Physics 1 (Algebra-Based) / Unit 2

What you'll learn

• State and apply Newton's first, second, and third laws.
• Draw and use free-body diagrams for all forces on an object.
• Solve for net force, acceleration, or unknown forces using ΣF = ma.
• Analyze inclined-plane systems, including components of gravity, normal force, and friction.
• Distinguish static from kinetic friction and apply friction-force equations.
• Solve multi-body problems (pulleys, blocks in contact).
• Apply equilibrium conditions (ΣF = 0) to static or constant-velocity systems.

1. Newton's First Law: Inertia and Equilibrium

Newton's first law states that an object at rest stays at rest, and an object in motion continues at constant velocity in a straight line, unless acted on by a net external force. This property — resistance to a change in velocity — is called inertia, and it scales with mass: more massive objects are harder to start moving, stop, or redirect.

The law is really a statement about equilibrium: whenever the net force on an object is zero, \(\Sigma \vec F = 0\), the object's velocity does not change. Critically, this includes constant nonzero velocity, not just rest. A common exam trap is assuming a moving object needs a forward force to "keep it going" — in reality, a net force of zero is exactly what constant velocity requires. Any time a problem says "moving at constant velocity" or "in equilibrium," that is your signal to set the sum of forces (in each direction) equal to zero.

Multi-force equilibrium problems — often a hanging sign, banner, or traffic light supported by two or more cables at different angles — are solved by resolving every force into x- and y-components and setting each sum to zero independently: \(\Sigma F_x = 0\) and \(\Sigma F_y = 0\). This gives two equations, enough to solve for two unknowns (typically two tensions, or one tension and one angle).

2. Newton's Second Law and Free-Body Diagrams

Newton's second law is the workhorse of dynamics: \(\Sigma \vec F = m\vec a\). The net force on an object equals its mass times its acceleration, and both force and acceleration are vectors — they share the same direction.

A free-body diagram (FBD) isolates a single object and represents every force acting on it as an arrow, drawn from the object's center, with length roughly proportional to magnitude. Forces are never combined into one arrow on the diagram; each individual force (gravity, normal, tension, applied, friction) gets its own arrow, and the algebra of adding them happens afterward, not on the picture.

Standard method: (1) isolate the object and sketch it in isolation, (2) identify and draw every force acting on it, (3) choose a coordinate system — often tilted along the direction of motion for incline problems, (4) resolve every force into components along those axes, (5) write \(\Sigma F_x = ma_x\) and \(\Sigma F_y = ma_y\) separately, (6) solve. When a problem involves an applied force at an angle, remember that its component perpendicular to the motion changes the normal force — which in turn changes friction, since \(f = \mu N\).

3. Newton's Third Law: Force Pairs

For every force one object exerts on a second object, the second object exerts a force of equal magnitude and opposite direction back on the first: \(\vec F_{A\text{ on }B} = -\vec F_{B\text{ on }A}\). The two forces in a third-law pair always act on two different objects, are always the same type of force (both contact forces, or both gravitational), and never cancel each other, since they act on different bodies.

This is a frequent source of exam confusion: on a book resting on a table, the normal force (table pushes up on book) and the weight (Earth pulls down on book) are equal and opposite — but that is a coincidence of equilibrium, not a third-law pair, since both forces act on the same object (the book) and are different types of force. The true third-law partner of the table's normal force on the book is the book's normal force pushing down on the table.

Third-law pairs explain propulsion: a swimmer pushes water backward, and the water pushes the swimmer forward; a rocket expels exhaust downward, and the exhaust pushes the rocket upward; when you walk, your foot pushes backward on the ground, and the ground pushes forward on you. In every case, the two forces act on different objects — the swimmer and the water, the rocket and the exhaust, the foot and the ground.

4. Identifying Forces and Building Correct Free-Body Diagrams

Most FBD mistakes come from misidentifying which forces are actually present. Weight \(F_g = mg\) always points straight down, toward Earth's center, and acts on every object regardless of what else is happening. Normal force \(N\) is always perpendicular to the contact surface and adjusts its magnitude to prevent objects from passing through each other — it is not always equal to \(mg\); an applied force with a vertical component, or motion along an incline, changes it. Tension \(T\) acts along a string, rope, or cable, always pulling (never pushing) on whatever it's attached to; for an ideal massless string over an ideal massless, frictionless pulley, the tension has the same magnitude throughout. Friction acts parallel to the contact surface, opposing sliding or the tendency to slide.

