Basics of mechanics

Mechanics is the branch of physics that studies the motion of objects and the forces that cause them to move. In addition to the linear and circular motion of rigid objects, mechanics also covers the deformation of non-rigid objects. A key concept in this field is pressure, which is discussed here along with density and hydrostatics. For all types of motion, this text describes their characteristic properties, the forces involved, and the associated work and power. These concepts are further explored through specific applications, such as the pendulum and the centrifuge.

The goal of physics is to describe natural processes and make them predictable using a few fundamental equations. The most important equations from mechanics that healthcare professionals may encounter are compiled in the following sections, illustrated with simple calculation examples. A solid understanding of the basic formulas is sufficient, as they can be easily rearranged or combined as needed. For a review of the necessary mathematical skills, see the article "Mathematics."

Newton's laws of classical mechanics

Classical mechanics is based on the work of Sir Isaac Newton, who formulated three fundamental laws of motion:

• Newton's first law of motion: An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. This is also known as the law of inertia.
• Newton's second law of motion: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. The acceleration occurs in the direction of the net force.
• Newton's third law of motion: For every action, there is an equal and opposite reaction. If object 1 exerts a force on object 2 (action), then object 2 exerts an equal and opposite force on object 1 (reaction).

Superposition of forces

Newton also described the superposition principle, which states that if multiple forces act on an object, they can be summed to find a net force. It is essential to remember that forces are vectors, meaning they have both a magnitude and a direction. For simplicity, the calculations below are shown without vector notation. However, the properties of force vectors are illustrated here with simple diagrams:

Force direction	Formula	Vector representation
Same direction	Fnet = F1 + F2
Opposite direction	Fnet = F1 - F2
Different directions	Fnet = F1 + F2

Motion is the change in an object's position over time. Forces can cause objects to accelerate or decelerate. The two main types of motion are linear motion, called translation, and rotational motion. There are several parallels between them. One way to transfer force from one body to another is through a collision, which involves the transfer of momentum.

• Force (physics): an influence that can change the motion of an object; a force will accelerate a body according to its mass
  • Formula: F = m × a
    • Unit: N (newton); 1 N = 1 (kg × m)/s²
    • F = force (N), m = mass (kg), a = acceleration (m/s²)
• Gravitational force (weight): the force that pulls an object toward the Earth; it is proportional to the object's mass and the acceleration due to gravity, g (≈ 9.81 m/s²)
  • Formula: F_g = m × g
    • Unit: N (newton)
    • F_g = gravitational force (N), m = mass (kg), g = acceleration due to gravity (≈ 9.81 m/s²)
• Momentum (physics): a measure of the quantity of motion of an object; since momentum has a direction, it is a vector quantity
  • Formula: p = m × v
    • Unit: Ns (newton-second) or kg⋅m/s
    • p = momentum (Ns), m = mass (kg), v = velocity (m/s)
• Conservation of momentum: In an isolated system (one with no external forces), the total momentum remains constant.
  • Example: In a rear-end collision, the car that strikes from behind slows down, and the car in front is pushed forward. The change in momentum of the rear car is equal in magnitude and opposite in direction to the change in momentum of the front car.
• Collision: an event where two or more bodies exchange momentum
  • Elastic collision: a collision in which both momentum and kinetic energy are conserved; any deformation that occurs is temporary
  • Inelastic collision: a collision in which momentum is conserved, but kinetic energy is not; permanent deformation occurs

What is commonly called "weight" is actually the gravitational force exerted on an object by the Earth. It is not an inherent property of the object itself but depends on its location in a gravitational field. This should not be confused with mass, which is an intrinsic property of an object and a measure of its inertia, or its resistance to acceleration.

Law of conservation of momentum: In an isolated system, the total momentum of all objects interacting within the system remains constant.

Example calculation

A metal ball with a mass of 5 kg is accelerated from rest to a velocity of 20 km/h in 3 seconds. What force is required?

