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Newton's Laws & Forces
Why things move: forces, friction, and the laws that govern them. 50 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Dynamics asks why objects move the way they do. Newton's three laws link forces to changes in motion, introduce inertia and mass, and explain how contact forces such as friction, tension and normal force act on bodies. Free-body diagrams are the main tool for turning a physical situation into equations you can solve.
How the topics connect
Dynamics asks why things move as they do. Newton's laws connect forces to changes in motion, and this area builds from single forces and free-body diagrams to friction, circular motion and rotating frames. Kinematics tells you what happens; dynamics explains why.
Real-world uses
Calculate the forces in cables, supports and brakes so that a structure or vehicle stays in equilibrium or decelerates safely.
Explain why a sprinter pushes backwards on the blocks to move forward, and how friction limits grip in a turn.
Work out the tension in ropes and the acceleration of a lift or counterweight using Newton's second law.
Use inertia and force to understand why seatbelts and crumple zones reduce injury in a collision.
Engineers use friction, banking and centripetal force to predict how fast a car can take a bend safely.
Civil and mechanical engineers draw free-body diagrams to check that every support, cable and joint can carry its loads.
What is a force?
A push or pull that can change an object's motion or shape.
F = maA force is an interaction that can change a body's velocity or deform it. It is a vector, measured in newtons, and it always comes from another object. The net force, the vector sum of all forces, determines the acceleration.
A 2 N push and a 3 N push in the same direction give 5 N; in opposite directions, 1 N.
Forces are not carried by moving objects; "force of motion" is not a thing.
Contact and non-contact forces
Contact forces need touching; gravity, electric and magnetic act at a distance.
Contact forces (normal, friction, tension, drag) act where bodies touch. Field forces (gravity, electric, magnetic) act across empty space through fields. Microscopically, contact forces are electromagnetic.
Gravity pulls a falling apple with no contact, while a table supports a book by contact.
Contact force on a surface has two components: normal (perpendicular) and friction (parallel).
Newton's first law
A body keeps its state of rest or uniform motion unless acted on by a net force.
A body at rest stays at rest and a body in motion keeps moving at constant velocity unless a net external force acts. Motion doesn't need a force; changing motion does.
A hockey puck on smooth ice keeps gliding almost unchanged.
Everyday objects stop because friction is a force, not because "motion fades".
Inertia
The resistance of a body to changes in its motion; measured by mass.
Inertia is the tendency to resist changes in motion. Mass is its measure: the larger the mass, the larger the force needed for a given acceleration.
A loaded truck needs a longer distance to stop than an empty one at the same speed.
Inertia isn't a force; it's a property.
Newton's second law
Net force equals mass times acceleration.
ΣF = maThe net force on a body equals its mass times its acceleration, ΣF = ma, and acceleration points along the net force. Apply it separately to each axis.
A 5 kg block pushed by 20 N with 5 N friction: a = 15/5 = 3 m/s².
Use the net force, including friction, not just the applied force.
Units of force
One newton accelerates one kilogram at one metre per second squared.
1 N = 1 kg·m/s²One newton gives 1 kg an acceleration of 1 m/s². Weight of an apple (~0.1 kg) is about 1 N. A kilogram-force is 9.8 N, an older unit.
A 70 kg person weighs about 686 N.
Mass in grams must be converted to kg before using F = ma.
Newton's third law
Forces come in equal and opposite pairs acting on different bodies.
F_AB = −F_BAIf A pushes on B, B pushes back on A equally and oppositely. The two forces are of the same type and act on different bodies, so they never cancel in a single free-body diagram.
A rocket pushes gas back; the gas pushes the rocket forward.
Weight and the normal force on a resting book are not a third-law pair.
Weight and normal force
Weight pulls down; the surface pushes back perpendicular to it.
N = mgWeight W = mg pulls down. The normal force is the surface's push perpendicular to the contact, adjusting to whatever stops penetration. On a level surface at rest N = mg, but it changes on slopes and in accelerating lifts.
On a 30° slope a 10 kg block has N = 98 cos30° ≈ 85 N.
N isn't always equal to mg.
Free-body diagrams
Sketch all forces on one body to apply Newton's laws.
