Unit 2 Beta

Force and Translational Dynamics: the one-page sheet

2.1 Systems and Center of Mass

Choose a system and its boundary; when its insides do not matter, model it as one point at its center of mass, the mass-weighted average position x_cm = Σm·x ÷ Σm, which lies nearer the heavier parts and on any line of symmetry.

A system is the object or objects you choose to study; everything outside the boundary is the surroundings.
Mass is a scalar measure of how much matter there is, in kilograms; masses add.
The center of mass is the mass-weighted average position: x_cm = (m₁x₁ + m₂x₂ + …) ÷ (m₁ + m₂ + …). Do x and y separately.
For a uniform, symmetric object the center of mass is on every line of symmetry. It can be in empty space (a ring).

they form the system, and everything else is the surroundings you can model the whole system as one point object that point, the center of mass, sits nearer the heavier parts: x_cm = Σm·x ÷ Σm its center of mass still moves like a single point object, so you can track one point instead of every part

system (physics): The object or group of objects you choose to study, inside a boundary you draw. Everything else is the surroundings.
surroundings: Everything outside the system you chose. Objects in the surroundings can push or pull on the system.
mass: A scalar measure of the amount of matter in an object, in kilograms (kg). The masses of parts add to the mass of the whole.
internal structure: The parts inside a system and how they are arranged and move relative to each other. When it does not affect the question, the system can be modeled as one object.
center of mass: The mass-weighted average position of a system: x_cm = Σmᵢxᵢ ÷ Σmᵢ (the same for y). It lies on any line of symmetry of a uniform object, and the system moves as if all its mass were there.

2.2 Forces and Free-Body Diagrams

A force is an interaction between two objects. To draw a free-body diagram, list every object touching your object plus Earth, and draw each force as a whole arrow from a dot, labeled with type and agent; the net force is their vector sum and is never drawn as an extra arrow.

A force is a push or pull between two objects, measured in newtons (1 N = 1 kg·m/s²). It is a vector.
Contact forces (normal force, tension, friction) need touching. Weight, F_g = mg, is Earth's pull at a distance; near Earth g = 9.8 m/s² (10 m/s² is accepted).
A free-body diagram: a dot for the object, one arrow per force starting on the dot, labeled type and agent ("F_N, floor on box"), no components, no velocity, no net force arrow.
The net force is the vector sum of all the forces. Find it from the diagram; never draw it on the diagram.

each exerts a force on the other: a push or pull with a size and a direction you have every force on it, each with an agent you get a free-body diagram: whole forces, labeled with type and agent, lengths in proportion you get the net force, which is a result of the diagram, not one of its arrows

interaction: Two objects acting on each other: pushing, pulling, rubbing or attracting. Every force is one side of an interaction.
force: A push or pull exerted on an object by another object, with a size and a direction (a vector). Unit: the newton, 1 N = 1 kg·m/s².
contact force: A force between objects that touch: the normal force, tension and friction. At the scale of atoms these are electric forces between the surfaces.
force at a distance: A force between objects that do not touch. In this course the only one is gravity, such as Earth's pull on a falling ball.
agent of a force: The object that exerts a force. A free-body diagram label names the type and the agent: "F_N, floor on box".
free-body diagram: A diagram of one object (a dot at its center of mass) with every force exerted on it by other objects drawn as a whole arrow starting on the dot, labeled with its type and agent, lengths in proportion. It never shows velocity, components or a net force arrow.
normal force: The push of a surface on an object touching it, perpendicular to the surface and away from it (F_N). Its size is whatever the situation requires, not always mg.
tension: The pull of a rope, string or cable on an object attached to it, directed along the rope away from the object (F_T).
weight: The pull of Earth (or another planet) on an object, F_g = mg, pointing down toward the planet's center. Near Earth, g = 9.8 m/s² (10 m/s² is accepted). Mass stays the same everywhere; weight depends on where you are.
net force: The vector sum of all the forces on an object, ΣF. It is found from the free-body diagram and is not itself one of the forces.

