Source: ENGR 216 Lectures, PHYS 206 Concurrent
Tags: Newton's laws, first law, second law, third law, inertia, F=ma, friction, kinetic friction, static friction, coefficient of friction, inclined plane, ENGR 216, PHYS 206
Newton's three laws are the foundation of classical mechanics. The first law defines inertia, the second law connects force to acceleration (F = ma), and the third law pairs every force with an equal-and-opposite reaction. Friction, governed by the coefficient of friction and the normal force, is the most common non-ideal force you will encounter in lab and exam problems.
Newton's first law (law of inertia)
An object at rest stays at rest, and an object in motion stays in uniform motion, unless acted upon by a net external force. This defines inertial reference frames.
Inertia
The tendency of an object to resist changes to its state of motion. Mass is the quantitative measure of inertia.
Newton's second law
The net force on an object equals its mass times its acceleration: ΣF = ma. This is a vector equation, so it applies independently in each direction (ΣF_x = ma_x, ΣF_y = ma_y).
Newton's third law
For every action force, there is an equal and opposite reaction force. These two forces act on different objects. They never cancel each other on a single FBD because they appear on different diagrams.
Coefficient of static friction (μ_s)
A dimensionless ratio that sets the maximum friction force before an object begins to slide: f_s,max = μ_s · N. Depends on the material pair, not on contact area.
Coefficient of kinetic friction (μ_k)
A dimensionless ratio that determines the friction force during sliding: f_k = μ_k · N. Always less than μ_s for the same surfaces.
Normal force
The perpendicular contact force exerted by a surface. On a flat horizontal surface, N = mg. On an incline at angle θ, N = mg cos θ.
Apparent weight
The normal force a scale or surface exerts on you. In an accelerating lift, apparent weight differs from true weight: N = m(g + a) when accelerating upward, N = m(g − a) when accelerating downward.
This is the standard workflow for dynamics problems:
Draw a free body diagram for each object of interest.
Choose a coordinate system. For inclined planes, tilt axes so one axis runs along the surface.
Resolve all forces into components along your chosen axes.
Write ΣF = ma for each axis direction.
If multiple objects are connected (e.g. by a rope over a pulley), write equations for each object and link them through shared acceleration or tension.
Solve the system of equations for the unknowns.
An object on a ramp at angle θ to the horizontal:
Component of gravity along the ramp (down the slope): mg sin θ
Component of gravity into the ramp (perpendicular): mg cos θ
Normal force: N = mg cos θ (from the perpendicular equilibrium)
Friction force along the ramp: f = μN = μmg cos θ
For an object sliding down with friction: ma = mg sin θ − μ_k mg cos θ, so a = g(sin θ − μ_k cos θ).
For the critical angle at which an object just begins to slide: tan θ_c = μ_s. This is a clean way to measure the coefficient of static friction experimentally.
When two masses are connected by a string over a frictionless pulley:
The tension is the same throughout the string (massless string assumption).
The magnitudes of acceleration are equal for both masses (inextensible string assumption).
Write ΣF = ma for each mass separately, then solve simultaneously.
For a simple Atwood machine with masses m₁ > m₂: a = (m₁ − m₂)g / (m₁ + m₂) T = 2m₁m₂g / (m₁ + m₂)
A common source of confusion: third-law force pairs act on different objects. The force the Earth exerts on you (your weight) and the force you exert on the Earth are a third-law pair. The normal force of the floor on you and your weight are not a third-law pair; they are both forces acting on you and happen to be equal only when you are in vertical equilibrium.
Once an object is sliding, kinetic friction applies:
f_k = μ_k · N (this is an equality, not an inequality)
Direction: always opposes the direction of relative sliding
Kinetic friction is constant for a given normal force and surface pair
For objects at rest, you must check whether the required static friction is within the limit. If it is, the object stays put. If the required friction exceeds μ_s · N, the object accelerates.
Newton's second law: ΣF = ma (vector), or ΣF_x = ma_x and ΣF_y = ma_y
Weight: W = mg
Inclined plane components: Along slope: mg sin θ , Perpendicular: mg cos θ
Normal force on incline: N = mg cos θ
Friction: f_s ≤ μ_s N , f_k = μ_k N
Critical slip angle: θ_c = arctan(μ_s)
Atwood machine: a = (m₁ − m₂)g / (m₁ + m₂) T = 2m₁m₂g / (m₁ + m₂)
⚠️ Newton's second law is a vector equation. Always apply it component by component, not as a single scalar.
⚠️ Third-law pairs act on different objects. If both forces are on the same FBD, they are not a third-law pair.
⚠️ On an incline, the normal force is mg cos θ, not mg. Using mg for N on an incline is one of the most common errors.
⚠️ When an object is in equilibrium on an incline (not sliding), friction is whatever value is needed to keep ΣF = 0 along the slope. It is not automatically equal to μ_s N.
⚠️ The "spaceship landing" lecture topic applies Newton's second law to variable thrust and gravity: the net upward force must produce the deceleration needed to bring the spacecraft to rest at the surface.
Q: A 5 kg block is pulled across a horizontal surface at constant velocity by a 20 N horizontal force. What is the coefficient of kinetic friction?
A: Constant velocity means a = 0, so f_k = F_applied = 20 N. N = mg = 49.05 N. μ_k = f_k / N = 20 / 49.05 ≈ 0.41.
Q: State Newton's third law and give an example.
A: For every force one object exerts on a second object, the second object exerts an equal-magnitude, opposite-direction force on the first. Example: when you push a wall, the wall pushes back on your hand with the same force.
Q: A block sits on an incline at 30°. If μ_s = 0.6, does it slide?
A: The critical angle is θ_c = arctan(0.6) ≈ 31°. Since 30° < 31°, the block does not slide.
Q: Two masses (3 kg and 5 kg) are connected over a frictionless pulley. What is the acceleration?
A: a = (5 − 3)(9.81) / (5 + 3) = 2(9.81)/8 ≈ 2.45 m/s².
Q: Why does kinetic friction remain constant while static friction can vary?
A: Kinetic friction depends only on μ_k and N, which are fixed for a given situation. Static friction is a reactive force that matches the applied force up to a maximum of μ_s N, so it varies depending on what other forces are present.
Newton's first law, Newton's second law, Newton's third law, F=ma, inertia, mass, force, friction, static friction, kinetic friction, coefficient of friction, normal force, inclined plane, ramp, Atwood machine, pulley, tension, apparent weight, free body diagram, ENGR 216, PHYS 206, mechanics, Texas A&M