Biophysics Exam Notes

Biophysics Recap

1001AHS End of Trimester Examination

• Date & Time: Saturday 7th June

• Duration: 2 hours (+ 10-min perusal)

• Format: 40 multiple choice questions

  • 40 marks; 40% of course grade

  • Theoretical and problem-based questions included; formula sheet provided

  • Section A: theoretical/concept questions (23 marks)

  • Section B: calculation-based questions (17 marks)

• Coverage: Module 3 (Biophysics only)

• Materials: Pencil, eraser, non-programmable scientific calculator, ruler

• Provided: 1x mark sense sheet; 1 x blank lined booklet
  *Note that the details above are for the main sitting of the exam. Dates and/or venues for Accessible and Alternate exam sittings may differ

Key Theoretical Concepts

• Kinematics

• Kinetics

• Rotational Kinematics and Kinetics

• Energy, Work, Power, and Momentum

• Fluid Dynamics

Topic 1: Kinematics

Definition

• Kinematics describes motion in linear and rotational terms without regard for the forces that underpin it.

Scalar vs Vector

• Human movement kinematics often described using two quantities:

  • Scalar

  • Vector

• Scalar quantities only have magnitude.

• Vector quantities have magnitude and direction.

Kinematic measurements

• Common kinematic measurements:

  • Scalar: Distance (x, m), Speed (v, ms⁻¹), Time (t, s)

  • Vector: Displacement (x̄, m), Velocity (v̄, ms⁻¹), Acceleration (ā, ms⁻²)

Distance vs Displacement

• Distance: Total length of the path traveled by an object, regardless of direction.

• Displacement: Straight-line distance and direction from an object's starting position to its final position.

Speed and Velocity

• Speed: How quickly we change our position.

• Speed is a scalar; it does not describe the direction of motion.

  • v_{avg} = \frac{total\ distance}{total\ time} = \frac{x}{t}

• Velocity: Describes how quickly we change position and has direction.

• Velocity is a vector.

  • \vec{v}_{avg} = \frac{total\ distance}{total\ time} = \frac{x}{t}

Acceleration

• The rate of change of velocity.

Acceleration

• Acceleration describes how quickly we change our velocity.

• Acceleration is a vector, it has a direction (sign).

  • \vec{a} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}f - \vec{v}i}{\Delta t}

Acceleration vs Deceleration

• Acceleration and velocity vectors can point opposite to each other.

• If velocity and acceleration have opposite signs, you are slowing down in the direction of motion.

• If velocity and acceleration have the same sign, you are speeding up in the direction of motion.

Interpreting displacement-time graphs

• Displacement-time graphs can be used to determine velocity and direction of motion.

Kinematic Equations of Uniform Motion

• These relationships allow us to calculate an unknown based on knowledge of other variables.

• Uniform motion = constant acceleration.

• Problems are solvable when there is only one unknown in the equation.

  • Vf = Vi + at

  • x = \frac{1}{2} (Vi + Vf) t

  • x = V_i t + \frac{1}{2} a t^2

  • Vf^2 = Vi^2 + 2ax

  • V_{avg} = \frac{x}{t}

Motion in 2D: Projectile motion

• A common example of 2D motion.

• Parabolic path due to gravity acting in the vertical direction.

• Assume no air resistance, so vertical acceleration is constant.

• Horizontal velocity is constant – initial is the same as final.

Velocity components during projectile motion

• Horizontal velocity v_x does not change.

• Resultant velocity v_{total} is completely horizontal at the apex (but still has a velocity).

• Vertical velocity is the same magnitude at the start and end, but has an opposite sign.

• Only the acceleration vector is vertical downward (gravity).

Topic 2: Kinetics

Definition

• Kinematics describes motion in linear and rotational terms, while kinetics describes the forces that underpin motion.

Net force

• A net force is the vector sum of all forces acting on a body.

• If several forces are acting:

  • If there is no change in motion, then the net force is zero, i.e., the forces are balanced. \sum F = 0

  • If there is a change in motion, then the net force is greater than zero, i.e., the system is unbalanced. \sum F = 5N\ Right

Net force

• If an object is moving at a constant velocity, one should generally assume that the net force acting on the object is zero.

Newton's Laws

• An object continues in a state of rest, or at a constant speed in a straight line, unless acted on by an unbalanced net force (Inertia).

• The acceleration of an object depends on the mass of the object and the amount of force applied (Acceleration).

• Whenever one object exerts a force on another, the second object exerts an equal and opposite force (law of action and reaction).

• “For every action, there is an equal and opposite reaction.”

Vector addition in 2D

• If vectors are facing in the same direction, we can just add them together!

  • Resultant = L + \frac{L}{2} = L + 0.5L = 1.5L

• If vectors are facing in opposite directions, we can just subtract them!

  • Resultant = L - \frac{L}{2} = L - 0.5L = 0.5L

• If vectors are perpendicular to each other (at a right angle), we can use Pythagoras' theorem to get the resultant.

