Difficulty: Introductory | Prerequisites: Kinematics (position, velocity, acceleration). Basic trigonometry for resolving force components.
Newton's laws connect motion to its cause: force. Once you understand kinematics (how things move), forces explain why they move that way. This unit covers the relationship between force, mass, and acceleration, how to analyse systems with multiple objects, and what happens when forces act at angles. Circular motion introduces a new direction for acceleration and force. If kinematics felt comfortable, this is the natural next step. If kinematics was rough, revisit it before diving in here.
Newton's second law (F_net = ma) is the central equation: the net force on an object equals its mass times its acceleration. An object moving at constant velocity has zero net force. In circular motion, the net force points toward the centre (centripetal), even when the speed is constant.
Force
A push or pull on an object, measured in newtons (N). In simple terms, a force is anything that can change an object's velocity.
Net force (F_net)
The vector sum of all forces acting on an object. This is the force that determines the acceleration. Think of it as what is left over after all pushes and pulls are added together.
Newton's first law (law of inertia)
An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted on by a net external force. In simple terms, things keep doing what they are doing unless something pushes or pulls them.
Newton's second law
F_net = ma. The acceleration of an object is proportional to the net force and inversely proportional to the mass.
Newton's third law
For every action there is an equal and opposite reaction. If object A pushes on object B, then B pushes back on A with the same magnitude but opposite direction.
Normal force (N or F_N)
The contact force exerted by a surface perpendicular to that surface. Think of it as the surface pushing back to prevent an object from falling through.
Weight (W or F_g)
The gravitational force on an object: W = mg, directed downward.
Centripetal acceleration
The acceleration directed toward the centre of a circular path, given by a_c = v squared / r. In simple terms, even at constant speed, turning requires acceleration because the direction of velocity is changing.
Centripetal force
The net inward force that causes circular motion: F_c = mv squared / r. This is not a new type of force; it is the label for whatever force (gravity, tension, friction, normal force) points toward the centre.
The net force on an object equals its mass times its acceleration: F_net = ma.
When multiple objects move together (like a sister on a sled), treat the system as a single object.
Total mass = sum of all masses in the system.
a = F_net / m_total.
Example: a force of 120 N accelerates a sled (7.4 kg) plus sister at 2.5 m/s². Total mass = 120 / 2.5 = 48 kg. Sister's mass = 48 - 7.4 = 40.6 kg, which rounds to 41 kg.
When lifting an object upward with acceleration, the applied force must overcome both gravity and provide the extra force for acceleration.
F_applied = m(g + a).
Example: a grocery bag tears at 51.5 N, lifted with a = 1.25 m/s². Maximum mass = 51.5 / (9.8 + 1.25) = 51.5 / 11.05 ≈ 4.66 kg.
The direction of the net force always matches the direction of the acceleration.
An object moving in a circle at constant speed is accelerating because its direction is continuously changing.
The acceleration points toward the centre of the circle (centripetal).
This requires a net inward force, also directed toward the centre.
A car on a circular track at constant speed: the net force is directed inward, toward the centre of the track. The common mistake is thinking the net force is zero because the speed is constant, but speed is not the same as velocity. The velocity vector is changing direction.
Explaining why constant-speed circular motion counts as acceleration: acceleration is the rate of change of velocity. Velocity is a vector (magnitude and direction). Even if the magnitude (speed) stays the same, a change in direction means the velocity vector is changing, so there is acceleration.
When a force is applied at an angle, break it into horizontal and vertical components.
Horizontal component: F cos θ.
Vertical component: F sin θ.
For an object on a flat surface, the vertical forces must balance if there is no vertical acceleration.
Normal force + vertical component of applied force = weight (if the applied force has an upward component).
Example: a suitcase weighing 225 N is pulled at 25° above horizontal. Normal force = 180 N. The upward vertical component of the pull = Weight - Normal = 225 - 180 = 45 N.
On an inclined plane, resolve forces parallel and perpendicular to the surface rather than into x and y.
Newton's first law: if an object moves at constant velocity (including zero velocity), the net force on it is zero.
This applies even to individual parts of a system.
Example: two stacked blocks A and B, both moving at constant speed to the right. Block A (on top) is moving at constant velocity, so the net force on block A is zero. The hand pushes block B, and friction between the blocks pushes A forward, but the total on A still sums to zero because there is no acceleration.
"Constant speed" is the key phrase. If you see it, set F_net = 0 and work from there.
