Stone Skipping Tom as Cruz Lu s Cebola Stone Shape Angle With - PowerPoint PPT Presentation

Stone Skipping Tom´ as Cruz Lu´ ıs Cebola

Stone Shape

Angle With Water

Spin

Children’s Game?

Children’s Game?

Physics of Everyday Life “Physicists have figured out some extremely fine details of the universe, from the radius of black holes to the behavior of subatomic particles - neither of which we can even see. It may surprise you to learn, then, that they lack explanations (or have only recently stumbled upon them) for many common phenomena we observe in daily life.” − Natalie Wolchover

Physics of Everyday Life “Physicists have figured out some extremely fine details of the universe, from the radius of black holes to the behavior of subatomic particles - neither of which we can even see. It may surprise you to learn, then, that they lack explanations (or have only recently stumbled upon them) for many common phenomena we observe in daily life.” − Natalie Wolchover The Cheerios Effect Why your breakfast cereal tends to clump together or cling to the sides of a bowl of milk? - See Vella and Mahadevan, The ’Cheerios effect’ .

Getting a Kick From Water

Getting a Kick From Water

A Simple Model 1 Reaction Force ν ∼ 10 5 ⇒ Inertial regime. ◮ R e = aV ◮ We = ρ V 2 z ∼ 100 ⇒ Neglect surface tension. σ ◮ Fr = V 2 zg ≫ 1 ⇒ Neglect buoyancy. F = 1 +1 2 C l ρ w V 2 S im f ( θ, β ) n 2 C d ρ w V 2 S im f ( θ, β ) t � �� � � �� � Lift Drag 1 Based on: Rosellini et al, 2005 e Boqcuet L, 2002.

A Simple Model 1 Stone Shape: Reaction Force ν ∼ 10 5 ⇒ Inertial regime. ◮ R e = aV ◮ We = ρ V 2 z ∼ 100 ⇒ Neglect surface tension. σ ◮ Fr = V 2 zg ≫ 1 ⇒ Neglect buoyancy. F = 1 +1 2 C l ρ w V 2 S im f ( θ, β ) n 2 C d ρ w V 2 S im f ( θ, β ) t � �� � � �� � Drag Lift Immersed Surface � � � � � � � � � 2 1 − s 1 − s 1 − s , s = | z | S im = R 2 arccos − 1 − sin θ R R R

A Simple Model Spin?: What is the effect of spinning?

A Simple Model Spin?: What is the effect of spinning? 1. Lift force is applied in the � I n − I t � 2 ( θ − θ 0 ) = M θ ¨ ˙ immersed surface - θ + φ I t I t destabilizing torque; 2. Only a specific range of θ is M θ I t favourable to skipping; ( θ − θ 0 ) ∼ ˙ φ 2 ( I n − I t ) 2 3. Gyroscopic effect - Stabilizing To work ˙ torque; φ � 20 Hz .

A Simple Model To address the problem of the angle we need to go deeper in the model � x 2 + ˙ x = − 1 z 2 ) S im ( z ) sin( θ + β )( C l sin θ + C d cos θ ) M ¨ 2 ρ (˙ x 2 + ˙ z = − Mg + 1 z 2 ) S im ( z ) sin( θ + β )( C l cos θ − C d sin θ ) M ¨ 2 ρ (˙ � − ˙ � z Notice that: β = arctan and θ ∼ Cte. x ˙

A Simple Model To address the problem of the angle we need to go deeper in the model � x 2 + ˙ x = − 1 z 2 ) S im ( z ) sin( θ + β )( C l sin θ + C d cos θ ) M ¨ 2 ρ (˙ x 2 + ˙ z = − Mg + 1 z 2 ) S im ( z ) sin( θ + β )( C l cos θ − C d sin θ ) M ¨ 2 ρ (˙ � − ˙ � z Notice that: β = arctan and θ ∼ Cte. x ˙

Model expansion and simulation C d ≪ C l ≃ 1 On air: ( z > 0) On water: ( z < 0) � x 2 + ˙ � x = − 1 z 2 ) S im ( z ) sin( θ + β ) sin θ M ¨ 2 ρ (˙ M ¨ x = 0 x 2 + ˙ z = − Mg + 1 z 2 ) S im ( z ) sin( θ + β ) cos θ M ¨ 2 ρ (˙ M ¨ z = − Mg Immersed Area  � � � � � 2 R 2 � 1 − s � � 1 − s � 1 − s arccos − 1 − , z < 0 & | z | < 2 R sin θ   R R R  S im = π R 2 z < 0 & | z | > 2 R sin θ   0 , z > 0  ρ = 1 g / cm 3 M = 0 . 1 kg g = 9 . 8 m / s 2 R = 0 . 025 m

Simulation: Stone Position

Simulation: Stone Position

Simulation: Energy Loss ◮ We can try a little analytical trick: � t coll ∆ E ≃ 1 x 2 f − 1 x 2 2 M ˙ 2 M ˙ 0 = F x ˙ xdt 0 � M sin θ ≃ − 1 . 4 NMg 2 π ≃ − 1 . 928 J C ρ R But in the end the most precise result come from the model:

Simulation: Best Angle Angle With Water: ◮ ’Magic angle’ ≃ 20 o ◮ Best angle ≃ 15 o

Experiment 1 - Rosellini Set up: Results:

Experiment 1 - Rosellini Set up: Results:

Experiment 2 - Hewitt

Contests and curiosities A little bit of history ◮ Record of stone skipping as a sport goes back at least to 1583. ◮ Tradition holds that the sport was begun by an English king who skipped stones across the Thames. ◮ Nowadays there are tens of tournaments and championships on stone skipping. Stones manufactured for skipping The current record for stone skipping is Russ Byars with a throw of 51 skips.

Feeling Lucky?

51 skips

Simulation: Mass

Simulation: Radius