Draw the body alone as a dot and draw every force acting on it with arrows, then choose axes and write ΣF = ma for each. Include only forces on that body, not forces it exerts.
A block on a slope: weight, normal, friction; resolve along and perpendicular to the slope.
Do not draw the acceleration or "force of motion" as a force.
Net force and equilibrium
Zero net force means zero acceleration.
ΣF = 0If the net force is zero the acceleration is zero: the body is at rest or moving at constant velocity (translational equilibrium). All forces form a closed polygon.
A 10 kg lamp hung by two cables is in equilibrium when the vertical components of the tensions sum to 98 N.
Equilibrium doesn't mean no forces, only balanced forces.
Tension
Pulling force transmitted along a rope or string.
Tension is the pulling force transmitted through a rope or string. For a light, inextensible rope over a smooth pulley the tension is the same everywhere along it. It pulls on whatever the rope is attached to.
A 3 kg mass hung at rest has T = 29.4 N.
Tension changes if the rope has mass or rubs a rough pulley.
Atwood machine
Two masses over a pulley accelerate by the difference in weights.
a = (m₁−m₂)g/(m₁+m₂)Two masses on either side of a light pulley accelerate at a = (m₁ − m₂)g/(m₁ + m₂), the heavier going down. The tension is T = 2m₁m₂g/(m₁ + m₂).
3 kg and 1 kg: a = 2·9.8/4 = 4.9 m/s², T = 14.7 N.
T is not equal to either weight when accelerating.
Connected bodies
Bodies linked by strings share acceleration in the same direction of motion.
For bodies joined by a rope, they share the same acceleration magnitude. Write ΣF = ma for each body separately, or for the whole system when the internal tensions can be ignored.
A 4 kg block pulls a 2 kg block with 12 N: a = 12/6 = 2 m/s², tension on the second block 4 N.
Don't forget that the tension acts on both bodies in opposite directions.
Elevator problems
Apparent weight changes with the elevator's acceleration.
N = m(g+a)Apparent weight is the normal force on you: N = m(g + a) for upward acceleration a, m(g − a) for downward. In free fall N = 0 (weightlessness).
70 kg in a lift accelerating up at 2 m/s²: N = 70·11.8 = 826 N.
Your mass doesn't change; your apparent weight does.
Pulley systems
Ideal pulleys change force direction; movable pulleys give mechanical advantage.
Ideal pulleys are frictionless and massless: they change the direction of the tension but not its size. A movable pulley supports a load with two rope segments, halving the force but doubling the rope pulled.
A system with a movable pulley lifts 100 N with 50 N, pulling twice the length.
Energy is not gained: force advantage costs distance.
Motion on an inclined plane
Component of weight along the plane drives motion.
a = g sinθResolve weight into mg sinθ along the slope and mg cosθ perpendicular. Without friction a = g sinθ; with friction a = g(sinθ − μ cosθ).
At 30° frictionless: a = 4.9 m/s².
N = mg cosθ, not mg.
Static friction
Friction that prevents sliding, up to a maximum value.
f_s ≤ μ_s NStatic friction opposes impending sliding and adjusts up to a maximum fs,max = μₛN. Below that value it exactly balances the applied force parallel to the surface.
A 10 kg crate with μₛ = 0.4 needs more than 39 N to start moving.
Static friction is not always μN; it equals the applied force until it reaches the limit.
Kinetic friction
Friction during sliding, usually less than the static maximum.
f_k = μ_k NWhen sliding, friction is fk = μₖN, usually smaller than the static maximum, and independent of speed and contact area (to a good approximation). It always opposes the relative motion.
A 10 kg crate with μₖ = 0.3 feels 29.4 N while sliding.
Friction does not depend on area in the simple model.
Angle of friction and repose
On an incline, sliding starts when tanθ exceeds μ.
tanθ = μThe angle of repose is the steepest slope at which an object stays at rest: tanθ = μₛ. Beyond it the block slides. It tells you μ by tilting a plane.
A block starts sliding at 25°: μₛ = tan25° ≈ 0.47.
Mass doesn't affect the angle.
Rolling friction
Smaller resistance of rolling wheels compared with sliding.
A rolling wheel deforms the surface slightly, giving a resisting force much smaller than sliding friction (Frr = μᵣN with μᵣ ≈ 0.01). This is why wheels are so useful.
