Newton's First Law
Newton’s first law identifies inertial frames and states the equilibrium condition: when the vector sum of external forces is zero, acceleration is zero and velocity is constant. Rest is only one special case.
Topic sequence · 4 of 10
Learning objectives
Objective 1
Translate “at rest” or “constant velocity” into a⃗=0 and component equations ΣF_x=0, ΣF_y=0 (and ΣF_z=0 when needed).
Objective 2
Distinguish zero net force from zero individual forces and static equilibrium from constant-velocity dynamic equilibrium.
Objective 3
Use free-body diagrams and vector components to solve equilibrium constraints without using the direction of velocity as a force clue.
Objective 4
Recognize that Newton’s first law defines the class of inertial frames in which the standard Newton-law model applies directly.
- Topic 2.2 free-body diagrams and vector force addition.
- Topic 1.2 acceleration as the derivative of velocity.
- Topic 1.4 inertial reference frames and frame choice.
Definitions first
Representations you must read and create
Representation 1
A free-body diagram with every real external force labeled and a vector sum of zero.
Representation 2
A velocity-time graph with zero slope; the horizontal line may be above, below, or on v=0.
Representation 3
A position-time graph that is linear when velocity is constant and horizontal only when the constant velocity is zero.
Representation 4
Component equations that show forces can balance independently along perpendicular axes.
Core relationships and calculus
Variables: external forces in N; acceleration in m/s²
Use: translational equilibrium in an inertial frame
Variables: velocity in m/s
Use: kinematic consequence of equilibrium
Variables: force components in N
Use: two-dimensional equilibrium equations
Variables: r in m; v in m/s
Use: calculus form of first-law motion
Newton’s first law can be written as the zero-right-hand-side differential equation m d²r⃗/dt²=0 for constant mass. Integrating once gives v⃗=v⃗₀, and integrating again gives r⃗=r⃗₀+v⃗₀t. The same equilibrium condition is enforced component by component through ΣF_x=0, ΣF_y=0, and if required ΣF_z=0.
In an inertial frame, start from ΣF⃗_ext=m a⃗. Set ΣF⃗_ext=0 to obtain d²r⃗/dt²=0. Integrate over time: dr⃗/dt=v⃗₀, a constant vector. Integrate again: r⃗(t)=r⃗₀+v⃗₀t. Thus zero net force predicts constant velocity, not necessarily zero velocity.
Apply the standard first-law equilibrium model in an inertial reference frame. Constant speed alone is not enough; the full velocity vector must be constant for a⃗=0.
Physical examples
Example 1
A puck on nearly frictionless ice can move at constant velocity while the net horizontal force is approximately zero.
Example 2
An elevator moving upward at constant speed has zero acceleration, so the upward cable force balances the downward weight.
Example 3
A book at rest on a table has normal force and weight that sum to zero; “rest” does not mean “no forces.”
Example 4
A car moving at constant speed around a circular track is not in equilibrium because its velocity direction changes and it has inward acceleration.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Translate “constant velocity” immediately into a⃗=0 before writing force equations.
Reasoning pattern 2
Do not infer net-force direction from the direction of motion; force follows acceleration, not velocity.
Reasoning pattern 3
If one axis has zero acceleration and another does not, impose equilibrium only on the zero-acceleration component.
Reasoning pattern 4
Check the reference frame. A braking car, accelerating elevator frame, or rotating platform is not automatically an inertial frame.
Reasoning pattern 5
When solving a multi-force equilibrium problem, draw the free-body diagram first and let the component equations determine unknown magnitudes.
Deeper calculus and limiting-case check
Newton’s first law is more than the special case ΣF⃗=0: it provides the operational criterion for an inertial frame. Once an inertial frame is chosen, the zero-force case integrates to linear position in time. A frame that accelerates relative to an inertial frame will not make every isolated object appear to maintain constant velocity.
Equilibrium is a vector condition, not a motion label
Static equilibrium
v=0 and a=0. Forces can still be large; they simply sum to zero. A hanging sign and a book on a table are standard examples.
Dynamic equilibrium
v is constant and nonzero while a=0. A gliding puck or constant-speed elevator can satisfy the same ΣF=0 equations.
Not equilibrium
Constant speed with changing direction is not constant velocity. Uniform circular motion has inward acceleration and therefore nonzero net force.
Partial equilibrium
One component can have a_i=0 while another has a_j≠0. Write Newton’s law separately by axis instead of giving the whole system one label.
Force-balance workflow
Choose the system
Identify the object or system whose acceleration is being analyzed. Only external forces on that system belong on its free-body diagram.
