A number of years ago, I tutored a teammate who was taking a first semester physics class. The following was the final exam I set for her, in preparation for her real final (which she aced, no doubt due to the superior (but modest) tutoring she received). As it is now finals time once again, I offer up this exam as a diversion for the readers of rec.sport.fencing. I hope that everyone will take it in the spirit intended, and not find offense at the use of the names of real fencers.

Ann McBain Ezzell
USFA Secretary
amcbainezzell@alum.mit.edu

### FENCING PHYSICS FINAL 27 April 1989 - updated 11 December 1994

[Disclaimer: All similarities between real fencers and characters in this exam are purely intentional and completely without malice.]

Instructions: Answer all questions. Be sure to show your work (including, where appropriate, free body diagrams). Don't screw up the math. Except as noted, you may neglect air resistance and friction.

1. A 2m tall Italian epee fencer loses his last repechage bout by being pushed off the end of the strip (standard 14m length). He knocks his mask straight into the air and simultaneously kicks his reel, which had been positioned at the end line, towards the other end of the strip. The mask just touches the 6m high gym ceiling before starting its downward descent. The fencer sees the reel barely clear the head of the 1.75m tall referee, who is standing in front of the scoring table recording the result. Just as he is knocked unconscious by his plummeting mask, he sees the reel land at the feet of the chairman of the Directoire Technique, who had been watching the bout from the far end of the strip.
a) How long does it take the reel to reach the ground?
b) Calculate the initial magnitude and direction of the reel.
c) How long will it take after the fencer regains consciousness until he is expelled from the competition?

2. Claus Block is bouncing up and down two meters from his opponent's end of the strip. His reel has slipped to 1.5 meters in front of his end line, and the reel cord is attached to his waist 1m above the ground. The mass of the exposed portion of the reel cord is 500g. A standing wave of three loops is being produced in the reel cord.
a) If Claus hits the ground 10 times per second (it's the finals), what is the tension in the reel cord?
b) Assume that the tension in the reel remains as calculated in part (a). Where would Claus have to stand and bounce, relative to his initial position, to produce a standing wave with only two loops?

3. A brand-new Uhlmann epee point is constructed such that the total travel is exactly 1.5mm, and it just passes the 0.5mm shim test. When a test weight of 750g is gently dropped onto the tip, the scoring machine light comes on. After the machine resets, the light remains off. However, any further depression of the tip causes the light to come on.
a) Calculate the spring constant (k) for the point spring (you may neglect the mass of the tip).

A Russian point is dimensionally identical to the Uhlmann point, but friction in the point produces an extra 1N of resistive force. Since its owner cannot readily fix his weapons, the point spring must be strong enough to lift 2kg (as above), to ensure that his weapons will never fail on the strip.
b) Calculate the spring constant (k') required for this point spring.

The two weapons are fixed horizontally, tip to tip, then the retaining screws are removed to allow free movement of the tips. The two tips are displaced 0.5mm from their equilibrium position and then released.
c) Calculate the frequency of the resultant SHM. (Assume that the mass of 1 tip is 1g and that both tips move together.)

4. Yuri Rabinovich and his long-lost identical twin brother Pavel (each with mass 65 kg) are fencing sabre. With weapon arms half-extended, they launch simultaneous fleche attacks and lock bell guards in mid air. Just before impact, each is traveling at a speed of 5m/s. When their bodies pass, the centers of mass are 1m apart. The bell guards remain locked and their arms extend to full length (adding 1m to the distance between the centers of mass).
a) What is the angular momentum of the resultant tangle immediately following the collision?
b) When the arms are extended, what is their rotational frequency in revolutions per second?

5. In the midst of a team free-for-all, Frank MacKenzie (mass 90 kg) picks up Lara Tomasso (mass 65 kg) and attempts to hold her at arm's length (this would put her center of mass 1m from his center of mass). Frank has enough upper body strength to support a mass of 25kg in this manner.
a) Frank, being an engineer, starts to spin. After accelerating for 5 seconds at a constant rate, his arms are forming an angle of 5 degrees with the horizontal. Find his angular acceleration.
b) At this same acceleration, how long will it take until his arms are 2.5 degrees from the horizontal?
c) How long before his arms are perfectly horizontal?
d) How long will it be before Lara throws up?

6. a) The maximum length of a foil blade from tip to bell guard is 90cm. Taking the pivot point to be at the bell guard, calculate the torque produced by a force of 20N applied perpendicular to the blade at the following distances from the tip of the foil:
1) 85 cm
2) 50 cm
3) 10 cm
b) If you are able to produce a torque of 10Nm around your own bell guard, calculate the resultant torque around your opponent's bell guard if your blades are pushing at right angles to each other and the intersection point is 10 cm from your bell guard and 45 cm from your opponent's bell guard.

7. Assume that a foil blade (not including the tang) is a uniform rod of length 90cm, diameter 5mm and mass 150g. Your opponent beats your blade sharply 40cm from the tip, breaking the blade. She then immediately does a circle disengage and hits the free end of the broken piece with a 20N force for .01 second. Calculate the rotational frequency of the broken piece of blade as it spins off end over end. (The rotational inertia, I, for a uniform rod of length L is 1/12mL^2, with the axis of rotation at the center of the length of the rod.)

8. A golf ball of mass 46g hangs from an ideal string 1m in length. A diligent epee fencer practicing point control strikes the ball with sufficient force to cause the string to form an angle of 15 degrees with the vertical.
a) What is the velocity of the golf ball immediately following impact?
b) How long after impact will it take the ball to reach the point where it is closest to the fencer?

9. Peter Westbrook (mass 70kg), having temporarily forgotten the end-of-strip rules in the heat of the finals, retreats rapidly off the end of a raised piste 0.30m high. Fortunately for Peter, the regulation run-off incline of 2m has been included.

Unfortunately, he trips and ends up rolling ignominiously the entire length of the incline. Assume that Peter's body approximates a cylinder of 50cm diameter as he rolls without slipping down the incline. Further assume that he is not moving horizontally when he hits the top of the ramp.
a) If Peter is making 2 revolutions per second when he reaches the bottom of the incline, what was his angular momentum when he hit the top of the incline?
b) What torque is required to stop Peter's rolling at the bottom of the ramp in 1 second?

10. Isabelle Hamori shrieks in the heat of combat at 13,000 Hz. The gym is set up with pairs of two meter wide strips three meters apart, with six meters between each pair.
a) If Isabelle is fencing in the middle of strip 11 at the far end of the gym from the Bout Committee table, which is 10 meters from strip 1, how much longer will it take the Chairman of the Bout Committee to wince than Isabelle's referee, who is standing halfway between strips 10 and 11? (This is at the 1988 Chicago Nationals, where the ambient temperature is approximately 40 degrees C. Take the speed of sound in air at 20 degrees C to be 340 m/s and remember that the speed of sound is related to the square root of the temperature in degrees Kelvin.)
b) Isabelle's opponent is MJ O'Neill, also known for her dulcet tones on the strip. MJ screeches while fleching at Isabelle, who attempts to retreat, at full voice. The referee, who is maintaining his original position relative to Isabelle, notices that the combined shrieking is producing 2 beats per second. If MJ screeches at 12,980 Hz, what is her minimum velocity relative to Isabelle?