavy, hot, or long an object is. These measurements are always made using specific instruments including meter rules, vernier calipers, beam balances, stopwatches, thermometers, and ammeters. Every measurement result consists of two necessary parts: a number and a unit of measurement.

Several critical terms must be understood within the context of measurement. Uncertainty is the doubt that exists about the result of any measurement, representing the quantification of doubt which requires an interval and a confidence level to be expressed properly. Error is defined as the difference between the measured value and the true value. Precision refers to the quality, condition, or fact of being exact and accurate, where being exact indicates there is no deviation in the measurement carried out. A unit is the name or symbol given to a standard of measurement. In physics, units must be consistent and not haphazard, which often necessitates the use of conversion factors. This course, Physics for Biology, is essential because certain aspects of biology require precise observations and measurements. Versatility in physics enhances the optimum performance of individuals in diverse areas of applied biology by teaching them how to observe and measure quantities effectively.

Fundamentals and Types of Motion

Motion, or movement, is a vital aspect of living things and occurs when a body changes its position with reference to time, which is essentially a change in separation between two points over time. A body is in motion if its position is changing with respect to a fixed position and time, covering a distance. In humans, it is essential that the majority of the body, including organs and systems, is able to move. Noticeable motions occur in parts such as the eye, neck, fingers, hands, toes, legs, lungs, intestines, and the heart. Stiffness in any part of the human body results in discomfort and necessitates interventions to ensure mobility, which is the ability to move without hindrance. Several parameters are used to determine motion, including position, distance, speed, velocity, and acceleration.

There are various distinct types of motion. Random motion, also referred to as zig-zag motion, is the motion of an object with no regular or definite pattern, such as the motion of gas molecules, butterflies, perfume particles, or smoke. Rectilinear motion is motion along a straight-line path, exemplified by a light ray traveling from one point to another. Translational motion occurs in one direction, such as a car moving from one town to another. Rotational or circular motion involves a body moving in a circle or ellipse and rotating about an axis, like the rotation of the Earth or the blades of an electric fan. Oscillatory or vibratory motion is the to-and-fro movement of an object about a fixed point, such as a simple pendulum, plucked guitar strings, or the vibration of molecules in a solid.

Vectors and Motion Parameters

Cartesian coordinates are used to locate positions on a plane using two coordinates $(x, y)$ or in space using three coordinates $(x, y, z)$. Distance is covered when there is a change in position, while displacement is the distance covered in a specified direction, often illustrated using bearings measured from true North. Speed is the rate of change of distance over time, reflecting how fast an object moves, whereas velocity is the change of displacement with time. Various types of velocity include instantaneous, relative, uniform, initial, final, and average velocity. Average velocity $V_{avg}$ is the ratio of displacement $\triangle x$ to the time interval $\triangle t$. Acceleration is the change of velocity with time, and uniform acceleration occurs when this change is constant over a given period.

A vector quantity is any physical quantity that has both a magnitude and a direction, such as force and velocity. Vectors are integral to human daily activities, including breathing—where diaphragm muscles exert force with magnitude and direction—walking, running, jumping, and sporting activities like throwing a javelin or football. A vector $A$ in a two-dimensional plane can be resolved into components $A_x = A \cos(\theta)$ and $A_y = A \sin(\theta)$. The magnitude is found using the Pythagorean theorem where $A = \sqrt{A_x^2 + A_y^2}$, and the angle is calculated as $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$. Vector addition involves summing components to find a resultant vector $R$, where $R = (A_x + B_x)i + (A_y + B_y)j$. The scalar or dot product is defined as $A \cdot B = AB \cos(\theta)$, while the vector or cross product is $A \times B = C$, with a magnitude of $C = AB \sin(\theta)$ and direction determined by the right-hand screw rule.

