17.4 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
vCalc Companion Formulas | |
vCalc Formulary | 17.4 Summary |
`T=2pisqrt(m/k)` | Time Period |
periodic motion — motion that repeats itself over and over
period — the time required for one cycle of a periodic motion
frequency — the number of cycles per second, the inverse of the period
amplitude — the amount of vibration, often measured from the center to one side; may have different units depending on the nature of the vibration
simple harmonic motion — motion whose `x?t` graph is a sine wave
`T` — period
`f` — frequency
`A` — amplitude
`k` — the slope of the graph of `F` versus `x`, where `F` is the total force acting on an object and `x` is the object's position; for a spring, this is known as the spring constant.
`nu` — The Greek letter `nu`, nu, is used in many books for frequency.
`?` — The Greek letter `?`, omega, is often used as an abbreviation for `2pif`.
Periodic motion is common in the world around us because of conservation laws. An important example is one-dimensional motion in which the only two forms of energy involved are potential and kinetic; in such a situation, conservation of energy requires that an object repeat its motion, because otherwise when it came back to the same point, it would have to have a different kinetic energy and therefore a different total energy.
Not only are periodic vibrations very common, but small-amplitude vibrations are always sinusoidal as well. That is, the `x?t` graph is a sine wave. This is because the graph of force versus position will always look like a straight line on a sufficiently small scale. This type of vibration is called simple harmonic motion. In simple harmonic motion, the period is independent of the amplitude, and is given by
k / Problem 4.
1. Find an equation for the frequency of simple harmonic motion in terms of `k` and `m`. (answer check available at lightandmatter.com)
2. Many single-celled organisms propel themselves through water with long tails, which they wiggle back and forth. (The most obvious example is the sperm cell.) The frequency of the tail's vibration is typically about 10-15 Hz. To what range of periods does this range of frequencies correspond?
3. (a) Pendulum 2 has a string twice as long as pendulum 1. If we define `x` as the distance traveled by the bob along a circle away from the bottom, how does the `k` of pendulum 2 compare with the `k` of pendulum 1? Give a numerical ratio. [Hint: the total force on the bob is the same if the angles away from the bottom are the same, but equal angles do not correspond to equal values of `x`.]
(b) Based on your answer from part (a), how does the period of pendulum 2 compare with the period of pendulum 1? Give a numerical ratio.
4. A pneumatic spring consists of a piston riding on top of the air in a cylinder. The upward force of the air on the piston is given by `F_(air)`=`ax`^?1.4, where `a` is a constant with funny units of `N*m^1.4`. For simplicity, assume the air only supports the weight, `F_W`, of the piston itself, although in practice this device is used to support some other object. The equilibrium position, `x_0`, is where `F_W` equals `?F_(air)`. (Note that in the main text I have assumed the equilibrium position to be at `x=0`, but that is not the natural choice here.) Assume friction is negligible, and consider a case where the amplitude of the vibrations is very small. Let `a`=`1.0 N`?`m^1.4`, `x_0`=1.00 `m`, and `F_W`=?`1.00 N`. The piston is released from `x`=`1.01 m`.Draw a neat, accurate graph of the total force, `F`, as a function of `x`, on graph paper, covering the range from `x=0.98 m` to `1.02 m`. Over this small range, you will find that the force is very nearly proportional to `x?x_0`. Approximate the curve with a straight line, find its slope, and derive the approximate period of oscillation. (answer check available at lightandmatter.com)
5. Consider the same pneumatic piston described in problem 4, but now imagine that the oscillations are not small. Sketch a graph of the total force on the piston as it would appear over this wider range of motion. For a wider range of motion, explain why the vibration of the piston about equilibrium is not simple harmonic motion, and sketch a graph of `x` vs `t`, showing roughly how the curve is different from a sine wave. [Hint: Acceleration corresponds to the curvature of the `x?t` graph, so if the force is greater, the graph should curve around more quickly.]
l / Problem 7.
6. Archimedes' principle states that an object partly or wholly immersed in fluid experiences a buoyant force equal to the weight of the fluid it displaces. For instance, if a boat is floating in water, the upward pressure of the water (vector sum of all the forces of the water pressing inward and upward on every square inch of its hull) must be equal to the weight of the water displaced, because if the boat was instantly removed and the hole in the water filled back in, the force of the surrounding water would be just the right amount to hold up this new “chunk” of water. (a) Show that a cube of mass `m` with edges of length `b` floating upright (not tilted) in a fluid of density `rho` will have a draft (depth to which it sinks below the waterline) `h` given at equilibrium by `h_0=m`/`b^2?`. (b) Find the total force on the cube when its draft is `h`, and verify that plugging in `h?h_0` gives a total force of zero. (c) Find the cube's period of oscillation as it bobs up and down in the water, and show that can be expressed in terms of and `g` only. (answer check available at lightandmatter.com)
7. The figure shows a see-saw with two springs at Codornices Park in Berkeley, California. Each spring has spring constant `k`, and a kid of mass `m` sits on each seat. (a) Find the period of vibration in terms of the variables `k`, `m`, `a`, and `b`. (b) Discuss the special case where `a=b`, rather than `a>b` as in the real see-saw. (c) Show that your answer to part a also makes sense in the case of `b=0`. (answer check available at lightandmatter.com)
8. Show that the equation `T=2pisqrt(m/k)` has units that make sense.
9. A hot scientific question of the 18th century was the shape of the earth: whether its radius was greater at the equator than at the poles, or the other way around. One method used to attack this question was to measure gravity accurately in different locations on the earth using pendula. If the highest and lowest latitudes accessible to explorers were `0` and `70` degrees, then the the strength of gravity would in reality be observed to vary over a range from about `9.780` to `9.826 m`/`s^2`. This change, amounting to `0.046 m`/`s^2`, is greater than the `0.022 m`/`s^2` effect to be expected if the earth had been spherical. The greater effect occurs because the equator feels a reduction due not just to the acceleration of the spinning earth out from under it, but also to the greater radius of the earth at the equator. What is the accuracy with which the period of a one-second pendulum would have to be measured in order to prove that the earth was not a sphere, and that it bulged at the equator?
Equipment:
Place the cart on the air track and attach springs so that it can vibrate.
1. Test whether the period of vibration depends on amplitude. Try at least one moderate amplitude, for which the springs do not go slack, at least one amplitude that is large enough so that they do go slack, and one amplitude that's the very smallest you can possibly observe.
2. Try a cart with a different mass. Does the period change by the expected factor, based on the equation `T=2pisqrt(m/k)`?
3. Use a spring scale to pull the cart away from equilibrium, and make a graph of force versus position. Is it linear? If so, what is its slope?
4. Test the equation `T=2pisqrt(m/k)` numerically.
17.4 Summary by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.