• Wanted: force F (N)
• Given: mass m (kg), final velocity v (m/s²), time interval Δt (s)
  • v = 20 km/h = (20,000 m) (3,600 s) ≈ 5.56 m/s
  • a = Δv/Δt ≈ (5.56 m/s)/3 s ≈ 1.85 m/s²
  • F = m × a ≈ 5 kg × 1.85 m/s² ≈ 9.3 N

Mathematical description of motion

• Velocity: a measure of the rate of change of an object's position
  • Formula (for constant velocity): v = s/t
    • Unit: m/s
    • v = velocity (m/s), s = distance (m), t = time (s)
    • The velocity vector points:
      • In linear motion: along the line of motion
      • In curved motion: tangent to the curve at any given point, in the direction of motion
  • Special case free fall: neglecting air resistance, the velocity of a body in free fall starting from rest can be calculated using the acceleration due to gravity, g
    • Formula: v = g × t
    • v = velocity (m/s), g = acceleration due to gravity (≈ 9.81 m/s²), t = time (s)
• Acceleration: the rate of change of velocity
  • Formula: a = Δv/Δt
    • Unit: m/s²
    • a = acceleration (m/s²), Δv = change in velocity (m/s), Δt = change in time (s)
    • The acceleration vector always points in the same direction as the net force vector acting on the body.
  • Special case uniform acceleration: when an object's velocity changes at a constant rate
    • Distance-time relationship (from rest): s = ½ a × t²
      • s = distance in meters, a = acceleration (m/s²), t = time (s)
    • Velocity-time relationship (from rest): a = v/t
      • a = acceleration (m/s²), v = final velocity (m/s), t = time (s)
    • For uniform acceleration from rest or to a stop: a = v²/(2 × s)
      • Unit: m/s²
      • a = acceleration (m/s²), v = final velocity (m/s), s = distance (m)

Graphical representation: distance-time graph

Translational motion can be visualized using a distance-time graph (y-axis: distance s [m]; x-axis: time t [s]), which provides key information about the motion:

• Types of motion and their corresponding curves
  • Stationary object: a horizontal line parallel to the x-axis (see curve a)
  • Object with constant velocity: a straight line with a positive (or negative) slope (see curve b)
  • Object with constant acceleration: a parabolic curve (see curve c)
• Information from the graph
  • Instantaneous velocity: the slope of the tangent to the curve at any point
  • Average velocity: the total distance traveled divided by the total time elapsed
    • Uniform motion
      • For motion at a constant velocity (a linear curve), the average velocity is equal to the instantaneous velocity.
      • The acceleration is zero.
    • Non-uniform motion
      • The instantaneous velocity varies over time.
      • Acceleration is positive for speeding up and negative for slowing down (deceleration).

A rotation is the movement of an object around an axis. Many concepts from translational motion have analogous counterparts in rotational motion. Important applications of rotational principles include the balance scale, the lever, and the centrifuge.

• Torque: the rotational equivalent of force, which causes an object to rotate
• Formula (force perpendicular to lever arm): M = r × F
  • Unit: Nm (newton-meter)
  • M = torque (Nm), r = lever arm radius (m), F = force (N)
• The torque vector is perpendicular to the plane formed by the force and lever arm vectors.
• Rotational speed (frequency): the number of rotations per unit of time
  • Formula: n = ΔN/Δt
    • Unit: 1/s or Hz (hertz)
    • n = rotational speed (Hz), ΔN = number of rotations, Δt = time duration (s)
• Rotational power: the rate at which work is done to rotate an object
  • Formula: P = M × 2π × n
    • Unit: W (watt)
    • P = power (W), M = torque (Nm), n = rotational speed (Hz or 1/s)
• Angular momentum: the rotational equivalent of linear momentum
  • Formula: L = I × ω
    • Unit: kg×m²/s
    • L = angular momentum (kg·m²/s), I = moment of inertia (kg·m²), ω = angular velocity (rad/s)
• Center of gravity: the single point from which the force of gravity appears to act on an object
  • If an object is suspended freely from a single point, it will come to rest with its center of gravity vertically below the point of suspension.