Confirm the frame
Use an inertial frame for the standard first-law model. A vehicle frame that is braking, turning, or otherwise accelerating needs different treatment.
Draw real forces
Include weight, normal, tension, friction, spring, or other real interactions as appropriate. Do not draw velocity as a force.
Use acceleration information
If the prompt says constant velocity, translate that to a=0. Then impose ΣF_x=0 and ΣF_y=0 where applicable.
Check the solution
Verify directions, units, and whether the computed force magnitude is physically plausible for the geometry.
Graph cross-checks for Newton’s first law
| Representation | Equilibrium signature | Common trap |
|---|---|---|
| Force diagram | Vector sum of all external forces is zero | Pairing forces merely because arrows look opposite |
| a–t graph | a=0 over the interval | Confusing zero acceleration with zero velocity |
| v–t graph | Horizontal line at any constant value | Assuming only the line v=0 represents equilibrium |
| x–t graph | Straight line; horizontal only if v=0 | Calling every non-horizontal x–t graph accelerated motion |
Interactive balancing-force solver
Enter two known force vectors. The tool returns the third force required for translational equilibrium.
FRQ and laboratory evidence
Calculation evidence
Show a labeled free-body diagram, write component force sums, set only the zero-acceleration components equal to zero, and solve symbolically before numbers when practical.
Explanation evidence
Say why acceleration is zero from the motion information, then connect a=0 to ΣF=0. Avoid circular wording such as “forces balance because the forces balance.”
Experimental evidence
For a cart coast interval, compare the force-sensor mean with zero and the v–t slope with zero. Include uncertainty rather than declaring exact equality from noisy data.
Frame evidence
If measurements are taken in a moving frame, explain whether it is inertial. Constant frame velocity is acceptable; changing frame velocity is the warning sign.
Three classic equilibrium patterns
Suspended object
Resolve angled tensions into components. Horizontal components may cancel while vertical components add to balance weight. Geometry determines tension; “two cables” does not imply T=mg/2.
Object on an incline
Choose axes parallel/perpendicular to the surface. If the object is at rest, both component accelerations are zero; the normal force balances only the perpendicular force component, not automatically the full weight.
Constant-speed elevator
Velocity may point up or down, but if speed and direction are constant then a=0. Cable tension equals weight only under the simple two-force constant-velocity model.
First law versus second law: use the wording carefully
For numerical equilibrium calculations, ΣF=ma with a=0 produces the same component equations as ΣF=0. The conceptual role of the first law is still distinct: it says isolated motion remains constant and identifies inertial reference frames. On an explanation question, stating only “because F=ma” can miss the requested reasoning about inertia or frame choice; connect the motion evidence, the inertial-frame assumption, and the zero-net-force conclusion explicitly.
Experimental connection
Use a low-friction cart with a motion sensor and, if available, a force sensor. Give the cart a short push, then analyze only the coast interval after contact ends. Test whether the measured net horizontal force is consistent with zero and whether the slope of v(t) is consistent with zero within uncertainty. Repeat with added mass to distinguish equilibrium from inertia-related misconceptions.
Connections across Physics C
Connection 1
Topic 1.4 explains how inertial frames can move at constant velocity relative to one another.
Connection 2
Topic 2.5 extends the model to nonzero net external force and nonzero acceleration.
Connection 3
Topic 2.10 is a high-value counterexample: constant speed in a circle is not constant velocity and therefore not equilibrium.
Connection 4
Unit 5 introduces rotational equilibrium, which additionally requires the net torque about an appropriate axis to vanish.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Treating zero net force as “no forces act.” Several forces can cancel exactly.
Avoid this
Assuming an object with zero net force must be at rest rather than moving with constant velocity.
Avoid this
Using constant speed as proof of equilibrium even when the direction of velocity changes.
Avoid this
Using the sign of velocity to choose the sign of net force.
Avoid this
Writing equilibrium equations in an accelerating frame without recognizing the frame issue.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Can an object move when the net force is zero?
Yes. In an inertial frame it moves with constant velocity, which may be nonzero.
Does constant speed mean equilibrium?
Only if the velocity vector is constant. Circular motion has constant speed but changing velocity, so the net force is not zero.
Are weight and normal force always equal?
No. They are equal only in situations where those are the relevant vertical forces and the vertical acceleration is zero.
Why is Newton’s first law needed if the second law exists?
The first law identifies inertial frames—the frames in which the standard Newton-law equations apply without extra frame-acceleration terms.
Can forces balance in one direction but not another?
Yes. Apply ΣF_i=m a_i component by component; one axis may be in equilibrium while another accelerates.