The Physics of Tennis

by Blake Sharin

Introduction

• The Sweet Spots• Vibration Node• Centre of Percussion• The Dead Spot• Coefficient of

Restitution

The Sweet Spot

• Two Sweet Spots– Vibration Node– COP

• Force Transmitted to hand– Motions of the handle

• Rotation• Translation• Vibration

• Maximum Speed of Ball

Vibration Node

• Behaves like a Uniform Beam

• Fundamental Mode• Frequency

– 100Hz for flexible frame– 140Hz for stiff frame

• Two nodes– Near the center of strings– Near the handle

Vibration Node (Contd.)

• Frequency is 2.75 times the fundamental frequency

• Not excited with any significant amplitude– Duration, T=5ms– Peaks at Zero at

f=1.5/T=300Hz• Close to second mode

frequency

Centre of Percussion

• Known as the impact point

• Conjugate points– Impact near tip

• Axis of rotation is about ½ way between the end of the handle and the CM

– Impact near throat• Axis of rotation is

beyond the end of the handle

Power Swinging

• More power when you swing at the ball near throat of racquet.

• When ball hits near the throat of racquet, the ball will have more ball speed.

• Heavier the racquet, more speed applied to the ball

• Many types of racquets

Coefficient of Restitution

• Main factor in the formation of a tennis racquet

• Ratio of rebound height to incident height of the ball

• COR varies when the ball bounces off a certain spot on the racket

• Maximum power is when COR is greatest• COR=square root of Rebound height/initial

height

COR Data

• Tip of Head• 1st set: hd = 20 cm, hr = 12 cm• 2nd set: hd = 40 cm, hr = 24 cm• 3rd set: hd = 60 cm, hr = 36 cm• 4th set: hd = 80 cm, hr = 48 cm• 5th set: hd = 100 cm, hr = 60 cm

• COR• COR (center of mass): .74162• COR (near the throat): .63246• COR (tip of the head): .7746

• Center of Mass• 1st set: hd = 20 cm, hr = 11 cm• 2nd set: hd = 40 cm, hr = 22 cm• 3rd set: hd = 60 cm, hr = 33 cm• 4th set: hd = 80 cm, hr = 44 cm• 5th set: hd = 100 cm, hr = 55 cm

• Near Throat• 1st set: hd = 20 cm, hr = 8 cm• 2nd set: hd = 40 cm, hr = 16 cm• 3rd set: hd = 60 cm, hr = 24 cm• 4th set: hd = 80 cm, hr = 32 cm• 5th set: hd = 100 cm, hr = 40 cm

hr = rebound height, hd = Initial height

COR Data Analysis

• Rebound height remains constant

• Best spot to hit the ball– Near tip– Center of mass– Disregard throat

The Dead Spot

• Spot near the tip where ball doesn’t bounce at all– All of the energy is given to

the racquet– Racquet doesn’t give any

energy back to the ball• Effective mass of the racquet

at that point is equal to the ball

• Effective mass is F=ma, therefore m=F/a

• When serving, the best place to hit the ball is at the dead spot

• When returning, dead spot is the worst

Strings on the Racquet

• Act as a medium– Absorb much of the ball’s kinetic energy– Returns some of that energy back to the ball

• Tighter strings produce slower ball speeds

• Loose strings– Leads to slightly higher rebound velocities

• More power

Conclusion

• Performance of the racquet– Strings– Size of the head

• Larger the head = more speed applied to ball

– Game depends on two main things

• How well you hit with the racquet

• The “spot”

Bibliography

• Brody, H. (1979) Physics of the tennis racket. American Journal of Physics 47, 482-487

• Brody, H. (1981) Physics of the tennis racket II: The sweet spot. American Journal of Physics, 49, 816-819.

• Brody, H. (1995) How would a physicist design a tennis racket?. Physics Today, 48, 26-31

• Pallis, Jani. "The Flight of the Tennis Ball" www.Tennisserver.com