Both translational and rotational motions can be periodic, such as the swinging of a pendulum or the oscillation of a spring. Such repeating phenomena are called oscillations. Oscillations that propagate through space are called waves. Simple oscillations can be described using a sine function. The maximum displacement from the equilibrium position is called the amplitude.

• Harmonic oscillation: a type of periodic motion where the restoring force is directly proportional to the displacement
  • Formula: s(t) = s₀ × sin(2πt/T) = s₀ × sin(ωt)
    • s(t) = displacement at time t (m) , s₀ = amplitude (m), t= time (s), T = period (s), ω = angular frequency (rad/s)
    • The angle of oscillatory motion in physics is often expressed in radians.
  • Important quantities
    • Amplitude (s₀): the maximum displacement from the equilibrium position
    • Period (T): the time required for one complete oscillation
    • Frequency (f = 1/T): the number of complete oscillations per second
      • Unit: Hertz (Hz), where 1 Hz = 1 oscillation per second
    • Phase (φ): describes the position of a point in time on a waveform cycle
      • Measured in: radians or degrees
      • In phase: a state where two waves have the same frequency and their crests and troughs align, resulting in constructive interference
      • Out of phase: a state where two waves do not align perfectly
        • Phase difference of 180° (π radians): the crests of one wave align with the troughs of the other, resulting in destructive interference
    • For progressive waves
      • Wavelength (λ): the spatial period of the wave, or the distance over which the wave's shape repeats (see also: "Optics" and "Acoustics")
      • For waves, the transmitted energy is proportional to the square of the amplitude (E ∝ A²).
      • Wave propagation speed: the speed at which a wave travels through a medium
        • Formula: v = f × λ
        • v = propagation speed (m/s), f = frequency (Hz), λ = wavelength (m)
  • Types of waves
    • Transverse waves: the displacement of particles in the medium is perpendicular to the direction of wave propagation
      • Examples: light waves, ripples on the surface of water
    • Longitudinal waves: the displacement of particles in the medium is parallel to the direction of wave propagation
      • Characterized by areas of compression and rarefaction
      • Example: sound waves

All motion requires an expenditure of energy, which can be quantified as work (work is the energy transferred by a force). Power is the rate at which work is done.

Energy (physics)

Energy can be converted from one form to another. For example, friction converts kinetic energy into thermal energy, and a pendulum converts kinetic energy into potential energy and back.

• Kinetic energy: the energy an object possesses due to its motion
  • Formula: E_kin = ½ × m × v²
    • Unit: J (joule)
    • E_kin = kinetic energy (J), m = mass (kg), v = velocity (m/s)
• Potential energy: the energy stored in an object due to its position in a force field, such as a gravitational field
  • Formula: E_pot = m × g × h
    • Unit: J (joule)
    • E_pot = potential energy (J), m = mass (kg), g = acceleration due to gravity (m/s²), h = height (m)
• Thermal energy: the internal energy of a system associated with the random, microscopic motion of its constituent particles (Brownian motion, see also: "Thermodynamics")
• Friction (physics): a force that resists motion between surfaces in contact. This process converts kinetic energy into thermal energy.
  • Internal friction (viscosity): occurs between particles within a fluid
  • External friction: occurs at the interface between two surfaces
    • Sliding friction: the resistive force that acts when one body slides over another
    • Static friction: the force that must be overcome to initiate motion between two stationary surfaces
• Conservative forces: forces for which the work done moving an object between two points is independent of the path taken
  • Net work done by a conservative force over a closed path is zero
  • Associated with potential energy
• Non-conservative forces: forces for which the work done depends on the path taken
  • Dissipate mechanical energy from a system, often as heat
  • Examples: friction, air resistance

Work (physics)

Work is the energy transferred to or from an object by applying a force along a displacement. Two common forms of work in mechanics are lifting work and acceleration work.

