Harvard Natural Sciences Lecture Demonstrations

The difference between static and kinetic friction can cause the car to skid when braking.

What it shows:

The automobile is a radio-contolled car (the same one we use for the Reactionary Roadbed). When the front wheels are locked so that they can't turn, the car simply slides to a stop. This is because the static friction of the rolling rear wheels is greater than the kinetic friction of the sliding front wheels. If instead the rear wheels are locked, the car goes into a dramatic tailspin as it comes to a stop. In this case the lesser friction of the rear wheels alllows the rear of the car to slide forward ahead of the front of the car. This is known as a tailspin.

How it works:

To prevent them from rolling and simulating "locking the breaks,", the wheels are secured to the body of the car with duct tape. The car is allowed to roll down a tilted ×ばつ8' sheet of styrofoam. The car rolls straight down without any brakes or if the front wheels are locked. In contrast, locking the rear wheels (but not the front) results in a 180 degree spin as it slides down the hill.

Setting it up:

Use aluminum sheathed styrofoam. One end rests on the floor and the other end on a stool.

Comments:

The demo is a good opportunity to explain anti-locking brake systems found on all modern cars.

If you're feeling pressure to do something fun, consider the vacuum cannon...

See slow-motion footage of the Vacuum Cannon in action (https://youtu.be/gex5Fq5oUtc)

What it Shows

Air rushing into an evacuated tube pushes a ping-pong ball down the tube with such a force and speed that it blasts out the other end at 690 mph and easily passes through three or four cans of soda (they're empty), embedding itself into the fifth can.

We learn two things. First, a modest pressure difference of 1 atmosphere, acting acting over a distance of a couple of meters, can impart a considerable amount of kinetic energy to a small object. Second, the impulse from a small light object traveling at a high speed can impart a surprising amount of damage to the object it strikes—a vivid lesson in kinematics.

One is reminded of the tragic 2003 space shuttle accident in which falling foam at liftoff caused the damage that doomed Columbia. Engineers at NASA thought lightweight foam could not harm the seemingly tough composite panels and privately predicted that the foam would bounce off harmlessly. Then researchers at the Southwest Research Institute in San Antonio shot a 1.7-pound piece of foam at a mock-up shuttle wing at 531 mph, roughly the speed of the chunk of foam that hit the Columbia wing about 81 seconds after liftoff. It was a jaw-dropping demonstration—people's intuitive sense of physics is sometimes way off.

How it Works

The tube is a clear, 8-ft-long, 11⁄2" diameter PVC pipe.^[1] A ping-pong ball^[2] is placed inside and the ends are sealed off with a small piece of thin Mylar^[3] held in place by PVC couplings that simply slip over the Mylar. The tube is evacuated with a vacuum pump:

Once evacuated, the pump is shut off and the Mylar is punctured with an Exacto knife, causing the Mylar to catastrophically rupture. If desired, an infra-red photogate at the other end may be used to measure the velocity of the ball as it blasts out. We have measured it to be 308±20 m/s. The ping-pong ball easily passes through three soda cans.

Setting it Up

C-clamp both the vacuum cannon and the soda can holder to the lecture bench.

References

R. Peterson, B. Pulford, and K. Stein, "The Ping-Pong Cannon: A closer Look," TPT 43(1), 22-25 (January 2005)
E. Ayars and L. Buchholtz, "Analysis of the vacuum cannon," AJP 72, 961-963 (July 2004)

The cannon was designed with the help of the collective wisdom and experience of the tap-L community...many thanks

[1] schedule 40; Grainger #1AAZ7. I.D.=1.610" O.D.=1.900"

[2] 40 mm nominal diameter, measured to be 1.562" or 39.7 mm in diameter and has a mass of 2.6 grams

[3] PrimaCare Foil Mylar Rescue Blanket (1.5 mil thick) sold as Emergency Mylar Thermal Blankets

The tension force in a rope grows exponentially with the number of turns the rope makes around a pole.

