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Introduction to mechanics

Learn how things move and why. Explore forces, energy, rotation, gravity and oscillations.

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Physics / University introductory course

68small concepts
11 subtopics · your own pace
SUBTOPIC 01 · 4 concepts

Foundations: measurement and models

Quantities, dimensional checks, measurement precision and the scope of an idealization.

  • Quantities, dimensions and units
    Report a mechanical quantity as number times unit, distinguish dimension from unit, and convert using equal-quantity conversion factors. Read SI prefixes and scientific notation.
  • Checking equations by dimensions
    Require matching dimensions for quantities added or equated. Dimensional consistency is necessary but does not establish numerical factors, signs or physical validity.
  • Measurement uncertainty and reported precision
    Distinguish a measured value with uncertainty from an exact count. Interpret x±δx, distinguish accuracy from repeatability, and avoid reporting more precision than the measurement supports.
  • A physical model and its assumptions
    Choose whether a particle or rigid-body model fits a question and state the idealizations that justify omitted effects. A model’s prediction is conditional on those assumptions.
SUBTOPIC 02 · 4 concepts

Foundations: vector operations

Components, addition, subtraction and both vector products.

  • Scalars, vectors and signed components
    Distinguish a scalar from a vector and express a vector in perpendicular unit-vector directions. Resolve signed components and reconstruct magnitude and direction; the zero vector has no unique direction.
  • Adding, subtracting and scaling vectors
    Calculate vector sums, differences and scalar multiples componentwise. Explain a−b=a+(−b), and how a negative scalar reverses direction.
  • Dot products and projections
    Calculate a·b from components or |a||b|cosθ. For nonzero vectors, its sign distinguishes acute, right and obtuse angles; a positive dot product alone does not imply parallelism.
  • Cross products and the right-hand rule
    Determine a×b using magnitude |a||b|sinθ and the right-hand direction. Explain anti-commutativity, zero product for parallel vectors, and the signed z-component a_x b_y−a_y b_x for planar vectors.
SUBTOPIC 03 · 3 concepts

Foundations: coordinates and change

Coordinate conventions, rates and accumulation.

  • Position, origin and reference axes
    Describe position relative to an origin and oriented axes. Distinguish a coordinate change from motion, and state a consistent positive direction before interpreting signs.
  • Average rates and local rates
    Interpret a secant slope as an average rate and a tangent slope as a local rate, with units. A negative derivative describes decrease at that instant, not a negative value or a guarantee of monotonic decrease over a whole interval.
  • Signed accumulation and initial values
    Interpret signed area under a rate graph as net change and use final value=initial value+net change. Distinguish signed accumulation from total unsigned amount.
SUBTOPIC 04 · 7 concepts

Describing motion

Connect position, velocity and acceleration in one and two dimensions.

  • Displacement
    Calculate displacement as final position minus initial position and distinguish it from distance traveled.
  • Average velocity
    Calculate average velocity as displacement divided by elapsed time, distinguishing it from average speed.
  • Instantaneous velocity
    Find instantaneous velocity as the time derivative of position, including componentwise derivatives for vector position.
  • Acceleration
    Calculate average acceleration from a change in velocity and instantaneous acceleration as the time derivative of velocity.
  • Motion with constant acceleration
    Use constant-acceleration relations to predict position or velocity, while checking that the acceleration is constant over the interval.
  • Independent components of projectile motion
    Solve ideal projectile motion by applying horizontal and vertical kinematics with a shared elapsed time and downward gravitational acceleration.
  • Acceleration in uniform circular motion
    Explain why constant-speed circular motion has inward acceleration of magnitude v²/r even though its speed does not change.
SUBTOPIC 05 · 10 concepts

Forces and Newton’s laws

Choose an object, draw its forces and predict its acceleration.

  • A force is an interaction
    Explain why each force on an object must be attributed to an interaction with another object, whether contact or noncontact.
  • A free-body diagram
    Draw a free-body diagram that isolates one chosen object and includes only forces exerted on it.
  • Net force
    Calculate the vector sum of forces on a single object using components.
  • Inertial frames and Newton’s first law
    Explain why an object with zero net force has constant velocity in an inertial frame, and why an accelerating frame needs special care.
  • Newton’s second law
    Apply the component equations of net force equals mass times acceleration to a constant-mass object in an inertial frame.
  • Newton’s third-law pairs
    Explain why equal-and-opposite interaction forces act on different objects and therefore do not cancel in a single object's free-body diagram.
  • Normal force
    Determine the normal force from the acceleration perpendicular to a contact surface.
  • Tension in an ideal rope
    Apply the same-tension constraint of an ideal rope to a connected pair of masses.
  • Static and kinetic friction
    Select and apply the static-friction bound or kinetic-friction model according to whether surfaces slip.
  • The net radial force
    Apply Newton’s second law in the radial direction to identify which actual forces provide the acceleration of circular motion.
SUBTOPIC 06 · 7 concepts

Work and energy

Choose a system and account for transfers and stored energy.