• Lifting work: the work done against gravity to lift an object
  • Formula: W = F_g × h
    • Unit: J (joule)
    • W = lifting work (J), F_g = gravitational force (N), h = height (m)
• Acceleration work: the work done to change the velocity of an object
  • Formula: W = F × s (= m × a × s)
    • Unit: J (joule) = Nm (newton-meter) = Ws (watt-second)
    • W = acceleration work (J), F = force (N), s = distance (m), m = mass (kg), a = acceleration (m/s²)
• Work done by a constant force: the product of the magnitude of the displacement and the component of the force parallel to the displacement
  • Formula: W = F × d × cos(θ)
    • W = work (J), F = magnitude of the constant force (N), d = magnitude of the displacement (m), θ = angle between vectors (degrees or radians)
    • Positive work (θ < 90°): the force has a component in the direction of displacement, transferring energy to the object
    • Negative work (θ > 90°): the force has a component opposite to the direction of displacement, removing energy from the object
    • Zero work (θ = 90°): the force is perpendicular to the displacement; no energy is transferred

Example calculation

How much work is done to lift a 2 kg stone to a height of 2 m?

• Wanted: lifting work W
• Given: mass m, height h
  • F_g = m × g => F_g = 2 kg × 9.81 m/s² = 19.62 N
  • W = F_g × h => 19.62 N × 2 m = 39.24 J

Power (physics)

Power is the rate at which work is done or energy is transferred.

• Power: work performed per unit of time
  • Formula: P = W/Δt
    • Unit: W (watt, = J/s)
    • P = power (W), W = work or energy (J), Δt = time interval (s)
  • Formula (for constant force): P = F × v
    • Unit: W (watt, = J/s = Nm/s)
    • P = power (W), F = force (N), v = velocity (m/s)

Example calculation

A force of 3 N is applied to lift a stone a height of 0.5 m over 5 minutes. What is the power generated?

• Wanted: power P
• Given: time t, height h, force F
  • W = F × h => 3 N × 0.5 m = 1.5 J
  • P = W/Δt => 1.5 J/300 s = 0.005 W

Forces acting on a body can cause it to deform. Common types of deformation include stretching, compression, bending, shearing (displacement of layers relative to each other), and torsion (twisting). When a body is stretched or compressed, it stores potential energy. A simple example of this is a spring.

• Strain: the fractional change in length
  • Formula: ε = Δl/l
    • ε = strain (unitless), Δl = change in length (m), l = original length (m)
  • Stretching (tension): an increase in length
    • Medical example: stretching of blood vessels (see also: vascular compliance)
  • Compression: a decrease in length
• Hooke's law: Within the elastic limit, the force required to stretch or compress an elastic object is directly proportional to the extension or compression.
  • Formula: F = E × A × ε
    • F = tensile force (N), E = modulus of elasticity (Pa or N/m²), A = cross-sectional area (m²), ε = strain (unitless)
  • Hooke's law can be used to calculate the extension of a spring when a certain force is applied. It can also be used to determine the force stored in a stretched spring if the extension is known.
• Young's modulus of elasticity: a measure of a material's stiffness, defined as the ratio of stress to strain
  • Formula: E = σ/ε
    • Unit: N/m² or Pa (pascal)
    • E = modulus of elasticity (Pa), σ = stress (Pa), ε = strain (unitless)
• Tensile stress: the internal force per unit area that an object resists when subjected to a stretching force
  • Formula: σ = F / A
    • Unit: N/m² or Pa (pascal)
    • σ = tensile stress (Pa), F = tensile force (N), A = cross-sectional area (m²)
• Elastic potential energy: the energy stored in a stretched or compressed elastic object, such as a spring
  • Formula: E = ½ D × s²
    • Unit: J (joule)
    • E = energy (J), D = spring constant (N/m), s = displacement (m)
    • The spring constant indicates the stiffness of a spring and depends on the spring's material and geometry.

Calculation example

A 60 cm long rubber band with a cross-sectional area of 2 cm² is pulled with a force of 1 N. What is the stress on the rubber band? If the band stretches by 5 cm, what is its modulus of elasticity?

