Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r, say , and some power of v, say v". Determine the values of n and m and write the simplest form of an equation for the acceleration. SOLUTION Write an expression for a with a dimensionless constant of proportionality k. (Use the following as necessary: v, r, k, n, and m.) Substitute the dimensions of a, r, and v. (Use the following as necessary: n, m, L, and T.) *-(+)" - [ Equate the exponents of L andT so that the dimensional equation is balanced: n+ m =1 and m = 2 Solve the two equations for n: Write the acceleration expression. (Use the following s necessary: v, r, and k.) From our discussion on uniform circular motion, we show that k = 1 if a consistent set of units is used. The constant k would not equal 1 if, for example, v were in km/h and you wanted a in m/s?. EXERCISE In physics, energy E carries dimensions of mass times length squared, divided by time squared. Use dimensional analysis to derive a relationship for energy in terms of mass m and speed v, up to a constant of proportionality. Hint O E = kmv2 O E = kmv O E = k OE= km* O E = km?v
Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r, say , and some power of v, say v". Determine the values of n and m and write the simplest form of an equation for the acceleration. SOLUTION Write an expression for a with a dimensionless constant of proportionality k. (Use the following as necessary: v, r, k, n, and m.) Substitute the dimensions of a, r, and v. (Use the following as necessary: n, m, L, and T.) *-(+)" - [ Equate the exponents of L andT so that the dimensional equation is balanced: n+ m =1 and m = 2 Solve the two equations for n: Write the acceleration expression. (Use the following s necessary: v, r, and k.) From our discussion on uniform circular motion, we show that k = 1 if a consistent set of units is used. The constant k would not equal 1 if, for example, v were in km/h and you wanted a in m/s?. EXERCISE In physics, energy E carries dimensions of mass times length squared, divided by time squared. Use dimensional analysis to derive a relationship for energy in terms of mass m and speed v, up to a constant of proportionality. Hint O E = kmv2 O E = kmv O E = k OE= km* O E = km?v
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
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Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text:Analysis of a Power Law
Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power ofr, say , and some power of v, say vm. Determine the values of n and m and write the simplest form of an equation for the acceleration.
SOLUTION
Write an expression for a with a dimensionless constant of proportionality k. (Use the following as necessary: v, r, k, n, and m.)
a =
Substitute the dimensions of a, r, and v. (Use the following as necessary: n, m, L, and T.)
=
- „(), -
Equate the exponents of L and T so that the dimensional equation is balanced:
n + m =1
and m = 2
Solve the two equations for n:
Write the acceleration expression. (Use the following as necessary: v, r, and k.)
a =
From our discussion on uniform circular motion, we show that k = 1 if a consistent set of units is used. The constant k would not equal 1 if, for example, were in km/h and you wanted a in m/s?.
EXERCISE
In physics, energy E carries dimensions of mass times length squared, divided by time squared. Use dimensional analysis to derive a relationship for energy in terms of mass m and speed v, up to a constant of proportionality.
Hint
O E = kmv2
O E = kmv
O E = km
v2
O E = km-
O E = km2v
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