David and Goliath: Strength and Power in Sport John D Barrow

Mass and Weight Mass = Force/Acceleration m= F/A Mass = Force/Acceleration m= F/A Weight is a force W = Mg Weight is a force W = Mg A= M/m  g A= M/m  g Expt: (M-m)/(M+m) < Expt: (M-m)/(M+m) < Implies m = M exactly (Einstein) Implies m = M exactly (Einstein) accelerate MM’ M If m  m’ gravity

Latitude and Gravity Weight of mass M is least at Equator and most at Poles Mg Equator < Mg Poles 100 Kg weighs 0.5% more at the Poles = 500 gm Mr 2 r

Throwing and Jumping V 2 /2g Maximum Horizontal Range: V 2 /2g V 2 /g Maximum vertical height: V 2 /g

The Best Place to Be a Weightlifter High altitude  greater distance from Earth’s centre  lower g Some of worst venues are Oslo or Helsinki, where m/s² and 100 kg weighs N Mexico City is the best venue to combine height and latitude gains g = m/s² so 100kg weighs N A good place to lose weight Mg = GMM E /R 2

Leverage Class 1 Class 2Class 3 press upsBarbell curlsrowing

Wrestling holds: class 2 > class 3 > class 1 Cael Sanderson 2004 Olympic champion

4.5-5m 2m 2.45m HJ 6.14m PV

Karate 3100 N Need 3100 N to split 20x30x1cm pine block 3200 N for same area brick 4cm thick Black belt strikes at V = 7m/s Arm mass is about m = 3.4 kg Momentum of arm is mV = 7 × 3.4 = 24 kg m/s contact time with the target is less than T = 5 millisec 4800 N Force  m x V/T = 24 ÷ = 4800 N [A 70kg person weighs 70 × 9.8 = 686 N] Easy !

The Giant Swing Bangkok Wat Suthat Temple, 30m high !

1910 Ceremonies stopped in 1932 due to fatal accidents To thank the god Shiva for the annual rice harvest

How big a force do you feel?

2.5m Angular velocity at top = w Angular velocity at bottom = W Centre of gravity

Rotation energy at bottom = Rotation energy at top + Potential Energy lost ½ IW 2 = ½ Iw 2 + 2Mgh Max force on gymnast is at the bottom Maximum force felt by the gymnast = Mg + MhW 2 = Mg + 4M 2 gh 2 /I I = Mk 2  Mh 2 Total force felt by the gymnast = Mg(1 + 4h 2 /k 2 +hw 2 /g) With a typical angular velocity at the top of 2-3 per s and g = 9.8 m/s 2 and a typical body size with h = 1.3m, Total force felt by the gymnast  Mg( )  6Mg Start from handstand only reduces it by 1g Head spins in circle h approx 1.3m only w W AngVely s -1 s -1

Naim Suleymanoglu (b.1967) lifts 3 times his body weight

Strength versus Size Strength is proportional to area R 2 Strength is proportional to area  R 2 Weight is proportional to volume  R 3

Strength and Weight strength  (weight) 2/3 Giants eventually break!

World Weightlifting Records (strength) 3  (weight) 2

Putting The Shot – Two Surprises World record metres

Max range isn’t achieved with 45 degree launch angle

Launching from above ground level h  2 m

The Second Surprise Top class shot putters use a launch angle of about 37 deg – not deg of about 37 deg – not degBecause… They can’t achieve the same launch speed at all launch angles at all launch angles

A Constrained Optimisation Typically degrees is best But is athlete dependent Launch speed falls as angle increases

R max = h tan(2  max ) 21.3 metres (= 70 ft) optimal angle is 42 deg metres (=50 ft) deg 10.7 metres (= 35 ft) deg

Rowing in Numbers How does the speed of the boat depend on the number of rowers? Drag on boat  V 2  wetted surface area of boat  V 2  L 2 Volume of boat  L 3  N – the number of crew Drag  V 2 N 2/3 Crew Power overcoming drag = N  P = V  Drag  V 3 N 2/3 P is the constant power exerted by each (identical rower) V  N 1/9 With a cox V  N 1/3 /[N + 1/3 ] 2/9 if cox is third the weight of a rower

Results of 1980 Olympic coxless races (N = 1, 2, 4) + coxed N = 8 Distance is 2000 metres N Time, T (sec) T = (2000) / V  N V  N 0.11  N 1/9 Compare with a cox N = 2, T = s N = 2, T = s N = 4,T = s N = 4,T = s (predicted if cox = 1/3 of a rower mass) (predicted if cox = 1/3 of a rower mass) Better without one!

Weight Classes Why do we have them in boxing, wrestling, judo and weightlifting? But not in Shot put, hammer, rowing, …..? Why not height classes in high jump and basketball?