What it shows:

Many people have probably observed that, by wrapping a rope around a post, a person can hold in check a much larger force than would ordinarily be possible. In this experiment a flexible thick rope is wound around a horizontal pipe. Due to the interaction of the frictional forces and tension, there can be a considerable difference in tension between the two ends of the rope. In the demonstration, one end of the rope supports a (heavy) load and the other end is held by (small) "holding" force. The demonstration models the principle by which a capstan works.

How it works:

The Euler-Eytelwein equation⁽¹⁾ relates the tension of the two ends of the rope: T₂ = T₁ e^μθ, where T₂ is the tension in the rope due to the load it's supporting, T₁ is the tension necessary to hold the load without slipping, μ is the coefficient of friction between the rope and the pipe, and θ is the total angle (measured in radians) made by all the windings of the rope (one full winding is 2∏ radians). The tension force increases exponentially with the coefficient of friction and the number of turns around the pipe. Note that the diameter of the pipe does not come into play. However, the pipe diameter cannot be too small because a significant amount of force would be lost in bending the rope around the pipe, especially if the rope is a little stiff.

Setting it up:

Secure a 2"-diameter steel pipe to the edge of the lecture bench with two C-clamps, as shown in the photographs. Use the 3/8"-diameter nylon rope. Our 65-lb wrecking ball makes an impressive heavy load. Tie one end of the rope to the ball and wrap several turns of the rope around the pipe. Upon slowly unwrapping the rope, you'll discover that 41⁄2 turns is enough to hold the ball with only the weight of the loose rope supplying the "holding" force. That length of loose rope weighs about 0.11 lb. With θ=9∏, this suggests that μ≈0.23 (not an unreasonable number). With 21⁄2 turns you can easily hold the ball with just your fingers. The 3rd photo shows a 100-lb dumbbell held with just 41⁄2 turns of rope and the fingers supplying the tension. Using μ≈0.23, we estimate a 0.15 lb finger holding force.

Comments:

Exponential relationships are always astounding and this is no exception. Many devices based on belt or wrap friction are used in rappelling, rock climbing (so-called top-roping in which one can hold (belay) a heavy person to prevent a fall), sailing, and rigging of equipment. The ubiquitus V-belt and pulley is another example, cleverly enhanced by the V shape. The relation between tightside and slackside tension for a V-belt is similar:

T₂ = T₁ e^{μθ/sin(β/2)}, where β is the angle of the V in the pulley (note that for β=180 degrees, we are back to the flat belt relationship).

footnotes:

(1) The equation is derived in most mechanical engineering textbooks. For example: R. Becker, Introduction to Theoretical Mechanics, (McGraw-Hill, NY, 1954) p 45-46. L. Goodman and W. Warner, Statics and Dynamics, (Wadsworth, Belmont CA, 1964) pp 308 - 314. Also see D. Morin, Introduction to Classical Mechanics, (Cambridge University Press, NY, 2007) pp 26-27.

A pulse of sound gets "inverted" when reflecting off the open end of a pipe, but does not get inverted when reflecting off the closed end.

What It Shows

Due to an impedance mismatch, a sound pressure pulse traveling down the length of a pipe suffers a reflection when it reaches the end. The nature of that reflection depends on whether or not the pipe is open or closed at the end—an open end results in an inverted reflection and a closed end produces a non-inverted reflection. The initial pulse and its reflection are detected by a microphone and are displayed on an oscilloscope.

How It Works

A short pulse of sound (200-500 μs long) is generated by a loudspeaker driven by a function/pulse generator. The sound pressure pulse enters an 8-ft long (2.45 m), 4-inch diameter, PVC pipe. A microphone placed at the mouth of the pipe detects the pressure pulse as it enters the pipe as well as its reflection from the opposite end. The opposite end is either open or capped closed.

The signal from the microphone is displayed on an oscilloscope. The first photo, below, shows the pulse (reflected from the end cap) arriving about 14 ms after the initial pulse has passed the mike. It is a positive pressure pulse followed by an overshoot. That negative overshoot is the reflection from the open end of the pipe where the microphone is located. You can demonstrate that that is so by moving the microphone further into the pipe. The two reflections superimposed in the display will separate. The second photo clearly shows the inversion of the reflected pressure pulse (the condensation becomes a rarefraction) when the end cap is removed and the pipe is open. Again, the overshoot is due to the reflection from the open end where the microphone is located.