• Wanted: stress σ, modulus of elasticity E
• Given: length l, cross-sectional area A, tensile force F, change in length Δl
  • σ = F/A → σ = 1 N/0.0002 m² = 5,000 N/m²
  • ε = Δl/l → ε = 5 cm/60 cm = 1/12
  • E = σ/ε → E = 5,000 N/m²/(1/12) = 60,000 N/m²

In addition to mass, density is another important intrinsic property of matter that relates the mass of a substance to its volume. Density is particularly important when working with fluids. Knowing a fluid's density allows for the calculation of the buoyant force on any object within it, which is related to hydrostatic pressure. This principle, attributed to Archimedes, states that the buoyant force on a body is equal to the weight of the fluid it displaces.

• Density: the ratio of an object's mass to its volume. It depends on how closely packed the atoms are in a material and their atomic mass.
  • Formula: ρ = m/V
    • Unit: kg/L or g/cm³
    • ρ = density (kg/m³), m = mass (kg), V = volume (m³)
    • The density of water is approximately 1000 kg/m³ or1 g/cm³ or 1 kg/L.
• Pressure (physics): the force applied perpendicular to the surface of an object per unit area
  • Formula: p = F/A
  • Unit: Pa (pascal = N/m² = kg/(m×s²))
    • mmHg (millimeters of mercury): an older unit of pressure still common in medicine, especially for blood pressure measurement; 1 mmHg ≈ 133.322 Pa
    • cmH₂O (centimeters of water): another older pressure unit used in mechanical ventilation; 1 cmH₂O ≈ 98.1 Pa ≈ 1 hPa
    • p = pressure (Pa), F = force (N), A = area (m²)
• Volume work: work done by or on a system when its volume changes against an external pressure
  • Formula (constant pressure): W = p × ΔV
    • Unit: J (joule)
    • W = work (J), p = pressure (Pa), ΔV = change in volume (m³)
• Hydrostatic pressure: the pressure exerted by a fluid at equilibrium due to the force of gravity
  • Formula: p = ρ_fluid × g × h
    • Unit: Pa (pascal)
    • p = pressure (Pa), ρ_fluid = density of the fluid (kg/m³), g = acceleration due to gravity (m/s²), h = depth (m)
• Buoyancy: the upward force exerted by a fluid that opposes the weight of an immersed object
  • Formula: F_B = G_fluid= m_fluid × g = ρ_fluid × V_submerged × g
    • Unit: N (newton)
    • F_B = buoyant force (N), G_fluid = weight of the displaced fluid (N), m_fluid = mass of the displaced fluid (kg), g = acceleration due to gravity (m/s²), ρ_fluid = density of the fluid (kg/m³), V_submerged = volume of the submerged part of the object (m³)
  • Specific gravity (SG): a dimensionless ratio of the density of a substance to the density of a reference substance, typically water
    • Formula: SG = ρ_substance / ρ_water
    • Application to buoyancy
      • If SG > 1, the object is denser than water and will sink.
      • If SG < 1, the object is less dense than water and will float.
      • The percentage of an object's volume that is submerged is equal to its specific gravity multiplied by 100.

Pascal's law

This principle states that a pressure change applied to an enclosed, incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.

• Principle: pressure is transmitted equally throughout the fluid (P₁ = P₂)
• Formula: F₁/A₁ = F₂/A₂
  • F₁ (N), F₂ (N) = forces applied to areas A₁ and A₂, respectively
• Application: forms the basis for hydraulic systems, where a small force applied to a small area generates a much larger force on a larger area, effectively multiplying the force

Measurement of central venous pressure
A simple way to estimate central venous pressure (CVP) uses the external jugular vein as a natural manometer. Because this vein runs roughly vertically above the superior vena cava, the blood level within it reflects the pressure in the large veins near the right atrium. In a healthy seated individual, the external jugular vein is collapsed and not visible. However, when lying down, it becomes filled to a certain level. The highest point of visible pulsation is used as a reference. The CVP in cmH₂O is approximately the vertical distance in centimeters between this pulsation point and the estimated level of the heart.

Balance scale (example of rotational motion)

A balance scale consists of a horizontal beam supported by a central pivot, or fulcrum. Pans are suspended at each end of the beam.

• m_left = m_right: the pans are balanced at the same height
• m_left ≠ m_right: the beam tilts toward the side with the greater mass

Lever (example of rotational motion)

A lever is a rigid bar that pivots around a fixed point called the fulcrum. For a lever to be in rotational equilibrium, the law of the lever must be satisfied.