A car tyre on road: μᵣ ≈ 0.015.
Pure rolling without slipping involves static friction at the contact point that does no work.
Friction on inclines with applied force
Include friction direction opposing impending motion.
Pushing a block up a slope, friction points down the slope; pushing it down, friction acts up the slope. Write along-slope balance: F = mg sinθ ± μmg cosθ.
To push a 10 kg block at constant speed up a 30° slope with μ = 0.2: F = 49 + 17 = 66 N.
The sign of friction depends on the direction of motion or tendency.
Block on block problems
Friction between stacked blocks sets maximum common acceleration.
For stacked blocks, friction between them provides the top block's acceleration, limited by μₛ m g. The top block slips when the common acceleration needed exceeds μₛg.
Top 2 kg, bottom 4 kg, μₛ = 0.5: the maximum shared acceleration is 4.9 m/s².
Analyse each block separately; friction pairs are equal and opposite.
Impulse
Force integrated over time, equal to change in momentum.
J = ∫F dt = ΔpImpulse is the force integrated over time and equals the change in momentum: J = ∫F dt = Δp. For constant force J = FΔt. The area under an F–t graph gives it.
A 0.15 kg ball hit from +20 to −30 m/s: Δp = 7.5 N·s.
Direction matters: reversing direction doubles the change.
Linear momentum
Mass times velocity.
p = mvMomentum p = mv is a vector measuring quantity of motion. It's what makes heavy, fast things hard to stop.
A 1000 kg car at 20 m/s has p = 20,000 kg·m/s.
Momentum depends on velocity, not speed.
Second law as rate of momentum change
Force is the time rate of change of momentum.
F = dp/dtNewton's original statement is F = dp/dt. It reduces to ma for constant mass but also covers changing mass (rockets).
A force of 10 N acting for 3 s changes the momentum by 30 kg·m/s.
F = ma only when m is constant.
Variable mass systems
Rockets and conveyor belts require the full dp/dt form.
F = m dv/dt + v dm/dtWhen mass enters or leaves a system, apply F = dp/dt carefully to a defined system including the added mass. Examples: a rocket, a leaking cart, rain falling in a moving wagon.
A coal-loaded belt needs extra force F = v dm/dt to accelerate new coal to belt speed.
Do not simply use m dv/dt without accounting for the mass change.
Rocket propulsion
Exhaust gas gives thrust and velocity change.
Δv = v_e ln(m₀/m)A rocket expels gas backward at speed vₑ relative to itself, gaining forward momentum. Thrust is F = vₑ dm/dt, and the final speed from the rocket equation is Δv = vₑ ln(m₀/m).
With vₑ = 3 km/s and mass ratio 10: Δv ≈ 6.9 km/s.
Rockets work in vacuum; they don't push against air.
Conservation of momentum
Total momentum of an isolated system stays constant.
Σp_i = Σp_fIn an isolated system the total momentum is constant, because internal forces cancel in pairs. It applies to collisions, explosions and recoil, even when energy isn't conserved.
A 60 kg skater pushes a 40 kg skater: velocities are in ratio 2:3 opposite.
External forces such as friction must be negligible during the short collision.
Circular motion dynamics
Net inward force provides centripetal acceleration.
F = mv²/rFor circular motion, the net force toward the centre is mv²/r. Identify which actual force(s) supply it (tension, friction, normal force or gravity) and set their inward sum equal to mv²/r.
A 0.5 kg stone on a 1 m string at 4 m/s: T = 0.5·16/1 = 8 N.
Centripetal force isn't a new force; it's the name for the net inward one.
Banking of roads
Bank angle supplies centripetal force without friction at design speed.
tanθ = v²/rgOn a banked curve, the horizontal component of the normal force provides the centripetal force. With no friction, tanθ = v²/(rg); with friction, vmax = √(rg(μ + tanθ)/(1 − μ tanθ)).
For v = 20 m/s and r = 100 m: tanθ = 0.41, θ ≈ 22°.
Don't resolve forces along the slope when the acceleration is horizontal; use horizontal and vertical.
Vehicle on flat circular turn
Friction provides centripetal force; maximum speed depends on μ.
v_max = √(μrg)On a flat turn the centripetal force is static friction: μmg ≥ mv²/r, so vmax = √(μrg). Wet roads lower μ and the safe speed.