  • Work by a constant force
    Calculate the work done by a constant force using its component along an object's displacement, including zero and negative work.
  • Work by a varying force
    Calculate work by integrating the component of a position-dependent force along a specified displacement.
  • Kinetic energy
    Calculate a particle's translational kinetic energy from its mass and speed.
  • The work–energy theorem
    Apply the work–energy theorem using the total work of all forces to find a change in speed.
  • Conservative forces and potential energy
    Use path independence to identify a conservative force and calculate its potential-energy change as the negative of its work.
  • Conservation of mechanical energy
    Determine whether mechanical energy is conserved and, when it is not, include nonconservative work in the energy balance.
  • Power
    Calculate average power from work and elapsed time, or instantaneous power from force and velocity.
SUBTOPIC 07 · 8 concepts

Momentum and collisions

Separate internal interactions from external impulses.

  • Momentum is a vector
    Calculate a particle's momentum from its mass and velocity, retaining the velocity's direction.
  • Impulse changes momentum
    Calculate impulse from a force–time history and use it to find a particle's change in momentum.
  • Choosing a system
    Choose a system boundary and classify forces as internal or external to it.
  • Conserving total momentum
    Explain why a system's total momentum remains constant when its external impulse is negligible, despite internal forces.
  • A perfectly inelastic collision
    Find the common final velocity of objects that stick together when external impulse during the collision is negligible.
  • An elastic collision
    Solve a one-dimensional elastic collision using conservation of both momentum and kinetic energy.
  • Center-of-mass position
    Calculate the center-of-mass position as a mass-weighted average.
  • Acceleration of the center of mass
    Predict center-of-mass acceleration from total mass and net external force.
SUBTOPIC 08 · 10 concepts

Rigid-body rotation

Relate torque, mass distribution and rotational motion.

  • Angular displacement
    Relate angular displacement in radians to arc displacement about a fixed axis.
  • Angular velocity
    Relate angular velocity to a point’s tangential speed about a fixed axis.
  • Angular acceleration
    Use angular acceleration as the rate of change of angular velocity to predict angular velocity and displacement when angular acceleration is constant.
  • Moment of inertia
    Calculate moment of inertia about a specified axis by summing mr² for discrete masses, accounting for each mass's distance from that axis.
  • Changing the rotation axis
    Apply the parallel-axis theorem to find a body's moment of inertia about an axis parallel to one through its center of mass.
  • Torque about an axis
    Calculate the signed torque about a fixed axis from an applied force, its point of application and the angle between force and radius.
  • Fixed-axis rotational dynamics
    Use the net external torque and moment of inertia to find a rigid body's angular acceleration about a fixed axis.
  • Rotational kinetic energy
    Calculate the kinetic energy of a rigid body rotating about a fixed axis using its moment of inertia and angular speed.
  • Rolling without slipping
    Apply the no-slip relation v_cm = Rω to connect a rolling body's translational and angular speeds.
  • Energy of a rolling body
    Use translational and rotational kinetic energy together with the no-slip constraint to solve a rolling energy-conservation problem.
SUBTOPIC 09 · 4 concepts

Angular momentum and equilibrium

Use torque balance and angular-momentum conservation.

  • Angular momentum of a particle
    Calculate a particle's angular momentum about a specified origin from its position and momentum, including its rotational direction.
  • Angular momentum about a fixed symmetry axis
    Calculate the component of a rigid body's angular momentum along its fixed symmetry axis using L = Iω.
  • Conserving angular momentum
    Apply angular-momentum conservation about a specified axis when the net external torque about that axis is negligible, distinguishing this condition from the absence of forces.
  • Static equilibrium
    Apply both zero net force and zero net torque to determine unknown forces on a stationary rigid body.
SUBTOPIC 10 · 5 concepts

Gravitation and orbits

Use force and energy to reason about planetary motion.

  • The inverse-square gravitational force
    Apply Newton’s gravitational force law to find the magnitude and direction of the force between two masses.
  • Gravitational potential energy
    Explain why inverse-square gravity has potential energy U(r) = −GMm/r when the zero is chosen at infinity.
  • A circular orbit
    Apply gravitational force and radial acceleration to determine the speed of a circular orbit around a much more massive body.
  • Escape speed
    Apply conservation of mechanical energy to find the minimum launch speed that allows an object to reach infinity with zero speed, neglecting other forces.
  • Kepler’s equal-area law
    Explain why a central gravitational force makes the line from an orbiting body to the attracting body sweep out equal areas in equal times.
SUBTOPIC 11 · 6 concepts

Oscillations

Recognize restoring forces and the limits of harmonic models.

  • A linear restoring force
    Apply F = -kx to find the magnitude and direction of a spring’s force relative to equilibrium, and distinguish a linear restoring force from one that is not proportional to displacement.
  • Simple harmonic motion
    Apply Newton’s second law to an ideal mass–spring system to obtain ω = √(k/m) and predict its sinusoidal motion from simple initial conditions.
  • The small-angle pendulum
    Explain how the small-angle approximation turns a pendulum’s tangential restoring force into a harmonic-motion model with ω ≈ √(g/L), and why that model is limited to small amplitudes.
  • Energy in an oscillator
    Apply mechanical-energy conservation to an ideal spring oscillator to relate amplitude, position, speed, and the exchange between spring potential and kinetic energy.
  • Damping removes oscillator energy
    Explain why a force opposing an oscillator’s motion does negative work, reducing its mechanical energy and amplitude rather than conserving them as in the ideal model.
  • Driven oscillations and resonance
    Explain why periodic forcing can produce a large response near an oscillator’s natural frequency and how damping limits the sustained response.