A normal force does not always point "up" — on a ceiling, it points down; on a vertical wall, it points horizontally. The rule is always: perpendicular to the surface, pointing away from it, into the object.

Applied forces with a downward-angled component increase the normal force above \(mg\), which increases friction — sometimes dramatically enough to nearly stall an object even though a forward-pushing force is being applied.

5. Friction: Static and Kinetic

Static friction acts on objects that are not sliding, and it is a responsive force: it takes on whatever value is needed to prevent motion, up to a maximum value \(f_{s,\max} = \mu_s N\). If the applied force is less than this maximum, the object stays put and static friction exactly equals the applied force — not \(\mu_s N\). Once the applied force exceeds \(f_{s,\max}\), the object begins to slide, and kinetic friction takes over: \(f_k = \mu_k N\), a roughly constant value (to good approximation independent of speed and contact area) as long as sliding continues.

Because it typically takes more force to start an object sliding than to keep it sliding, \(\mu_s\) is generally greater than or equal to \(\mu_k\) for the same pair of surfaces. A common exam question gives both coefficients and an applied force, and asks you to first determine whether the object moves at all before calculating anything else.

6. Inclined Planes

On an incline, the standard trick is to rotate the coordinate axes so that one axis points along the slope and the other perpendicular to it. Gravity, the one force that doesn't naturally align with these axes, is resolved into two components: \(mg\sin\theta\) along the slope (pulling the object down the incline) and \(mg\cos\theta\) perpendicular to the slope (pressing the object into the surface). When there's no other force perpendicular to the incline, the normal force balances this second component exactly: \(N = mg\cos\theta\).

Friction, if present, acts along the incline surface, opposing whichever direction the object slides (or tends to slide). For an object sliding down under gravity alone, the acceleration down the slope is \(a = g\sin\theta - \mu_k g\cos\theta\).

A related and frequently tested question: what is the minimum angle at which a stationary object on an incline just begins to slide? At that critical angle, static friction is at its maximum and exactly balances gravity's component along the slope: \(mg\sin\theta = \mu_s mg\cos\theta\), which simplifies to \(\tan\theta = \mu_s\) — notice the mass cancels out entirely.

7. Systems of Connected Objects

When two or more objects are linked — by a string over a pulley, or simply pushed together in contact — solve by drawing a separate free-body diagram for each object and writing Newton's second law for each one individually. The link between the objects provides a constraint: objects connected by an inextensible string share the same magnitude of acceleration, and for an ideal massless string over an ideal massless, frictionless pulley, the tension is the same on both sides.

For objects merely pushed together (not connected by a string), the contact force between them is an internal force — it appears as a reaction pair (Newton's third law) between the two free-body diagrams, equal in magnitude and opposite in direction on the two objects, and it can be solved for once the shared acceleration is known.

8. Apparent Weight and Accelerating Reference Frames

"Apparent weight" is what a scale actually reads — the normal (or support) force on an object — which is not always equal to the true weight \(mg\) when there is vertical acceleration, such as in an elevator. Applying Newton's second law to a person of mass \(m\) standing on a scale in an elevator accelerating with acceleration \(a\) (taking up as positive): \(N - mg = ma \Rightarrow N = m(g+a)\).

If the elevator accelerates upward, \(a > 0\) and the scale reads more than the true weight — you feel heavier. If it accelerates downward, \(a < 0\) and the scale reads less — you feel lighter. At constant velocity (up, down, or stationary), \(a = 0\) and the scale reads exactly \(mg\); velocity itself never affects apparent weight, only acceleration does.

The extreme case is free fall: if the elevator cable snaps, \(a = -g\), and \(N = m(g - g) = 0\) — the scale reads zero. This is the same physical situation as an astronaut in orbit (see the Circular Motion and Gravitation unit): gravity has not disappeared, but there is no normal force, so nothing pushes back against the person's weight, producing the sensation of weightlessness.

Key equations

• ΣF = ma — Any dynamics problem
• Ff = μN — Friction problems
• Fg = mg — Any problem needing weight
• Along incline: mg sinθ; N = mg cosθ — Inclined-plane setups
• Per-object ΣF = ma, solved as a system — Pulley / connected-block problems

Open interactive practice for this unit