By measuring the time delay between the two pulses (14.2 ms) and the round-trip travel distance (4.9 m), you can also use this opportunity to determine the speed of sound. If you wish, you can increase the time scale on the scope and see multiple reflections as the pulse keeps bouncing back and forth inside the tube (if you do this, push the microphone into the tube as far as it will go).

Setting It Up

The pulse generator (Wavetek model 801), bookshelf speaker, and oscilloscope (Tek TDS 3014B) reside on one cart and the long PVC pipe rests on another.

The pulse generator can drive the loudspeaker directly from its 50-Ω output. The sound is not very loud, but loud enough. If you desire a better signal to noise ratio than what you see in the photographs above, you can amplify the signal with a power amp. Adjust the pulse length to be between 200 and 500 μs long and a repetition rate between 2 and 5 HZ. Trigger the oscilloscope with the TTL output of the function generator. Not only is that going to give you reliable triggering, but you can also show the change in temporal pulse position on the scope display as you move the microphone closer or further from the speaker. (If you measure the actual distance between microphone and speaker, you can then determine the speed of sound.)

The microphone of choice is the Earthworks model M30 used with the Yamaha MG10/2 mixer. Other microphone and mixer combinations will do, but be sure it's not a directional microphone (it won't be sensitive to the initial pulse coming from behind).

Comments:

It's difficult for students to conceptualize the nature of reflections from closed and open pipes—especially open pipes. "How can the sound reflect off something that isn't there?" This is a convincing demonstration of the phenomenon.

A new optical illusion has been added to our geometric optics repertoire: A combination of four converging lenses creates a 3D cloaking effect.

What it shows

The demonstration is a replica of an experiment described by J.S. Choi and J.C. Howell (Univ of Rochester, NY) in their paper entitled Paraxial Ray Optics Cloaking.¹ Using standard lenses, the optical devise produces an image of the background with unity magnification. Objects, such as your hand, appear invisible when inserted into the devise. This is basically an optical illusion and happens because the light rays that make up the imaged background pass around the object as if it isn't there.

How it works:

Four lenses are used to produce a non-inverted image of a grid background with unity magnification. If one looks through the lenses and changes the viewing angle slightly, the image of the grid shifts accordingly and gives the illusion that one is simply looking through the lenses at the background. The lenses are configured so that the light rays that create the backgound image pass through a point on the optical axis. Thus, if an object is placed inside the lens configuration, it will not be visible as long as the object is located in an annulus-shaped space around the optical axis outside the bundle of rays. The cloaking space is depicted in this idealized illustration (not to scale) in which the background is very far away.

Setting it up

The two outside lenses are 20.0 cm focal length achromat doublets. The center two lenses are 7.5 cm focal length achromat doublets. The prescription given by Choi and Howell is that the separation of the first two and last two lenses be equal to the sum of their focal lengths: t₁ = f₁+f₂ = 27.5 cm. The two inside lenses should be separated by an amount equal to t₂ = 2f₂(f₁+f₂)/(f₁-f₂) = 33 cm. This assumes thin lenses and that the background grid is far away, conditions not met in this set-up. The background grid is approximately 2 meters away. The 20.0 cm lens is 1.05 cm thick and the 7.5 cm lens is 2.3 cm thick. Thus, adjustments to t₁and t₂ must be made to secure an image having unity magnification. For a background distance of 2 meters, t₁ = 25.6 cm and t₂ = 33.6 cm work well. These distances are between the closest surfaces of the lenses.

To avoid pincushion distortion, it is important to orient the lenses properly by following these general guidelines: when collimating a point source, the first air-to-glass interface should have the greater radius of curvature (e.g. the flatter side faces the point source). Conversely, when focusing a collimated beam, the air-to-glass interface with the smaller radius of curvature should face the incoming light.