μ = 0.7, r = 50 m: vmax ≈ 18.5 m/s.
The speed limit doesn't depend on the car's mass.
Conical pendulum
A bob moving in a horizontal circle with the string tracing a cone.
T = 2π√(L cosθ/g)A bob on a string moves in a horizontal circle. T cosθ = mg and T sinθ = mv²/r give tanθ = v²/(rg) and period 2π√(L cosθ/g).
L = 1 m, θ = 30°: T ≈ 1.87 s.
The tension is bigger than the weight.
Vertical circular motion
Tension and speed vary around the loop; minimum speed at the top sets the loop condition.
v_top ≥ √(gr)In a vertical circle the speed and tension vary: at the top T + mg = mv²/r, at the bottom T − mg = mv²/r. The string stays taut only if v_top ≥ √(gr).
A ball on a 1 m string must have at least 3.13 m/s at the top.
The tension at the top is smaller than at the bottom.
Loop-the-loop
Minimum speed at bottom to complete a loop is √(5gr).
v_bottom = √(5gr)To complete a vertical loop of radius r, the minimum speed at the top is √(gr) and so, from energy conservation, the minimum at the bottom is √(5gr).
r = 5 m: v_bottom ≥ 15.7 m/s.
At the top the normal force is zero at the critical speed; the object is not weightless in other senses.
Centrifugal force
A fictitious outward force in a rotating frame.
F = mω²rIn a rotating frame a body appears pushed outward with fictitious force mω²r. In the inertial frame there is no such force; the body simply tends to go straight.
Mud flies off a wheel because there's not enough force to keep it in a circle.
Don't draw centrifugal force in an inertial-frame diagram together with the centripetal force.
Pseudo forces
In accelerating frames add −ma to apply Newton's laws.
F_pseudo = −maIn an accelerating frame with acceleration a, every body feels an extra force −ma in that frame. Using it, Newton's laws apply as usual in the non-inertial frame.
In a braking car (a = −3 m/s²) a 60 kg passenger feels +180 N forward.
Pseudo forces have no reaction partner.
Drag force
Fluid resistance grows with speed.
F = ½ρC_dAv²Drag opposes motion through a fluid: F = ½ρC_dAv² at high speed (turbulent), growing with the square of speed. It depends on shape (C_d), frontal area A and density ρ.
Doubling the speed of a cyclist increases drag fourfold and the power needed eightfold.
At low speeds drag is closer to linear in v.
Stokes' law
Viscous drag on a small sphere at low speed.
F = 6πηrvFor a small sphere moving slowly through a viscous fluid, the drag is F = 6πηrv, proportional to speed. It governs the settling of fine particles and oil drops in Millikan's experiment.
A 1 mm steel ball in glycerol quickly reaches a steady slow speed.
It holds only for low Reynolds number (smooth flow).
Terminal velocity of a sphere
Reaches constant speed when weight, buoyancy and drag balance.
v_t = 2r²(ρ−σ)g/9ηAt terminal speed the weight equals the buoyancy plus drag: v_t = 2r²(ρ − σ)g/(9η). The speed grows with the square of the radius, so large drops fall faster than small ones.
Fog droplets with r = 10 μm fall a few millimetres per second.
This formula needs Stokes' law to be valid.
Spring force
A spring pulls back in proportion to its stretch.
F = −kxA spring exerts F = −kx, a restoring force proportional to its extension from the natural length, pointing opposite to the displacement. k (N/m) measures stiffness.
k = 200 N/m stretched 5 cm: force 10 N.
x is the stretch from the natural length, not the total length.
Springs in series and parallel
Series reduces stiffness; parallel increases it.
k_p = k₁+k₂Springs in parallel share the displacement: k = k₁ + k₂ (stiffer). Springs in series share the force: 1/k = 1/k₁ + 1/k₂ (softer).
Two 100 N/m springs: parallel 200, series 50 N/m.
Series is the opposite of resistors in series for the same-looking formula; watch which quantity is shared.
Hooke's law limits
Beyond the elastic limit, deformation is permanent.
Hooke's law holds only up to the proportional limit. Beyond the elastic limit the material deforms permanently, and finally breaks. Real springs and wires follow F = kx only for small extensions.
