Source: ENGR 216 Lectures, PHYS 206 Concurrent
Tags: forces, free body diagram, FBD, statics, moments, torque, equilibrium, Newton's laws, resultant force, vector, static equilibrium, ENGR 216, PHYS 206, mechanics
Forces are pushes or pulls described by vectors (magnitude and direction). A free body diagram isolates a single object and maps every force acting on it, which is the essential first step for solving any statics or dynamics problem. When all forces and moments on a body sum to zero, the body is in static equilibrium.
Force
A push or pull acting on an object, measured in newtons (N). Force is a vector quantity, meaning it has both magnitude and direction.
Free body diagram (FBD)
A simplified sketch of a single object, isolated from its surroundings, showing every external force and moment acting on it. The most important problem-solving tool in mechanics.
Normal force (N or F_N)
The contact force exerted by a surface perpendicular to that surface. It adjusts to prevent objects from passing through one another.
Weight (W)
The gravitational force on an object, W = mg, directed straight down toward the centre of the Earth. g ≈ 9.81 m/s².
Tension (T)
The pulling force transmitted through a rope, cable, or string when it is taut. For an ideal (massless) rope, tension is the same throughout its length.
Friction (f)
A contact force that opposes relative sliding between surfaces. Static friction (f_s ≤ μ_s · N) keeps objects stationary; kinetic friction (f_k = μ_k · N) acts on sliding objects. μ_s > μ_k for the same surface pair.
Resultant force (net force)
The single vector sum of all forces acting on a body: F_net = ΣF. If F_net = 0, the object is in translational equilibrium.
Moment (torque)
The rotational effect of a force about a point: M = r × F, or in scalar form M = rF sin θ, measured in N·m. The sign convention (positive counterclockwise) must be consistent throughout a problem.
Moment arm (lever arm)
The perpendicular distance from the line of action of a force to the pivot point. A longer moment arm produces a larger moment for the same force.
Static equilibrium
The condition where both the net force and the net moment about any point are zero: ΣF = 0 and ΣM = 0. The object is neither translating nor rotating.
Concurrent forces
Forces whose lines of action all pass through a single point. These produce no net moment about that point, so only force-balance equations are needed.
Distributed load
A force spread over an area or length rather than applied at a single point. Replaced by an equivalent resultant force at the centroid of the distribution for analysis.
Identify the body of interest and mentally "cut" it free from everything it touches.
Draw the body as a simple shape (dot, box, beam).
Add the weight vector acting downward through the centre of gravity.
At every contact point, add the appropriate reaction forces: normal forces perpendicular to surfaces, friction forces parallel to surfaces, tension along ropes/cables.
Add any applied loads or external forces.
Label every force with a symbol and, where known, a magnitude. Include angles relative to a chosen coordinate system.
Choose and mark your positive x and y directions.
A common early mistake is including internal forces, or forces that act on other objects rather than the body you have isolated.
Any force vector can be decomposed into perpendicular components along your chosen axes:
F_x = F cos θ
F_y = F sin θ
where θ is measured from the positive x-axis to the force vector. Pick axes that align with the most forces (e.g. along an inclined plane) to minimise the number of components you need to resolve.
For a body in static equilibrium, three independent equations are available in 2D:
ΣF_x = 0 (no net horizontal force)
ΣF_y = 0 (no net vertical force)
ΣM_point = 0 (no net moment about any chosen point)
This means you can solve for at most three unknowns in a single 2D FBD. If you have more unknowns than equations, the system is statically indeterminate.
Choosing the moment point wisely can simplify the algebra. Pick a point where one or more unknown forces act, so those forces drop out of the moment equation.
The moment of a force about a point is calculated as:
M = F · d, where d is the perpendicular distance from the pivot to the line of action.
Equivalently, M = rF sin θ, where r is the distance from the pivot to the point of application and θ is the angle between r and F.
Sign convention: counterclockwise is typically positive, clockwise negative. Be consistent.
Couples are pairs of equal and opposite forces separated by a distance. They produce a pure moment with no net force: M_couple = F · d. A couple produces the same moment about every point.
Static friction is a reactive force. It only appears when something is trying to cause sliding, and it adjusts up to a maximum value:
f_s ≤ μ_s · N (inequality, not equality, unless the object is on the verge of slipping)
f_k = μ_k · N (equality, once sliding occurs)
When a problem asks for the minimum force to start motion, set f_s = μ_s · N (the threshold condition). When it asks whether an object remains stationary under given loads, check that the required friction force is less than μ_s · N.
Force components: F_x = F cos θ , F_y = F sin θ
Resultant force magnitude: F_net = √(ΣF_x² + ΣF_y²)
Resultant direction: θ = arctan(ΣF_y / ΣF_x)
Moment about a point: M = r × F = rF sin θ = F · d
Static equilibrium (2D): ΣF_x = 0 , ΣF_y = 0 , ΣM = 0
Friction: f_s,max = μ_s · N , f_k = μ_k · N
⚠️ FBDs appear in nearly every mechanics exam question. If your diagram is wrong, everything downstream is wrong.
⚠️ Do not set friction equal to μN automatically. Static friction is an inequality; it equals μ_s · N only at the tipping point of motion.
⚠️ When summing moments, include the moment contributions of every force, not just the obvious ones. Weight, applied loads, and reaction forces all contribute.
⚠️ The direction of the friction force opposes the direction the object would slide if friction were absent. Think about which way the object "wants" to move.
⚠️ Units matter. Forces in newtons, distances in metres, moments in N·m. Mixing cm and m is a frequent source of factor-of-100 errors.
Q: An object is in static equilibrium. What two vector conditions must be satisfied?
A: The sum of all forces must equal zero (ΣF = 0) and the sum of all moments about any point must equal zero (ΣM = 0).
Q: A 10 kg block sits on a horizontal surface with μ_s = 0.4. What is the maximum horizontal force you can apply before the block starts to slide?
A: The normal force equals the weight: N = mg = 10 × 9.81 = 98.1 N. Maximum static friction is f_s,max = μ_s · N = 0.4 × 98.1 = 39.24 N. So the maximum horizontal force before sliding is approximately 39.2 N.
Q: Why is it useful to take moments about the point where an unknown force acts?
A: Because a force acting through the pivot point has zero moment arm and therefore contributes zero moment. This eliminates that unknown from the equation, simplifying the algebra.
Q: What is the difference between a force and a moment?
A: A force causes (or tends to cause) linear acceleration of an object. A moment causes (or tends to cause) rotational acceleration about a point. A moment is produced by a force acting at a distance from a pivot.
Q: On a free body diagram, should you include forces the object exerts on other objects?
A: No. An FBD shows only the forces acting on the isolated body, not forces the body exerts on its surroundings. This is one of the most common FBD mistakes.
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