Worked examples
Each example has the problem statement, an animation of that exact system that you can tune, and a step-by-step solution. Try each problem on paper first, then open one solution step at a time.
The five-step method
- Choose a coordinate system: define each coordinate, its positive direction and where it is measured from.
- Draw the free-body diagrams of every body, in a displaced position.
- Kinetics: write Newton–Euler equations ($\sum F=m\ddot x$, $\sum M_O=I_O\ddot\theta$) for each body.
- Kinematics: relate the coordinates through constraints (cables, rolling, links).
- Solve: eliminate the unknowns to reach one equation of motion, then answer the question.
Block on an incline driving a pulley
A spring-restrained block on a smooth incline is tied by a cable to a pulley carrying a hanging block.
Example 2Pulley system with a hanging spring
Block A on a smooth table pulls a pulley-hung block B (3m) that also sits on a spring.
Example 3Two carts linked by a pivoting bar
A massless bar pinned at its base links two spring-restrained carts at different heights.
Example 4Impact: a block sticks to a damped carriage
Block A (2m) hits a spring–damper-mounted carriage B (m) at 30 m/s and sticks.
Example 5Critical damping of a pinned bar
A homogeneous bar pinned at O, restrained by a spring at a and a damper at b, in a horizontal plane.
Example 6Rolling disk driven through a spring
A disk that rolls without slipping, tied to a wall by k and to a moving base by 4k.
Example 7Keeping a base-driven mass below a limit
A mass between a moving base and a wall. Which frequencies keep its amplitude under 1.5y₀?
Example 8Pulley system driven by a moving support
A moving support drives a mass through a spring, a cable and two pulleys; a second spring holds the mass from below.