When looking into the last lens, we wish the grid background to fill the lens, so one must not be too close to keep the exit pupil small. The other issue is depth-of-field which is influenced not only by the camera's lens focal length and aperture, but also by the distance from camera to object. Again, a simple rule: The closer you are to the object you are focusing on, the fuzzier everything else will be — or, put another way, the less your depth-of-field will be. Since we need to be far away to satisfy the exit pupil constraint, but want the image to be large, we need to use a telephoto lens. We gain depth-of-field by going farther away and lose depth-of-field by using a telephoto lens. A satisfactory compromise is the following.

To show the cloaking effect to the audience, set up a video camera approximately 5 meters away from the lenses — this ensures that the grid fills the lens. Use the 150 mm Cannon zoom lens and adjust it for a small aperture (large f-stop) to gain depth-of-field. Rely on the camera's auto-gain. Take care to align the camera along the optical axis of the cloaking lenses. Finally, to achieve the best overall focus, focus the camera lens a little beyond the hand you intend to image and take advantage of yet another rule-of-thumb: 1/3 of the depth-of-field is in front of the object you're focusing on and the other 2/3 is behind. So by focusing a little beyond the object of interest, the object itself will remain in focus and you have a better chance of bringing the background into focus.

Before demonstrating the cloaking, it might be worthwhile to walk in front of the grid in the background to convince the audience that the image in the lens is truly an image of the background.

Comments:

The illustrtation above depicts a cloaking region outside a 2 cm diameter bundle of light rays between the two center lenses. The last photo shows a test of that prediction; a 2 cm diameter tube is not visible when looking into the device (all the light rays pass through the inside of the tube). Cloaking aside, this demonstration can be used as an interesting example of ray tracing (a little more interesting than telescope ray tracing).

Notes:

J.S. Choi and J.C. Howell, "Paraxial Ray Optics Cloaking," Optics Express 22(24), pp 29465-29478 (2014).

Water moving around a cylindrical pendulum makes the cylinder swing at the vortex shedding frequency.

What it Shows:

When fluid flows around a cylindrical object, there is a range of flow velocities for which a von Karman vortex street is formed. The shedding of these vortices imparts a periodic force on the object. The force is quite small and not enough to accelerate the object to any significant amount, especially if the object is relatively massive. If the situation is such that the object can vibrate about a fixed position, we have the possibility of simple harmonic motion; and if the frequency of the periodic driving force matches the natural frequency of the oscillation, then resonance obtains and the amplitude of the oscillations can be significant.

The object is a cylinder attached to a physical pendulum whose frequency of oscillation is adjustable. The end of the cylinder is submerged in flowing water. When the frequency of the pendulum is adjusted to match the frequency of vortex shedding, the cylinder swings transverse to the direction of flow with a peak-to-peak amplitude of 2 to 3 centimeters.

How it Works:

Consider the flow of fluid around a smooth cylinder. For velocities exceeding laminar flow, the inertia of the fluid starts to become significant and, as the fluid stream passes the topmost part of the cylinder, it is unable to negotiate the rear half of the cylinder. Hence the fluid tends to separate from the top surface and peel off in a clockwise motion as it approaches the rear end of the cylinder, ending up as a shed vortex (it will peel off in a CCW motion from the bottom surface). For a given velocity of flow, this model suggests the vortex formation time will be proportional to the distance around the cylinder (or its diameter) and thus the frequency of vortex formation will be inversely proportional to the diameter. Furthermore, if the flow velocity increases, the frequency of vortex formation will likewise increase, leading to a direct relation between the two. This is what Strouhal found empirically in 1878 (ref 1).

The proportionality constant is called the Strouhal number and turns out to be a function of the Reynolds number. For that reason it is now known as the Strouhal-Reynolds number. It is very nearly equal to 0.2 for a large range of Reynolds numbers.

Strouhal's empirical formula for the frequency of vortex shedding is f = (1/5)(v/d)

where v is the velocity of flow and d is the diameter of the cylinder (a full and modern treatment of vortex shedding can be found in references 2-4). Thus, for water flowing past a 2.54 cm diameter cylinder at a velocity of 10 cm/s, one can expect a frequency of 0.8 s^-1.