Newton’s Laws and Equations of Motion

Newton’s laws of motion summarize the relationship between force and motion. The first law, the law of inertia, states that every body continues in its state of rest or uniform motion in a straight line unless compelled by an external force; inertia is the reluctance of an object to change its state. The second law states that the rate of change of momentum is proportional to the applied force and acts in the direction of the force, expressed as $F = ma$. The third law states that for every force applied, there is an equal and opposite reaction. These laws lead to three fundamental equations of motion: $v = u + at$, $s = ut + \frac{1}{2}at^2$, and $v^2 = u^2 + 2as$. These equations are derived from the definition of acceleration and average velocity, where $s = \frac{1}{2}(u + v)t$.

Mass is the quantity of matter in a body and measures inertia. Weight is the force of gravity on that mass. Momentum is the product of mass and velocity. The impulse-linear momentum theorem states that the change in linear momentum in a collision equals the impulse acting on the body, where impulse is the product of force and time. In any collision, total linear momentum is conserved if no external forces act. Elastic collisions conserve both momentum and kinetic energy, with the formula $\frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$. Inelastic collisions involve bodies sticking together and moving at a common velocity, where kinetic energy is not conserved. The work-kinetic energy theorem states that net work done on an object equals its change in kinetic energy: $W_{net} = K.E_f - K.E_i = \triangle K.E$.

Rotational Dynamics and Equilibrium

The turning effect of a force is the moment of force or torque $\tau$, calculated as the product of the force and the perpendicular distance from the pivot to the line of action: $\tau = F \times r$. For maximum effectiveness in muscle activity, the line of action should be as far from the joint as possible. A body is in equilibrium if the sum of all forces and moments is zero. Static equilibrium occurs at rest, while dynamic equilibrium occurs during uniform velocity where acceleration is zero. Conditions for equilibrium require that total upward forces equal downward forces, total components in one direction equal those in the opposite, and clockwise moments equal anticlockwise moments. The triangle law of forces states that three forces in equilibrium can be represented by the sides of a triangle.

Circular motion involves movement in a circular path, such as a car on a bend or planets in orbit. Angular displacement $\theta$ is the angle turned through, and angular velocity $\omega$ is the rate of this displacement: $\omega = \frac{d\theta}{dt}$. The relationship between linear velocity $v$ and angular velocity is $v = r\omega$. Angular acceleration $\alpha$ is $\frac{d\omega}{dt}$. Centripetal acceleration $a_c = \frac{v^2}{r}$ is directed toward the center, sustained by a centripetal force $F_c = \frac{mv^2}{r}$. Centrifugal force tends to throw a particle out of its orbit and must be balanced by centripetal force. The moment of inertia $I = \sum mr^2$ depends on mass, dimensions, and the axis of rotation. Torque is related to inertia by $\tau = I\alpha$. Angular momentum is $L = I\omega$, and rotational kinetic energy is $K.E = \frac{1}{2}I\omega^2$. For rolling bodies, total kinetic energy is the sum of translational and rotational energy: $K.E_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$. Angular momentum is conserved if no external torque acts.

Elasticity and Bone Physics

Elasticity is the phenomenon where a rigid body's dimensions change due to pulling, pushing, twisting, or compression. Tensile stress $\sigma$ is force per unit area $F/A$, and tensile strain $\epsilon$ is extension per unit length $e/L$. Young’s modulus $E$ is the ratio of stress to strain. Hooke’s law states that extension $x$ is directly proportional to applied load $F$: $F = kx$, where $k$ is the stiffness constant. A restoring force is denoted by $F = -kx$. Materials exhibit elastic behavior until the elastic limit, after which plastic deformation occurs, meaning the material does not return to its original length. Ductile materials lengthen considerably before breaking, while brittle materials break shortly after the elastic limit. Elastic moduli include Young Modulus for length, Shear Modulus for tangential force, and Bulk Modulus for volume change. Compressibility $K$ is the inverse of the Bulk Modulus $B$.