2.3 Newton's Third Law

Forces come in third-law pairs: equal in size, opposite in direction, same type, acting on two different objects, so they never cancel each other. Inside a chosen system the pairs are internal and add to zero; an ideal string carries one tension over an ideal pulley.

Newton's third law: if A pushes on B, B pushes on A with a force equal in size, opposite in direction, of the same type. Partners act on different objects.
To name a partner, swap the two objects and flip the direction: "road on tire, forward" ↔ "tire on road, backward".
Internal forces act between parts of the system and cancel in pairs; external forces come from the surroundings.
An ideal string is massless and does not stretch: one tension all along it. An ideal pulley is massless and frictionless: it turns the string without changing the tension.

B exerts a force on A of the same size, the opposite direction and the same type they appear on different free-body diagrams and never cancel each other the pair are internal forces and add to zero for the system only external forces can change how the system's center of mass moves

Newton's third law: Newton's third law: when object A exerts a force on object B, B exerts a force on A that is equal in size, opposite in direction and of the same type. The two forces act on different objects, so they never cancel each other.
internal force: A force that one part of a system exerts on another part. Internal forces come in third-law pairs that add to zero for the system.
external force: A force exerted on a system by an object outside it. Only external forces can change how the system's center of mass moves.
ideal string: A model string that has no mass and does not stretch, so the tension is the same all along it and the objects tied to it move together.
ideal pulley: A model pulley with no mass and no friction. It changes the direction of an ideal string without changing its tension.

2.4 Newton's First Law

Newton's first law: zero net force means constant velocity, including rest. Translational equilibrium is ΣF = 0, checked direction by direction; inertia is a property, not a force; and the law holds in inertial frames, such as the ground.

Newton's first law: if the net force on an object is zero, its velocity does not change. Rest is just a velocity of zero.
Translational equilibrium: ΣF = 0. Check each direction separately: ΣF_x = 0 and ΣF_y = 0. Forces can balance in one direction and not the other.
Inertia is an object's tendency to keep its velocity; mass measures it. Inertia is not a force.
An inertial reference frame is one in which the first law holds: the ground, or a vehicle at constant velocity.

the object is in translational equilibrium its velocity stays constant: at rest stays at rest, moving keeps the same speed and direction some net force (friction, air) acts on it; with that force removed it would keep going objects seem to lurch with no force, because that frame is not inertial; use the ground's frame

translational equilibrium: The state of an object whose net force is zero (ΣF = 0, in every direction). Its velocity stays constant, which includes staying at rest.
Newton's first law: Newton's first law: an object's velocity stays constant (it stays at rest or keeps moving in a straight line at the same speed) unless a nonzero net force acts on it.
inertia: The tendency of an object to keep its velocity. Mass measures inertia: the more mass, the harder it is to change the velocity. Inertia is not a force.
inertial reference frame: A reference frame in which Newton's first law holds, such as the ground or a vehicle moving at constant velocity. An accelerating vehicle is a noninertial frame.

2.5 Newton's Second Law

Newton's second law: the acceleration of a system's center of mass equals the net external force divided by its mass, a = ΣF/m, and points along the net force. Apply it one direction at a time; for connected objects, use the whole system for the acceleration and one object for the tension.

Newton's second law: a = ΣF / m, or ΣF = ma. The acceleration points along the net force, not necessarily along the velocity.
Apply it one axis at a time: ΣF_x = ma_x, ΣF_y = ma_y. Pick the axis along the acceleration.
For a system, use only external forces and the total mass: a_sys = ΣF_ext / m_sys.
For connected objects (cart and hanging block, Atwood machine), find the shared acceleration from the whole system, then the tension from one object.

there is a nonzero net external force ΣF the center of mass accelerates in the direction of the net force the acceleration is larger: a = ΣF / m internal forces (tension, contact pushes) drop out, and one equation gives the shared acceleration

Newton's second law: Newton's second law: the acceleration of an object (or of a system's center of mass) equals the net external force divided by the mass, a = ΣF/m, and points in the direction of the net force.
acceleration of a system: The acceleration of a system's center of mass, a_sys = ΣF_ext / m_sys: the net external force divided by the total mass. Internal forces do not enter.
connected objects: Objects joined by ideal strings or pushing on each other so that they share one acceleration, such as a cart pulled by a hanging block or an Atwood machine (two masses hung over one pulley).