• Law of the lever: a lever is in equilibrium when the sum of clockwise torques equals the sum of counter-clockwise torques. A torque is the product of the force and the perpendicular distance from the fulcrum (the lever arm).
  • Formula: Force₁ × lever arm₁ = Force₂ × lever arm₂
    • Force is measured in Newtons (N), and lever arm is measured in meters (m).

Mechanical advantage

A measure of the force amplification achieved by using tools such as levers, pulleys, or wedges

• Definition: the ratio of the output force (load) to the input force (effort)
• Formula: mechanical advantage (MA) = F_out/F_in
  • F_out = output force, F_in = input force
• Interpretation
  • MA > 1: the machine multiplies the input force, making it easier to move a heavy object
  • MA < 1: the machine requires more input force than the output force it produces, but provides an advantage in speed or range of motion
  • MA = 1: the machine only changes the direction of the force

Example calculation

• Wanted: force that a muscle must apply to hold a load
• Given: The muscle attaches 1 cm from the joint (fulcrum), and the 100 g load is held 30 cm from the joint.
  • Force of load (weight) F_g = m × g with g ≈ 10 m/s²→ 0.1 kg × 10 m/s² = 1 N
  • Force_muscle × arm_muscle = force_load × arm_load → force_muscle × 0.01 m = 1 N × 0.3 m
  • Force_muscle= (1 N × 0.3 m) / 0.01 m = 30 N

Centrifuge (example of rotational motion)

A centrifuge is a device used to separate mixtures, such as emulsions and suspensions, by spinning them at high rotational speed. This process relies on the different sedimentation rates of the components, which are influenced by factors like mass, density, and shape. The rapid circular motion creates large centripetal acceleration (toward the rotation axis); in the rotating frame, this is experienced as an outward centrifugal effect. Denser components with a higher buoyant mass settle faster (pellet) toward the bottom of the tube than less dense components.

• Centripetal force: the force that keeps an object moving in a circular path, directed toward the center of the circle
  • Formula: F_c = m × a_c = m × v²/r = m × r × ω²
    • Unit: N
    • F_c = centripetal force (N), a_c = centripetal acceleration (m/s²), m = mass (kg), v = tangential velocity (m/s), r = radius (m), ω = angular velocity (rad/s)
• Centrifugal force: fictitious or inertial force in the rotating frame that appears to push an object outward from the center of rotation. It is equal in magnitude and opposite in direction to the centripetal force in the rotating frame.
• Centripetal acceleration: the acceleration of an object toward the center of its circular path
  • Formula: a_c = v²/r = ω² × r
    • Unit: m/s²
    • a_c = centripetal acceleration (m/s²), v = tangential velocity (m/s), r = radius (m), ω = angular velocity (rad/s)
• Frequency: the number of revolutions per unit time
  • Formula: f = 1/T
    • Unit: Hz (revolutions per second) or rpm (revolutions per minute)
    • f = frequency (Hz), T = period (s)
• Relative centrifugal force (RCF)
  • The standard unit for centrifugation, measured in multiples of gravitational acceleration (×g).
  • This is used instead of RPM because it standardizes the force across centrifuges with different rotor sizes, making experiments reproducible.
  • Formula: RCF (in units of ×g) = 1.118 × 10^-5 × r(cm) × (RPM)²
• Sedimentation coefficient: a measure of a particle's sedimentation velocity per unit of applied acceleration, accounting for its mass, density, and shape; this value allows for comparison of particles independent of centrifuge conditions
  • Formula: s = v_s/a_c
    • Unit: S (Svedberg; 1S = 10^-13 s)
    • s = sedimentation coefficient (s), v_s = sedimentation velocity (m/s), a_c = centripetal acceleration (m/s²)
  • A higher S value means the particle is larger and/or more compact and sediments faster.
  • E.g., prokaryotic ribosomes are 70S, while eukaryotic ribosomes are 80S, meaning the eukaryotic ones are larger and sediment faster.
  • Biological application: sedimentation
    • Mass and density
      • More massive and denser particles (like nuclei or whole cells) have more inertia and pellet at lower RCFs.
      • Lighter, less-dense components (like ribosomes or small proteins) require much higher RCFs to pellet.
      • This is the basis for differential centrifugation, a technique used to separate organelles
        • Order of pelleting (from lowest to highest speed): whole cells (largest and densest) → nuclei → mitochondria → microsomes → ribosomes (smallest)
    • Shape: Compact, spherical particles experience less drag and sediment faster than long, irregularly shaped particles of the same mass.
    • Medium viscosity: Sedimentation is slower in a more viscous (thicker) solution, like a dense sucrose gradient. This is the principle behind density-gradient centrifugation.