In biology, bone is composed of collagen and bone salt (calcium, phosphorus, oxygen, and hydrogen). Bone salt crystallites increase the stiffness of bone significantly. Bones support weight, resist bending, and enable movement. Since weight varies with the cube of linear dimensions while cross-sectional area varies with the square of diameter, larger animals require relatively thicker legs. Wolff’s law states that living bone increases its strength in response to prolonged mechanical stress, which aids fracture healing. However, replacing bone with metal prosthesis can lead to fretting—damage caused by relative movement between materials with different elastic constants. Bone cement is used to spread the load and reduce this stress.

Forces in Fluids and Fluid Movement

Fluids include liquids and gases that flow and conform to container boundaries because they cannot sustain tangential forces. Water makes up about $70\%$ of the human body, with most as intracellular fluid and the rest as extracellular fluid (plasma, tissue fluid, lymph). Water movement is driven by differences in hydrostatic pressure (generated by fluid weight or gravity) and osmotic pressure (needed to prevent inward flow across a semi-permeable membrane). Osmosis is the passage of solvent from dilute to concentrated solutions. Diffusion is the mixing of fluids due to random molecular motion; Graham’s law states the rate of diffusion is inversely proportional to the square root of density. Cohesion is attraction between like molecules, while adhesion is between different molecules. Capillarity is the rise or fall of liquid in a tube.

Pressure is force per unit area $P = F/A$, measured in pascals ($N/m^2$). It increases with depth $h$ and density $\rho$, according to $P = \rho gh$. Absolute pressure is the sum of atmospheric and gauge pressure: $P_{abs} = P_a + \rho gh$. Siphons and syringes use atmospheric and liquid pressure differences. Fluid in motion follows Bernoulli’s principle, which states that for an incompressible non-viscous fluid, the sum of pressure, kinetic energy per unit volume, and potential energy per unit volume is constant: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$. This explains why pressure is low where velocity is high. Applications include filter pumps, aerofoil lift in planes, venturimeters, and suction effects near fast trains.

Kinetic Theory and Thermodynamics

Matter consists of molecules in constant motion. Brownian motion—the erratic movement of tiny particles in suspension—provides evidence for this molecular activity. The kinetic theory of gases assumes identical molecules in random motion with elastic collisions and negligible intermolecular forces except during collisions. Gas pressure is derived as $P = \frac{1}{3} \frac{N}{V} mv^2$, showing pressure is proportional to the number of molecules per unit volume and average translational kinetic energy. Temperature is a direct measure of average translational kinetic energy: $T = \frac{2}{3k_B} (\frac{1}{2}mv^2)$. The theorem of equipartition of energy states each degree of freedom contributes $\frac{1}{2}k_BT$ to the energy. The root-mean-square speed is $v_{rms} = \sqrt{\frac{3k_BT}{m}}$. Boltzmann’s distribution law describes the probability of molecules occupying specific energy states, while the Maxwell-Boltzmann distribution describes the range of molecular speeds.

Thermodynamics is the study of heat energy and its conversion to work. A system is in thermodynamic equilibrium if it is in mechanical, chemical, and thermal equilibrium. The Zeroth law establishes that if two systems are each in thermal equilibrium with a third, they are in equilibrium with each other, defining temperature. The First law is the conservation of energy: $Q - W = \triangle U$, where $Q$ is heat added, $W$ is work done, and $\triangle U$ is the internal energy change. Different processes include adiabatic (no heat exchange, $Q=0$), isobaric (constant pressure), isochoric (constant volume, $W=0$), and isothermal (constant temperature). The ideal gas law is $PV = nRT$ or $PV = Nk_BT$. In biology, environmental temperatures affect metabolic rates. Poikilotherms (lizards) have body temperatures following the environment, while homeotherms (birds, mammals) regulate constant body temperatures via the hypothalamus in the brain using mechanisms like sweating, breathing rates, fat insulation, and ruffling hair or feathers to trap air.