A paperclip pulled too far stays bent.
Never extrapolate F = kx for large stretches.
Equilibrium of concurrent forces
Lami's theorem links three forces in equilibrium.
F₁/sinα = F₂/sinβ = F₃/sinγThree forces in equilibrium on a point form a closed triangle, so Lami's theorem applies: F₁/sinα = F₂/sinβ = F₃/sinγ, with each angle opposite its force.
A 100 N weight hanging from two ropes at 45° each: each tension ≈ 70.7 N.
Lami's angles are between the other two forces, opposite the one in the numerator.
Torque and equilibrium
Balance both forces and torques for static equilibrium.
Στ = 0A rigid body in equilibrium needs ΣF = 0 and Στ = 0 about any point. Choose a pivot that eliminates unknown forces to simplify.
A 4 m plank, 20 kg, supported at its ends with a 10 kg load at 1 m: reaction at the far end by moments.
Moments must be taken about the same point for all forces.
Center of mass
The point where the system's mass behaves as concentrated.
x_cm = Σm x/ΣmThe centre of mass is the mass-weighted average position: x_cm = Σm x/Σm. Net external force accelerates it as if all mass were there, no matter how complex the internal motion.
Masses 1 kg at 0 and 3 kg at 4 m: x_cm = 3 m.
It may lie outside the body (a ring).
Motion of center of mass
The centre of mass moves as if all external forces act on it.
The centre of mass of a system moves as a point of mass M under the total external force: F_ext = M a_cm. Internal forces such as explosions don't change it.
A thrown wrench spins complexly but its centre of mass follows a parabola.
An exploding firework's centre of mass keeps its projectile path until things hit the ground.
Normal force on curves
Normal force combines weight component and centripetal requirement.
On a curved path the normal force N adjusts to both gravity and the centripetal need. At the top of a hill N = mg − mv²/r (lighter); at the bottom of a dip N = mg + mv²/r (heavier).
At 15 m/s over a 50 m radius crest, a 1000 kg car has N = 9800 − 4500 = 5300 N.
The car loses contact if v² ≥ gr at a crest.
Constraint forces
Forces that enforce geometric restrictions such as strings and surfaces.
Constraint forces (normal, tension) arise to enforce a geometric restriction and adjust their size to whatever is needed. They have no fixed formula and are found from the equations of motion.
A bead on a wire exerts whatever force is needed to keep it on the wire.
Constraint forces do no work if the constraint doesn't move.
Newton's laws and inertial frames
The laws hold in inertial frames; failing that, add pseudo forces.
Newton's laws hold exactly only in inertial frames. Earth is nearly inertial for short experiments. To use Newton's laws in rotating or accelerating frames, add fictitious forces.
A Foucault pendulum's rotating swing plane reveals Earth's rotation.
The laws don't claim that the frame is special, only that it is inertial.
Dimensions and limits of Newtonian mechanics
Breaks down at high speeds and small scales, replaced by relativity and quantum mechanics.
Newtonian mechanics works for everyday sizes and speeds but fails at speeds near c (relativity), at atomic scales (quantum mechanics) and in strong gravity (general relativity). It's still an excellent approximation within its domain.
GPS satellites need relativity but car engineering does not.
A theory's limits are a feature of science, not a failure.
Frequently asked questions
What is inertia?
Inertia is the tendency of an object to keep its current state of motion. The more mass an object has, the more inertia it has, and the harder it is to change its velocity.
Why does a heavier object not fall faster than a light one in a vacuum?
Gravity pulls harder on a heavier object, but it also has more inertia. The two effects cancel, so all objects fall with the same acceleration, g, when air resistance is negligible.
What is the difference between static and kinetic friction?
Static friction acts on objects that are not sliding and can take any value up to a maximum. Kinetic friction acts on sliding objects and is usually smaller and roughly constant.
What is the most common mistake in Newton's laws problems?
Drawing forces that the object does not experience, such as including the force it exerts on something else. Draw a free-body diagram showing only forces acting on the chosen object.
Is there a force that pushes an object forward when it moves?
No. Forward motion needs a net force in the direction of acceleration, but no separate forward force is needed to keep a body moving at constant velocity.
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