Speed is king: Force and acceleration are proportional to the square of the speed (Fc ∝ ω²). If you double the angular speed (ω), you quadruple the centripetal force (F_c)! If you triple the angular speed, the force increases by a factor of 9.

Radius matters: Force and acceleration are directly proportional to the radius (Fc ∝ r). If you double the radius of the centrifuge rotor, you double the centripetal force.

Mass is simple: Force is directly proportional to mass (Fc ∝ m). If you double the mass of the particle, you double the centripetal force acting on it.

In a centrifuge, denser or larger substances settle faster and farther toward the bottom of the tube.

Example calculations

Two proteins with sedimentation coefficients of 1 S and 4 S are to be separated in a centrifuge. What are their respective sedimentation velocities if the centrifuge has a radius of 15 cm and rotates at 5,000 rpm?

• Wanted: sedimentation velocities v_s
• Given: sedimentation coefficients s, radius r, rotational speed
  • Frequency f = 5,000 rev/min × (1 min/60 s) ≈ 83.33 rev/s (Hz)
  • Angular velocity ω = 2π × f ≈ 2π × 83.33 s^-1 ≈ 523.6 rad/s
  • Centripetal acceleration a_c = ω² × r ≈ (523.6 s^-1)² × 0.15 m ≈ 41,123 m/s²
  • Sedimentation velocity v_s = s × a_c
  • For protein 1: v_s = (1 × 10^-13 s) × 41,123 m/s² ≈ 4.11 × 10^-9 m/s
  • For protein 2: v_s = (4 × 10^-13 s) × 41,123 m/s² ≈ 16.45 × 10^-9 m/s

A researcher doubles the RPM of their centrifuge. To keep the RCF constant, what must they do to the radius of the rotor they use?

• Answer: Since RCF depends on (RPM)², doubling the RPM increases the RCF by 4x. To counteract this, you must decrease the radius by 4x (i.e., use a radius that is 1/4 of the original).

Pendulum (example of energy conversion)

A pendulum consists of a mass suspended from a pivot so that it can swing freely. When displaced from its resting position (equilibrium), it gains potential energy. This energy is due to its new position in the gravitational field. When released, the pendulum swings back toward its equilibrium position, converting potential energy into kinetic energy and gaining speed. As it passes the equilibrium position and swings up the other side, its kinetic energy is converted back into potential energy, and it slows down. The two energy states of the pendulum during its swing are described as follows:

• Law of conservation of energy: The total energy in an isolated system remains constant.
• Undamped oscillation: an ideal oscillation where total mechanical energy is conserved
  • Formula: E_kin + E_pot = constant
    • E_kin = kinetic energy (J), E_pot = potential energy (J)
• Damped oscillation: a realistic oscillation that gradually diminishes over time because energy is dissipated, typically as heat due to friction or air resistance
  • All real-world macroscopic oscillations are damped.

U-tube manometer

A U-tube manometer is a device that uses hydrostatic pressure to measure the pressure of a gas. It consists of a U-shaped tube partially filled with a liquid. When both ends are open to the atmosphere, the liquid levels are equal. If one end is connected to a container with an unknown pressure, the liquid column shifts:

• Pressure in container < atmospheric pressure: the liquid column moves toward the container until the pressures are balanced
• Pressure in container > atmospheric pressure: the liquid column is pushed away from the container until the pressures are balanced