Dynamics: practice problems and exam questions

Kinematics and kinetics of particles and rigid bodies: projectile motion, work and energy, friction and springs, and rotation about an axis. Exam problems reward choosing the right method, and energy methods are the fastest when only speeds and positions matter.

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Cram pack (PDF)Formula sheets, method steps and practice problems with full solutions, ready to print.

Topics in this course

  1. Kinematicsspeeding up · braking · vertical throwcalculations
  2. Newton’s lawspush with friction · inclined plane · table and pulleycalculations
  3. Projectile motionhorizontal launch · launch from ground · launch from heightcalculations
  4. Circular motioncentripetal force · friction on a curve · vertical circlecalculations
  5. Work & energyfree fall · slope with friction · ramp, friction and springcalculations
  6. Momentum & collisionsperfectly inelastic · elastic · impulse and forcecalculations
  7. Rigid-body rotationmoment of inertia · ΣM = Iα, work and power · rolling and angular momentumcalculations
  8. Gears & gear trainsspeed ratio · torque and power · compound gear traincalculations
  9. Springs & pendulumsmass on a spring · pendulum · speed, acceleration and energycalculations
  10. Mechanical vibrationsnatural frequency · damping ratio · springs in series and parallelcalculations

Exam tips for Dynamics

  • Split the motion into horizontal and vertical parts; only the vertical part feels gravity.
  • Use conservation of energy when time is not asked: it skips the forces and accelerations.
  • Friction always does negative work: subtract μ·N·d in the energy balance.

Worked examples

KinematicsEasy
t (s)v (m/s)20 s10

A train slows down uniformly from 15 m/s to 10 m/s in 20 s. How far does it travel in that time? Find s.

v₀ = 15 m/sv = 10 m/st = 20 s

Find s (m)

Show hints
  1. With constant acceleration the average speed is the mean of the initial and final speeds.
  2. s = (v₀ + v)/2 · t. It is the area under the v–t graph.
  3. average speed = 12.5 m/s.
Show the answer and the step-by-step solution

Motion with constant acceleration: v = v₀ + a·t, s = v₀·t + ½·a·t² and v² = v₀² + 2·a·s. Pick the equation that contains the three things you know and the one you want, and keep the signs consistent with one positive direction.

Average speed

(v₀ + v)/2 = (15 + 10)/2 = 12.5 m/s

Distance

s = 12.5·20 = 250 m

Answer

s = 250 m
Newton’s lawsMedium
25 kg35°F = 140 Nμ = 0.25

A 25 kg sled is pulled across level snow by a rope at 35° above the horizontal with a force of 140 N. The coefficient of kinetic friction is 0.25. Use g = 9.81 m/s². Find the acceleration a of the sled.

m = 25 kgF = 140 Nθ = 35°μ = 0.25g = 9.81 m/s²

Find a (m/s²)

Show hints
  1. Split the pull into F·cos θ (forward) and F·sin θ (up).
  2. Vertically: N + F·sin θ = m·g. Friction is μ·N.
  3. N = 164.9 N.
Show the answer and the step-by-step solution

Newton’s second law: the net force on a body equals its mass times its acceleration, ΣF = m·a. Draw a free-body diagram, choose a positive direction along the motion, and add every force in that direction: applied forces, friction μ·N against the motion, and the components of gravity.

Normal force

N = m·g − F·sin θ = 245.3 − 140·sin 35° = 164.9 N

Horizontal forces

m·a = F·cos θ − μ·N = 114.7 − 41.24
a = 73.44/25 = 2.938 m/s²

Answer

a = 2.938 m/s²

How practising works

  • “Practise this course” mixes problems from all its topics, like a real exam.
  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Mock exams are timed and have no hints, and you get your score and a full review at the end.

Calculators for this course

Gear ratio calculatorProjectile motion calculatorNatural frequency and damping calculatorBelt length calculator (open and crossed belts)Kinematics calculator (SUVAT)Centripetal force calculatorCollision and momentum calculatorPendulum and spring period calculatorSpring constant calculator (Hooke’s law)Kinetic energy calculator (and potential energy, work and power)Free fall calculator (time, height and impact speed)Torque calculator (force, power and rpm)Engineering unit converterAcceleration calculator (and average velocity)

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