The cylinder is only partially immersed in the water and does not rest on the bottom of the channel. Rather, it is supported by a threaded rod and is part of a physical pendulum. The pendulum's frequency of oscillation is adjustable and can be made to closely match the vortex shedding frequency.

One can see the vortex shedding by shadow projection. Similar to a ripple tank, if a point light source is set up over the tank, then shadow patterns of the water's motion can be seen on the bottom of the water channel. Thus, not only can one demonstrate resonance oscillations of the cylinder, but one can also see the relative motion of the vortex eddies and the cylinder—they move in opposite directions.

As with most fluid dynamics phenomena, the physics of vortex-induced vibrations is quite rich and very complicated—even in two dimensions. For example, various vortex wake modes are possible with fluid flow jumping from one mode to another. Additionally, the sideways motion of the object in the fluid affects the formation and/or shedding of vortices and can have a positive or negative feedback effect. Also, depending on the phase between object and fluid motion as well as their frequency difference, a lock-in or synchronization effect may or may not occur. Furthermore, the ratio of the object-to-fluid mass as well as damping forces have a significant effect, leading to parameters described in the literature as effective mass, critical mass, high-mass ratio, etc. Reference 4 is an excellent resource for those that wish to go deeper into the subject matter—and it can be quite deep indeed!

Setting it Up:

A 55-gallon aquarium has been converted into a flow tank. A barrier divides the aquarium into two halves. Water is pumped from one half of the aquarium to the other. When the water spills over the barrier, it returns to the other end via a wide flow channel. The flow rate in the channel can be adjusted by enlisting one or two pumps, as well as by changing the amount of water in the tank. A convenient flow rate for this demonstration is about 10 cm/second. The flow can be readily measured by placing a plastic meter stick (with decimeter markings) in the channel and letting a ping-pong ball float downstream, noting how long it takes to pass the markings.

The object is a 1" dia. ×ばつ 2" long aluminum cylinder constrained to move transverse to the flow. The cylinder is attached to a 15" long, 10-32 threaded rod which, in turn, is suspended from a knife-edge. The combination is a physical pendulum whose period of oscillation can be changed by its length, or screwing machine nuts onto the threaded rod. The pendulum can be suspended over any part of the water flow channel, allowing one to choose a location where the flow rate is optimal for what one wants to demonstrate.

Comments:

Vortex-induced vibrations are important in that they can have a strong influence in countless situations ranging from tethered structures in the ocean, pipes bringing oil from the ocean floor to the surface, aeolian harps, tall buildings, and chimneys, to name but a few. For example, the tallest building in the world, the Burj Khalifa in Dubai, UAE, incorporates a variation in cross section with height to help ensure that vortices are not shed coherently along the entire height of the building (reference 5). The Tacoma Narrows Bridge collapse is discussed in practically every introductory physics course as a dramatic example of resonance. Although vortex shedding is often cited as being the culprit, Billah & Scanlan say that this is oversimplified physics and posit that the real culprit was flutter—a non-linear phenomenon in which the motion of the bridge was the source of self induced periodic impulses (reference 6).

References:

1. V. Strouhal, "Ueber eine besondere Art der Tonerregung," Ann. Physik. Chem (Leipzig) 5(10), 216-251 (1878).

2. C.H.K. Williamson, "Vortex Dynamics in the Cylinder Wake," Annual Review of Fluid Mechanics 28, 477-539 (1996).

3. U. Fey, M. König, and H. Eckelman, "A New Strouhal-Reynolds-number relationship for the circular cylinder in the range 47<Re&l×ばつ10⁵," Physics Fluids 10(7), 1547-1549 (1998).

4. C.H.K. Williamson and R. Govardhan, "Vortex-Induced Vibrations," Annual Review of Fluid Mechanics 36, 413-445 (2004).

5. P.A. Irwin, "Vortices and tall buildings: A recipe for resonance," Physics Today 63(9), 68-69 (2010).

6. K.Y. Billah and R.H. Scanlan, "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks," Am. J. Phys. 59(2), 118-124 (1991).