Newton's second law: F_net = ma
Weight: W = mg (g = 9.8 m/s²)
Centripetal acceleration: a_c = v² / r
Centripetal force: F_c = mv² / r
Force components at angle θ: Fx = F cos θ, Fy = F sin θ
Vertical equilibrium (flat surface, no vertical acceleration): F_N + F_y = mg
System acceleration: a = F_net / (m₁ + m₂ + ...)
Lifting with acceleration: F_applied = m(g + a)
Newton's second law is how engineers size motors, brakes, and rocket engines: you need to know the force required to produce a given acceleration for a given mass. Centripetal force keeps satellites in orbit (gravity provides the inward pull) and keeps a car on a banked curve (the normal force and friction combine to point toward the centre). The grocery-bag problem is a practical version of what happens when you jerk a heavy shopping bag upward too quickly.
Students often think that an object moving at constant speed has no forces acting on it. Forces may be present, but they balance to give zero net force.
Students confuse centripetal and centrifugal. Centripetal force is the real inward force. "Centrifugal force" is a fictitious force felt in a rotating reference frame; it does not appear in Newton's laws in an inertial frame.
Students sometimes think a heavier object needs more force to keep it in circular motion at the same speed and radius. This is correct (F_c = mv²/r), but the common mistake is thinking heavier objects move in tighter circles, which conflates several variables.
Students assume the normal force always equals the weight. It does not when there is a vertical component from another force, or when the surface is inclined, or when the object is accelerating vertically.
⚠️ System problems (multiple objects accelerating together) are a favourite. Identify the total mass, find the acceleration, then isolate individual objects if needed.
⚠️ The "constant speed" trigger: if the problem says constant velocity or constant speed, set the net force to zero immediately.
⚠️ Circular motion conceptual questions appear frequently. Know that the net force points inward and that constant speed does not mean zero acceleration.
⚠️ Force-component problems with angled pulls or pushes require careful free-body diagrams. Draw them before writing equations.
⚠️ Lifting problems (like the grocery bag) combine Newton's second law with the condition that force must exceed mg by enough to also provide the upward acceleration.
True or false: if an object is at rest, no forces act on it. (False. Forces may act, but they balance to zero net force.)
Fill in the blank: the net force on an object moving in a circle at constant speed points ______. (toward the centre)
True or false: the normal force on a flat surface always equals the weight. (False. Other vertical forces can change it.)
Fill in the blank: if you double the mass but keep the same net force, the acceleration is ______. (halved)
True or false: Newton's third law pairs act on the same object. (False. They act on two different objects.)
Q: You pull a sled (7.4 kg) with your sister on it across frictionless ice with a force of 120 N, producing an acceleration of 2.5 m/s². What is your sister's mass?
A: Total mass = F/a = 120/2.5 = 48 kg. Sister's mass = 48 - 7.4 = 40.6 ≈ 41 kg.
Q: A car travels at constant speed around a circular track. What is the direction of the net force?
A: Inward, toward the centre of the track. Constant speed does not mean zero force; the direction of velocity is changing.
Q: A suitcase (weight 225 N) is pulled at 25° above horizontal. The normal force is 180 N. What is the vertical component of the pulling force?
A: F_vertical = Weight - Normal = 225 - 180 = 45 N upward.
Q: Two stacked blocks move together at constant speed when pushed. What is the net force on the top block?
A: Zero. Constant velocity means zero acceleration, which means zero net force.
Q: A grocery bag tears at 51.5 N. If lifted upward at 1.25 m/s², what is the maximum mass it can hold?
A: m = F / (g + a) = 51.5 / (9.8 + 1.25) = 51.5 / 11.05 ≈ 4.66 kg.
Q: Explain why an object moving in a circle at constant speed is accelerating.
A: Acceleration is the rate of change of velocity. Velocity is a vector quantity with both magnitude and direction. In circular motion, the direction of the velocity continuously changes even though the speed (magnitude) stays the same. A change in direction means the velocity vector is changing, which constitutes acceleration. This acceleration points toward the centre of the circle.
Newton's second law connects directly to work and energy: force applied over a distance does work, which changes kinetic energy. Momentum is the product of mass and velocity, and Newton's second law can be rewritten as F = dp/dt (the rate of change of momentum). Understanding forces in circular motion prepares you for topics like orbital mechanics and rotational dynamics later in the course.
Newton's laws, F = ma, net force, free body diagram, FBD, normal force, weight, centripetal force, centripetal acceleration, circular motion, constant velocity, equilibrium, force components, inclined plane, friction, frictionless surface, system of objects, PHYS 114, general physics, introductory mechanics, University of Washington