You shouldn’t use a heavy racquet if you have a light arm. You can see why in these

300 fps double pendulum movies. An adult forearm  weighs about 1.5 kg. I used a

406 g forearm here. A 52 g “racquet”  is then OK but a 272 g racquet is too heavy for

this very light arm. Note the double pendulum legs in the background!




Film of a 100 mph kick serve shows double pendulum action of the forearm and the racquet.

When the upper arm slows down, it exerts a torque on the forearm and causes the forearm to

rotate faster. When the forearm slows down, it exerts a torque on the racquet and causes the

racquet to rotate faster. That way, rotational energy is transferred from one segment to the

next in sequence. Does the racquet rotate the wrist or does the wrist rotate the racquet? 

Double pendulum calculations show that in golf, the club rotates the wrist after the wrist

relaxes. In tennis, the player needs to use the wrist to rotate the racquet.






Sideways stretch of tennis strings can increase the outgoing ball spin, provided

the strings return to their original position before the ball bounces. The ball

was incident at 25 m/s.        String1    String2    String3      String4





Here are two nice topspin backhand shots showing how topspin

is generated if the racquet head is rising (and moving forward)

when it strikes the ball.


      Shot1       Shot2  (rising ball problem)




The force on the arm can be reduced by using softer strings or hitting the ball at the

sweet spot. The result here shows that the centre of percussion is NOT a sweet spot

since the axis of rotation is through the wrist regardless of the impact point.

See also impact at throat where the axis is close to the butt end of the handle.





A ball impacting near the edge of the frame causes the racquet to

rotate about its long axis. What effect does that have on the ball?

Watch in slow motion.    Movie1     Movie2      Movie3

What if the rotation angle is much larger?  See Movie 4



The explanation of Movie 4 is probably that the racquet rotates by only 1 or 2 degrees during the first half of the collision so the ball compresses in a direction perpendicular to the strings. It then expands back along the same path, the resultant force, R, on the ball being perpendicular to the original position of the string plane (see drawing). The force on the ball has a component N perpendicular to the strings and a static friction force component F parallel to the strings. That also explains the rotation of the ball after it bounces.



Compare MovieA with MovieB. The incident speed (22 m/s) spin (3900 rpm)

and angle (40 deg) is the same but the bounce is quite different since the impact

point is different. If the racquet rotates then the ACOR is low ie low perpendicular

bounce speed. That happens with most tennis shots, even with the pros. Note change in

spin direction. That is how topspin is generated, viewed in the racquet reference frame.





The Tennis Racquet Theorem.  Racquets flip over when

tossed in the air.  So do books and a packet of cornflakes. 

But tennis racquets work best. Watch in slow motion.




Low speed tennis ball bounce

filmed at 1000 frames/s



High speed tennis ball filmed at 3000 frames/sec, bouncing on a clay court

to measure the court “speed”.   Bounce 2 is on a grass court.




Bounce off a moving surface, to simulate the bounce off a racquet.

The spin changes in       Bounce1       Bounce2      Bounce3




Measuring the friction force F and the normal reaction

force N on a ball bouncing off a glass block.

Ball incident without spin         Ball incident with spin





How to measure the balance point

of a tennis racquet (ie the location

of the centre of mass).




How to measure the swing weight of a tennis racquet

(ie the moment of inertia about an axis 10 cm from

the end of the handle). It’s easier to first measure

about a different axis.




How to measure the ACOR of a tennis racquet.

ACOR = (outgoing ball speed/incoming ball speed)

when the racquet is at rest. The ACOR is lowest at

the tip and largest in the throat region.



Real racquets don’t vibrate enough to see the vibrations.

Here is a vibrating cardboard racquet with two strings.

The node point in the “middle” of the strings can be

seen, as well as the node points at 2 and 10 o’clock

(towards bottom end of racquet).