Part A:
Problem 1
It is inconvenient to write out numbers such as 1,530,000,000 and 0.000087. These same numbers can be written better in scientific notation as 1.53×109 and 8.7×10-5, respectively. Write the following numbers in scientific notation:
a) 123,456,789
b) 0.00000004923
c) eight hundred ninety-three million seven hundred twenty-six thousand four hundred thirty-two
d) 0.15
Problem 2
To multiply numbers in scientific notation, we add exponents. To divide, we subtract exponents. For example, (3.1×105)(2.2×102) = (3.1×2.2)×105+2 = 6.8×107 and (6×105) ÷ (3×103) = 2×102. Perform the following calculations:
a) (103)(104)
b) (5×102)(3×105)
c) (3×10-3)(2×105)
d) (4×105) ÷ (2×104)
Problem 3
The logarithm to the base 10 of a number x is equal to the power to which 10 must be raised in order to equal x. That is, if x = 10y, then y = log x. The following identities are useful for performing calculations with logarithms:
log AB = log A + log B
log A/B = log A - log B
log An = n log A
The logarithms of some numbers are as follows:
|
x
|
log x
|
x
|
log x
|
|
1
|
0
|
6
|
0.778
|
|
2
|
0.301
|
7
|
0.845
|
|
3
|
0.477
|
8
|
0.903
|
|
4
|
0.602
|
9
|
0.954
|
|
5
|
0.699
|
10
|
1.000
|
Find the following logarithms using the logarithms of the numbers 1-10 and the three identities given.
a) log 50
b) log 0.5
c) log 2*1010
d) log 16
Given log x, find the number x in each case.
a) log x = 0.3
b) log x = 3.0
c) log x = 1.3
d) log x = -0.3
Part B: Standing Waves on a Stretched String
The purpose of this experiment is to examine the relationship between wavelength λ, frequency f, and velocity v of standing waves in a stretched string.
1. What is the relationship between the length of a single "loop" portion of the vibrating string and the wavelength of the standing wave?
2. What happens to the amplitude of the loops as n increases?
3. Why are there only whole loops and no half loops visible?
4. What happens if you use a frequency that is not at one of the resonances?
Standing Waves Worksheet
1. A guitar string of length 0.648 m is fixed at both ends. The string has a mass of 4.50 g and is under a tension of 74.0 N.
a) Calculate the speed of waves in the string using v = √(T/μ). Note that the mass density μ = mass/length.
b) Sketch the first four allowed standing waves in the string.
c) For each of the first four allowed standing waves, calculate the wavelength and the frequency. Use the formulas λn = 2L/n and fn = nv/2L.
2. An organ consists of pipes that are open at one and end and closed at the other. Consider a particular pipe of length 1.50 m. Assume the speed of sound in air is v = 343 m/s.
a) Sketch the first four allowed standing waves in the pipe.
b) For each of the first four allowed standing waves, calculate the wavelength and the frequency. Use the formulas λn = 4L/n, fn = nv/4L = nf1, n=odd.