DYNAMICS · CONNECTED SYSTEMS

Pulleys and Connected Bodies

Two bodies connected by an ideal inextensible rope. Study an Atwood machine or a block on an inclined plane connected to a hanging mass. Observe tension, friction, acceleration and stopping events.

Physical Parameters

Guided Experiments

Configuration

Mass m₁

Mass m₂

Gravitational Acceleration g

m/s²

Initial Conditions

Available Travel L

Initial Coordinate q₀

Initial Velocity v₀ (positive: m₂ descends)

m/s

Maximum Duration

Sign Convention: q increases as m₂ moves downward and m₁ upward. In the Atwood system, the coordinates along the two branches, measured from their travel references, are q for m₂ and L−q for m₁. On the incline, q measures how far m₁ has travelled from the foot of the slope, together with the corresponding downward displacement of m₂. Small gaps around blocks and the pulley in the drawing do not represent additional physical distances.

Examples: Atwood: Unequal Masses · Equilibrium · Motion Reversal · Incline: Sliding · Static Friction · Block Sliding Down

Simulation and Force Diagrams

True-Scale Geometry 1:1: the same number of pixels represents one metre horizontally and vertically. Incline angles are not distorted.

Connected System

Force Diagrams for Both Bodies

tension weight normal force friction applied force

Playback Speed

Time 0.00 s

Position q—

Common Velocity v—

Common Acceleration a—

Tension T—

The rope remains taut during free motion. The end of travel is an inelastic stop with permanent locking, not a rebound: after the impact, rope tension cannot be calculated from the free-motion equation.

Forces and Motion Conditions

Weight m₁g—

Weight m₂g—

Normal Reaction N—

Maximum Static Friction—

Actual Friction—

Signed Driving Force—

Net Force Along Rope—

Motion Regime—

Synchronized Graphs

Click a graph or enter an exact time: the selection updates both bodies, forces, results and explanations. Abrupt acceleration changes indicate physical regime transitions.

Position q(t)

Common Velocity v(t)

Common Acceleration a(t)

Tension T(t)

Values Table

t (s)	q (m)	v (m/s)	a (m/s²)	T (N)	Friction (N)	Regime

SI units, comma-separated values and decimal points.

Model Assumptions

The two masses are treated as point masses. The rope is massless, always taut and inextensible; the pulley has no mass, friction or rotational inertia. Rope tension therefore has the same magnitude throughout the free-motion phase.

In the Atwood case, both masses move vertically in opposite directions. In the inclined-plane system, m₁ moves along the slope and m₂ moves vertically. Both bodies have equal acceleration magnitudes along the constrained coordinate.

Static friction acts only while the system is at rest and adjusts up to its maximum value. When the block slides, kinetic friction acts opposite the velocity of m₁ along the slope.

Rope elasticity, extension, lateral oscillations, slipping over the pulley, collisions between masses, axle inertia and friction, and body geometry are not modelled. The end stop adds an extra constraint: tension after impact is undetermined in this model and is shown as “—”.

FROM FORCE COMPONENTS TO MOTION

Step-by-Step Calculations

Separate equations for each mass, the kinematic constraint, friction conditions and numerical calculations with SI units throughout.

Values are rounded only for display, not during calculations. Every simulation uses the same geometric scale in both directions.

Other Experiments: Forces and Newton · Inclined Plane · Projectile Motion · Physics Lab