1. The Physics of Tennis
by Blake Sharin

2. Introduction The Sweet Spots Vibration Node Centre of Percussion
The Dead Spot
Coefficient of Restitution

3. The Sweet Spot Two Sweet Spots Force Transmitted to hand
Vibration Node
COP
Force Transmitted to hand
Motions of the handle
Rotation
Translation
Vibration
Maximum Speed of Ball

4. 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

5. 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

6. 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

7. 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

8. 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

9. COR Data Tip of Head Center of Mass 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):
COR (near the throat):
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

10. COR Data Analysis Rebound height remains constant
Best spot to hit the ball
Near tip
Center of mass
Disregard throat

11. 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

12. 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

13. 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”

14. Bibliography
Brody, H. (1979) Physics of the tennis racket. American Journal of Physics 47,
Brody, H. (1981) Physics of the tennis racket II: The sweet spot. American Journal of Physics, 49,
Brody, H. (1995) How would a physicist design a tennis racket?. Physics Today, 48, 26-31
Pallis, Jani. "The Flight of the Tennis Ball"