TRANSVERSE VIBRATIONS 637 instructive experiments may be performed with this ap- paratus with an average error between 1 and 2 per cent. Both of these pieces are very rugged in construction and yet capable of yielding precise results in the hands of the careful student. They can be completely dismantled and stored in small space which is a factor to be appreciated in the average laboratory. They can be readily constructed at very low cost. Accessories such as support rods, clamps, and pulleys are of the type commonly used in the laboratory. VELOCITY OF TRANSVERSE VIBRATIONS IN STRINGS. By F. M. DENTON, The University of New Mexico, Albuquerque, N. Mex. Many text-books of Physics state the formula V=VF/nz without proof, remarking that the proof may be found in more advanced works. A few books, intended for University use, give the ingenious proof of Professor Tail in which the vibrating string is thought of as passing through a bent tube. I have worked out the following proof which, though brief, non-mathematical and, I believe, sound, appears to be unknown. {Let AB be a uniform flexi- __ _ ble string stretched by a ""^ force of F dynes between end posts A and B. Let m be the mass of the string per centimeter of its length. Experiment shows that any small transverse distortion imparted to the string travels along it with uniform velocity. It is required to show that this velocity is V==VF/m. Proof:Let the string be distorted at A into the shape ACB by being pressed up against the post for a short distance AC. On releasing the pressure at C the kink moves with uniform velocity V along the string towards B. We may regard the force F as causing this motion, and using Newton’s definition of Force, F=AM, may write: p ^ (Velocity given to a mass\ /Mass to which that\ \ in one second ) \ velocity is given ) It is the ^kink" to which velocity is being given and the mass constituting the kink suffers continual renewal at the rate (mV) grams per second. Thus the mass to which velocity V is imparted in each second is mV, and in the formula F = AM we have A = V and M=?nV, whence F==V(mV)=??zV2 and V==VF/W.

VELOCITY OF TRANSVERSE VIBRATIONS IN STRINGS

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Page 1: VELOCITY OF TRANSVERSE VIBRATIONS IN STRINGS

TRANSVERSE VIBRATIONS 637

instructive experiments may be performed with this ap-paratus with an average error between 1 and 2 per cent.Both of these pieces are very rugged in construction and

yet capable of yielding precise results in the hands of thecareful student. They can be completely dismantled andstored in small space which is a factor to be appreciatedin the average laboratory. They can be readily constructedat very low cost. Accessories such as support rods, clamps,and pulleys are of the type commonly used in the laboratory.

VELOCITY OF TRANSVERSE VIBRATIONS IN STRINGS.By F. M. DENTON,

The University of New Mexico, Albuquerque, N. Mex.Many text-books of Physics state the formula V=VF/nz

without proof, remarking that the proof may be found in moreadvanced works. A few books, intended for University use,give the ingenious proof of Professor Tail in which the vibratingstring is thought of as passing through a bent tube.

I have worked out the following proof which, though brief,non-mathematical and, I believe, sound, appears to be unknown.

{Let AB be a uniform flexi-__

ble string stretched by a

""^ force of F dynes betweenend posts A and B. Let m

be the mass of the string per centimeter of its length.Experiment shows that any small transverse distortion im-

parted to the string travels along it with uniform velocity.It is required to show that this velocity is V==VF/m.Proof:�Let the string be distorted at A into the shape ACB

by being pressed up against the post for a short distance AC.On releasing the pressure at C the kink moves with uniformvelocity V along the string towards B.We may regard the force F as causing this motion, and using

Newton’s definition of Force, F=AM, may write:

p ^(Velocity given to a mass\ � /Mass to which that\\ in one second ) \ velocity is given )

It is the ^kink" to which velocity is being given and the massconstituting the kink suffers continual renewal at the rate (mV)grams per second. Thus the mass to which velocity V is impartedin each second is mV, and in the formula F =AM we have A =Vand M=?nV, whence F==V(mV)=??zV2 and V==VF/W.