2.6 Gravitational Force

Every two masses attract with |F_g| = Gm₁m₂/r², measured center to center; the field g = F/m is about 9.8 N/kg at Earth's surface. A scale reads the normal force, the apparent weight, which differs from mg whenever the object accelerates and is zero in free fall.

Newton's law of universal gravitation: |F_g| = Gm₁m₂/r², attractive, along the line between centers, with r from center to center and G = 6.67 × 10⁻¹¹ N·m²/kg².
The gravitational field g = F_g/m = GM/R² at a planet's surface. Near Earth g ≈ 9.8 N/kg = 9.8 m/s², treated as constant.
Apparent weight is the normal force (what a scale reads). It equals mg only when the acceleration is zero; in free fall it is zero (weightless).
Inertial mass (resisting acceleration) and gravitational mass (feeling gravity) are equal by experiment, so all objects fall with the same g.

each pulls the other with a gravitational force Gm₁m₂/r², toward the other's center you get the gravitational field g = F/m, which near Earth's surface is about 9.8 N/kg and nearly constant the normal force differs from mg: F_N = m(g + a) going up, m(g − a) going down the normal force is zero, so the object is weightless even though gravity still acts

Newton's law of universal gravitation: Newton's law of universal gravitation: any two objects attract each other with a force of size |F_g| = Gm₁m₂/r², where r is the distance between their centers and G = 6.67 × 10⁻¹¹ N·m²/kg² is the gravitational constant.
gravitational force: The attractive force between two objects because of their masses, acting along the line between their centers. Near a planet's surface it is the object's weight, mg.
gravitational field: The gravitational force per unit mass at a point, g = F_g/m, in N/kg (the same as m/s²). At a planet's surface g = GM/R²; near Earth it is about 9.8 N/kg and treated as constant.
apparent weight: The force a scale or floor pushes up with: the normal force. It equals mg only when the object's acceleration is zero; it is larger when accelerating up, smaller when accelerating down, and zero in free fall (weightlessness).
gravitational mass: Mass measured by how strongly an object is pulled by gravity. Experiments find it equal to inertial mass (how strongly the object resists acceleration), which is why all objects fall with the same acceleration (the equivalence principle).

2.7 Kinetic and Static Friction

Friction acts along a surface. Kinetic friction, while sliding, is μ_k F_N and opposes the sliding; static friction, while gripping, is whatever prevents slipping, up to μ_s F_N. Usually μ_s > μ_k. Find the normal force first, since it is mg cos θ on a ramp and changes with tilted pulls.

Kinetic friction (sliding): f_k = μ_kF_N, opposing the sliding of the surfaces; independent of contact area and speed.
Static friction (gripping): f_s ≤ μ_sF_N. It equals whatever prevents slipping; it reaches μ_sF_N only at the point of slipping.
Usually μ_s > μ_k, so a stuck object lurches once it breaks free.
Find F_N from the perpendicular direction first: on a ramp it is mg cos θ, not mg. The equations sheet writes |F_f| ≤ |μF_N|.

they can exert friction on each other along the surface static friction takes whatever value prevents slipping, up to μ_s F_N the surfaces slip and friction drops to kinetic friction, μ_k F_N, opposing the sliding both the maximum static friction and the kinetic friction change in proportion

friction: A contact force exerted by a surface on an object, along the surface, that opposes the sliding (or the tendency to slide) of the two surfaces against each other.
coefficient of friction: A number with no unit, μ, that describes how strongly two particular surfaces grip: friction = μ × normal force (the maximum, for static friction). It depends on the materials, not on area or speed.
kinetic friction: Friction between surfaces that slide against each other: f_k = μ_k F_N, opposite the direction of sliding.
static friction: Friction between surfaces that do not slide against each other. It takes whatever value prevents slipping, up to a maximum of μ_s F_N; usually μ_s is larger than μ_k.