  • Resultant = \sqrt{L^2 + L^2}

• If vectors are not perpendicular to each other, we first need to resolve them into the appropriate components before addition.

Resolving vectors into components

• A vector can be resolved into vertical and horizontal components using trigonometric functions.

  • Vertical component: L \sin{\theta}

  • Horizontal component: L \cos{\theta}

Adding resolved vectors

• After resolving vectors into vertical and horizontal components, we can add them together.

Finding the resultant of perpendicular vectors

• Now we have two vectors which are perpendicular to each other, and we can use Pythagoras' theorem to find the resultant.

  • Resultant = \sqrt{(L + L \cos{\theta})^2 + (L \sin{\theta})^2}

Types of Forces

• Normal Force

• Friction Force

• Tension Force

• Weight Force

Weight force

• A gravitational force

• An everyday force that a mass experiences due to its gravitational attraction with the Earth.

• A vector quantity.

• SI units: kg \cdot m/s^2 = N

• Different from mass:

  • Mass is the amount of matter in an object.

  • A scalar quantity.

  • SI units: Kg

• W = mg where g = 9.81 m/s^2

Normal force

• Associated with Newton’s 3rd law.

• A reaction force exerted by a surface on a contacting object perpendicular to the surface.

• Indicates how much two objects are pressing against each other.

• The magnitude may or may not be equal to weight magnitude. F_n = -W = -mg

Friction force

• An object in contact with a surface, while moving or attempting to move, experiences frictional force parallel to the surface.

• Friction acts in the direction opposing the intended direction of motion.

Equation for friction

• F{fric} = \mu Fn

  • F_{fric}: Friction force

  • \mu: Coefficient of friction

  • F_n: Normal force

Tension forces

• Tension forces are applied externally by:

  • An extended rope, a cable, a ligament, a tendon, etc.

• Tension forces are transmitted UNDIMINISHED, even if the “rope” bends around a pulley.

Topic 3: Rotational kinematics and kinetics

Angular displacement (θ)

• Angular displacement = Angle swept out by a radial line perpendicular to the axis of rotation.

• Units are Radians (rads).

Defining Radians

• The units of radians describe how the radius of a circle fits around the circumference of a circle.

• The radius always fits 6.28 (2π) around a full circle (360°); this is independent of the magnitude of the radius.

  • C = 2\pi r

• The radius always fits 3.14 (π) times around a half circle (180°).

• 3. 14 radians is commonly known as pi π.

• 28 radians is commonly known as 2pi 2π.

• If we get the angular measurement in degrees, we can convert it to radians using:

  • rads = degrees * \frac{\pi}{180°}

  • \frac{x\ radians}{n\ degrees} = \frac{\pi}{180}

Angular velocity (ω)

• The angular equivalent of linear velocity.

• The rate of change of angular displacement.

• Units are rads/s or rads s⁻¹.

• Can be measured using a gyroscope.

  • \omega = \frac{\Delta \theta}{\Delta t}

Angular and linear relationships

• Linear velocity is directly proportional to the angular velocity.

• Linear velocity is known as the tangential velocity (it is always at 90° to the radius).

• It is directly related to the angular velocity and magnitude of the radius.

  • Note: All points on a radius can have the same angular velocity but will have different linear velocity because the direction of the linear velocity vector is constantly changing!

  • Linear velocity v = r\omega

Uniform circular motion (constant angular velocity)

• All joints will have the same angular velocities.

• Tangential velocity (v) will be greatest at the most distal segment.

• Remember conventions for whether it is positive or negative:

  • Anti-clockwise rotation is positive, clockwise rotation is negative.

• Tangential velocity of all segments is changing during the motion, not in magnitude but in direction!

  • v = r\omega

Circular motion and centripetal force

• The center-seeking force that keeps the object turning 'inwards'.

Centripetal Force

• Required to keep an object of mass (m) moving at a constant velocity (v) in a circular path of radius r.

• SI unit: Newton (N).

• F = ma_c

Centripetal Acceleration

• Always acting towards the center of the circular path.

• SI units: ms-2

  • a_c = \frac{v^2}{r}

Rotational kinematic equations of motion

• Assumes constant acceleration.

  • Angular: \omegaf = \omegai + \alpha t, \theta = \omegai t + \frac{1}{2} \alpha t^2, \omegaf^2 = \omega_i^2 + 2 \alpha \theta

  • Linear: Vf = Vi + at, x = Vi t + \frac{1}{2} a t^2, Vf^2 = V_i^2 + 2ax

Angular Kinetics

• The forces causing objects to rotate: Torque (T) or Moment.

Torque or Moment

• A force applied at some distance to the axis of rotation or CoG.

Application of Torque

• A net force is applied at some distance from the axis of rotation.

• The value of the torque is found by multiplying the force by the length of the lever arm.

• The lever arm is the perpendicular distance between the line of action of the applied force and the axis of rotation.

