Laws of Motion

The One Thing

Newton’s laws provide a unified method for analysing motion: identify the external forces acting on an object or system, determine their resultant, and relate that resultant to acceleration or change in momentum. Equilibrium occurs when the resultant force—and, where relevant, the resultant torque—is zero.

Reactions, Processes and Experiments

Key Terms

• Force: An interaction that can change the state of motion, direction, or shape of an object; force is a vector quantity.
• Inertia: The tendency of an object to resist any change in its state of rest or uniform motion; it depends on mass.
• Newton’s First Law: An object remains at rest or continues in uniform straight-line motion unless acted upon by a non-zero resultant external force.
• Momentum: The quantity of motion of an object, defined as the product of its mass and velocity: $p=mv$.
• Newton’s Second Law: The net external force equals the rate of change of momentum; for constant mass, $F_{\text{net}}=ma$.
• Newton’s Third Law: For every action force, there is an equal and opposite reaction force acting on a different object.
• Impulse: The product of force and the time for which it acts; impulse equals the change in momentum: $J=F\Delta t=\Delta p$.
• Equilibrium: A condition in which the resultant force on an object is zero, so its acceleration is zero; it may be at rest or moving with constant velocity.
• Weight: The gravitational force acting on an object, given by $W=mg$ and directed vertically downward.
• Normal Reaction: The contact force exerted by a surface on an object, acting perpendicular to the surface.
• Tension: The pulling force transmitted through a stretched string, rope, or cable, directed along the string.
• Friction: A contact force that opposes relative motion or the tendency of relative motion between surfaces.
• Static Friction: Friction acting when surfaces are not sliding; its value adjusts up to a maximum value: $f_s\le {\mu }_{s}N$.
• Limiting Friction: The maximum value of static friction just before sliding begins: $f_{\max }={\mu }_{s}N$.
• Kinetic Friction: Friction acting when surfaces slide over each other: $f_k={\mu }_{k}N$, usually with ${\mu }_k$ less than ${\mu }_s$.
• Coefficient of Friction: A dimensionless ratio that measures frictional interaction between two surfaces: $\mu =f/N$.
• Free-Body Diagram: A diagram showing all external forces acting on an isolated object, used to apply Newton’s laws.
• Inertial Frame of Reference: A reference frame in which Newton’s laws hold directly, such as a frame at rest or moving with constant velocity.
• Centripetal Force: The net inward force required for circular motion: $F_c=m{v}^2/r$; it is supplied by forces such as tension, friction, or gravity.
• Impulse-Momentum Theorem: The impulse delivered to an object equals its change in momentum: $J=\Delta p$.
• Resultant External Force: The vector sum of all external forces acting on an object or system; it determines the object’s acceleration.
• Torque: The turning effect of a force; when relevant, equilibrium also requires the resultant torque to be zero.
• Conservation of Linear Momentum: The principle that the total linear momentum of an isolated system remains constant when the net external force is zero.
• SI Unit of Force: The newton, with $1\mathrm{N}=1\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2}$.
• SI Unit of Momentum: $\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-1}$.

Newton’s laws were published in 1687 in a major work on the mathematical principles of natural philosophy.

Easily Confused

• Mass and weight: Mass measures an object’s inertia, whereas weight is the gravitational force $W=mg$.
• Resultant force and individual force: Acceleration is determined by the vector sum of all external forces, not by one force considered alone.
• Static friction and limiting friction: Static friction can take any required value up to ${\mu }_{s}N$; limiting friction is its maximum value immediately before sliding.
• Static friction and kinetic friction: Static friction acts without sliding, whereas kinetic friction acts while surfaces slide; $f_k={\mu }_{k}N$, usually with ${\mu }_k<{\mu }_s$.
• Newton’s Third Law pairs and forces in equilibrium: Third-law forces act on different objects, whereas forces that cancel in equilibrium act on the same object.
• Centripetal force and a separate force type: Centripetal force is the inward resultant force required for circular motion; it may be provided by tension, friction, gravity, or another existing force.
• Zero resultant force and zero velocity: Zero resultant force means zero acceleration, so velocity remains constant; the object need not be at rest.
• Impulse and force: Impulse is $F\Delta t$ for constant force and equals change in momentum; force alone does not specify the duration of the interaction.
• Normal reaction and weight: The normal reaction acts perpendicular to the surface, whereas weight acts vertically downward.
• Inertial and accelerating frames of reference: Newton’s laws hold directly in an inertial frame, such as one at rest or moving with constant velocity.

What Gets Asked

• State or apply Newton’s laws: Questions may ask for the law explaining inertia, the relation $F_{\text{net}}=ma$, or equal and opposite interaction forces. A common mark-losing error is treating a third-law pair as acting on the same object.
• Calculate force, acceleration, momentum, or impulse: Use $p=mv$, $F_{\text{net}}=dp/dt$, $F_{\text{net}}=ma$, or $J=F\Delta t=\Delta p$. The specific distinction between $F_{\text{net}}=dp/dt$ and $F_{\text{net}}=ma$ is that the latter assumes constant mass.
• Determine whether an object is in equilibrium: Apply $\Sigma F_x=0$ and $\Sigma F_y=0$. Zero resultant force means constant velocity, not necessarily zero velocity.
• Draw or interpret a free-body diagram: Include every external force and give each the correct direction. Common errors include omitting friction, drawing tension other than along the string, or drawing the normal reaction in a direction that is not perpendicular to the surface.
• Analyse friction: Distinguish $f_s\le {\mu }_{s}N$, $f_{\max }={\mu }_{s}N$, and $f_k={\mu }_{k}N$. Static friction is not automatically equal to ${\mu }_{s}N$.
• Resolve forces or analyse circular motion: On an inclined plane, use $mg\sin \theta$ parallel to the plane and $mg\cos \theta$ perpendicular to it; in circular motion, use $a_c=v^2/r={\omega }^{2}r$ and identify which existing force supplies the centripetal resultant.