2.8 Spring Forces

An ideal spring exerts a force proportional to its stretch or compression from its relaxed length, F_s = −kΔx, always back toward the relaxed length; the spring constant k (N/m) is the slope of force against stretch, and a hanging mass at rest stretches it by mg/k.

Hooke's law: F_s = −kΔx. Δx is the displacement of the spring's end from its relaxed length; the minus sign means the force points back toward the relaxed length.
A stretched spring pulls; a compressed spring pushes. The spring constant k, in N/m, measures stiffness.
An ideal spring is massless and obeys Hooke's law, so it pulls equally on whatever is at each end.
A graph of |F_s| against |Δx| is a straight line through the origin with slope k.

it exerts a force on whatever is attached to it, toward its relaxed length the force is proportional to the displacement: |F_s| = k|Δx| the spring is stiffer: more force for the same stretch, less stretch for the same force kΔx = mg, so measuring the stretch measures the weight (a spring scale)

spring force: The force a stretched or compressed spring exerts on an object attached to its end, always toward the spring's relaxed length: a stretched spring pulls, a compressed spring pushes.
Hooke's law: For an ideal spring, F_s = −kΔx: the spring force is proportional to the displacement Δx of its end from the relaxed length, and opposite to it. The spring constant k (N/m) measures stiffness.
ideal spring: A model spring that has no mass and obeys Hooke's law exactly, so it exerts forces of equal size at its two ends.

2.9 Circular Motion

Circular motion at constant speed is accelerated motion: a_c = v²/r toward the center, with v = 2πr/T. Real forces (tension, friction, normal force, gravity) must add to a net force mv²/r toward the center; for circular orbits this gives v = √(GM/r) and T² ∝ r³.

Uniform circular motion: constant speed v = 2πr/T on a circle; the acceleration a_c = v²/r points to the center. The period T is the time for one trip; the frequency f = 1/T, in Hz.
The centripetal net force, ΣF toward the center = mv²/r, is supplied by real forces: tension, friction, normal force, gravity, or their parts. Never draw it as an extra arrow.
Speeding up or slowing on a circle adds a tangential acceleration; the total is the vector sum.
Circular orbits: GMm/r² = mv²/r gives v = √(GM/r) and Kepler's third law, T² = (4π²/GM)r³.

its velocity direction changes continuously, so it accelerates even at constant speed the centripetal acceleration points to the center with size a_c = v²/r the real forces must add to a net force toward the center of size mv²/r the object moves off in a straight line along the tangent

uniform circular motion: Motion in a circle at a constant speed. The velocity changes direction continuously, so the object accelerates toward the center.
centripetal acceleration: The acceleration of an object moving in a circle, pointing toward the center, of size a_c = v²/r.
centripetal net force: The net force toward the center of a circular path, of size mv²/r. It is not a separate force: one or more real forces (tension, friction, normal force, gravity) supply it, and it is never drawn as its own arrow on a free-body diagram.
tangential acceleration: The part of an object's acceleration along its velocity, which changes its speed. On a circle it adds to the centripetal part as a vector.
period (motion): The time for one complete trip around a circle (or one cycle of a repeating motion), T, in seconds. For uniform circular motion, v = 2πr/T.
frequency (motion): The number of trips or cycles per second, f = 1/T, in hertz (Hz = 1/s).
banked curve: A curve whose road is tilted toward the center. On a frictionless banked curve the horizontal part of the normal force supplies the centripetal net force, at one speed: tan θ = v²/(gr). Curves that need friction are treated in words only.
vertical loop: A circle in a vertical plane, such as a loop-the-loop. At the top, gravity and the track's normal force both point to the center; the minimum speed there is when gravity alone supplies the net force, v = √(gr).
conical pendulum: A mass on a string swinging in a horizontal circle so that the string traces a cone. The vertical part of the tension balances the weight, and the horizontal part supplies the centripetal net force.
circular orbit: The circular path of a satellite or moon around a central body, with gravity supplying the centripetal net force: GMm/r² = mv²/r, so v = √(GM/r), independent of the satellite's own mass.
Kepler's third law: For circular orbits around the same central body of mass M, T² = (4π²/GM)r³: the square of the period is proportional to the cube of the orbit radius.