• If the line of action of the force is not at 90 degrees to the lever arm, we need to determine the components of the force.

• Note: direction matters!

• If the same force is applied, torque will be greater with a longer lever arm!

  • \tau = Fl

Equilibrium

• If the net torque on an object is zero, the object is not rotating OR is rotating at a constant angular velocity.

  • \sum \tau = 0

• Often applied in biomechanics to analyse torques about a joint during isometric exercise.

• If a net torque is acting, an angular acceleration will be observed, i.e., the state of motion will change.

Topic 4: Energy, work, power, and momentum

Types of energy

• Kinetic Energy:

  • When a mass is in motion, it has energy.

• Potential Energy:

  • Elastic (tendons etc.).

  • Gravitational (at a height).

Kinetic Energy

• This is the energy of motion: for an object with a mass = m and velocity = v.

• The units for energy are the Joule (J).

• Compared to the calorie, which is commonly used relative to available energy in food or energy expenditure.

• The kinetic energy of an object is a measure of the work that an object can do because it is moving.

• Mass is important!

  • KE = \frac{1}{2} m v^2

Gravitational Potential Energy

• The stored energy of an object due to its position (height).

  • PE = mgh

Conservation of Energy

• Energy is neither created nor destroyed, only transferred.

• Kinetic to elastic potential.

• Elastic potential to gravitational potential.

• Gravitational potential to kinetic energy.

• In a conserved system, KE and PE will change to maintain total energy.

Work

• This describes the transfer of energy.

• If something changes its state of motion, a net force is acting.

  • WORK is the product of a force and the distance over which it is applied, units are either Joule or N.m.

• Note:

  • It has the same unit as energy.

  • If there is zero displacement, then the work done is zero.

  • W = Fd

Work/Energy relationships

• When work is done on an object that changes its velocity.

• When work is done on an object that changes its height.

  • W = \Delta KE

  • W = \Delta GPE

Power

• Power can also be expressed with its relationship to the velocity of movement.

• Velocity indicates the rate of change of position.

• It is the rate at which a force is applied (or produced).

• If power is a product of force and velocity, then max power is produced at a moderate velocity and moderate force (the two do not tend to co-exist maximally in muscle contractions).

  • P = \frac{work}{time} = \frac{w}{t} = \frac{Fd}{t} = FV

Linear Momentum

• In linear terms, Momentum is an object's mass multiplied by its velocity (units: kg m/s).

  • p = mv

Conservation of Momentum

• In a closed system (no external forces), total momentum is conserved.

• “What you start with is what you finish with, but it can be transferred within the system.”

  • pi = pf

Topic 5: Fluid dynamics

Fluids

• Fluids take the shape of their container.

  • Liquid

  • Gas

Density

• Density ρ of a substance is its mass (m) divided by its volume (v).

• SI units: kg/m^3

  • \rho = \frac{m}{V}

Pressure

• Pressure p is a force (F) applied perpendicular to the surface divided by the area (a) over which the force is applied.

• SI Unit: Pascal

• Other units of measurement:

  • Atmospheres (atm)

  • BAR

  • lb/in2 (PSI)

  • mmHG

  • 1atm = 760mmHG = 105Pa

• Pa = \frac{N}{m^2}

• P = \frac{F}{A}

Pascal's principle

• Any changes in the pressure applied to a completely enclosed incompressible fluid are transmitted undiminished to all parts of the fluid and the enclosing walls.

• Incompressible fluid: a fluid whose volume does not change under pressure, e.g., liquids.

  • \Delta P = \rho gh\n

Archimedes principle

• An object immersed in fluid experiences buoyant force (FB) equal and opposite to the weight (Wfluid) of the fluid it has displaced.

• Hint: buoyant force (FB) is directed vertically upwards.

  • FB = W{fluid}

Archimede’s Principle

• Buoyant force equals the weight of displaced water (in magnitude).

• Buoyant force has nothing to do with submerged objects and everything to do with the volume of water displaced.

Volume flow rate

• Volume of fluid flowing per second.

• Area (A) = cross-sectional area, velocity (V) = fluid velocity.

• SI Units = m2m/s = m3/s.

• Remember this for Physiology and cardiac output, e.g., how much blood is flowing across an artery’s cross-section in a given time.

  • Q = Area \times Velocity

  • Q = \frac{Volume}{time}

Equation of continuity

• For non-viscous fluids, we need to maintain the same flow rate.

  • A1V1 = A2V2

Bernoulli’s equation (non-viscous fluids)

• Pressure (P), velocity (v), and height (h) of a fluid are related at any two points along a vessel.

• This is conservation of energy - but for fluids.

  • P1 + \frac{1}{2} \rho v1^2 + \rho gh1 = P2 + \frac{1}{2} \rho v2^2 + \rho gh2

Fluid Pressure Change

• If height does not change, then if velocity increases, the pressure must decrease to maintain the sum along the streamline.

  • P1 + \frac{1}{2} \rho v1^2 = constant