PH230 General Physics I, Week 4 Homework
In addition to the following problems do Problems 3.4 and 3.15
Each problem is worth 10 pts.
1. spring has a length of 11 cm when relaxed, and a length of 15 cm when
stretched by a force of magnitude 54 N. What is the stiffness k of this spring?
This same spring is then compressed by an unknown force so that the length of
the spring becomes 9 cm. What is the magnitude of this force?
2. One mole of gallium (6 1023 atoms) has a mass of 70 grams, as shown in the
periodic table on the inside front cover of the textbook. The density of gallium is 5.9
grams/cm3. What is the diameter of a gallium atom in a solid block of the material?
Make the simplifying assumption that the atoms are arranged in a "cubic" array, so
that an atom of diameter d occupies an amount of space d3 in the solid. Note that
the density is equal to the mass of one atom divided by d3. Of course, you need to
convert to SI units. 1 g = 1e-3 kg. 1 cm3 = (0.01 m)3 = 1e-6 m3.
3. (This is Problem 3.2, page 82.) Young's modulus for aluminum is 6.2 1010 N/m2.
The density of aluminum is 2.7 grams/cm3, and the mass of one mole (6 1023 atoms)
is 27 grams. If we model the interactions of neighboring aluminum atoms as though
they were connected by springs, determine the approximate spring constant of such
a spring. Repeat this analysis for lead: Young's modulus for lead is 1.6 1010 N/m2, the
density of lead is 11.4 grams/cm3, and the mass of one mole is 207 grams. Make a
note of these results, which we will use for various purposes later on. Note that
aluminum is a rather stiff material, whereas lead is quite soft.
4. You hang a heavy ball with a mass of 10 kg from a gold wire.5 m long that is 2
mm in diameter. You measure the stretch of the wire, and find that the wire
stretched 0.000993 m. Calculate Young's modulus for the wire. Y = N/m2. The
atomic mass of gold is 197 g/mole, and the density of gold is 19.3 g/cm3. Calculate
the interatomic spring stiffness for gold.
5. Suppose we hang a heavy ball with a mass m = 15 kg (weight mg) from a steel
wire 2.9 m long that is 2 mm in diameter (radius = 1 10-3 m). Steel is very stiff, and
Young's modulus for steel is unusually large, 2 1011 N/m2. Calculate the stretch L of
the steel wire. Note that the fractional change in the length of the wire is very small!
6. For a spring-mass oscillator, if you quadruple the mass but keep the stiffness the
same, by what numerical factor does the period change? That is, if the original period
was T and the new period is bT, what is b? It is useful to write out the expression for
the period and ask yourself what would happen if you quadrupled the mass. If,
instead, you quadruple the spring stiffness but keep the mass the same, what is the
factor b? If, instead, you quadruple the mass and also quadruple the spring stiffness,
what is the factor b? If, instead, you quadruple the amplitude (keeping the original
mass and spring stiffness),what is the factor b?
7. A ball whose mass is 1.2 kg is suspended from a spring whose stiffness is 4 N/m.
The ball oscillates up and down with an amplitude of 18 cm. (a) What is the angular
frequency? = radians/s (b) What is the frequency? f = hertz (cycles per second) (c)
What is the period? T = s (d) Suppose this apparatus were taken to the Moon, where
the strength of the gravitational field is only 1/6 of that on Earth. What would the
period be on the Moon? (Consider carefully how the period depends on properties of
the system; look at the equation.)
8. A certain spring has stiffness 110 N/m. The spring is then cut into two equal
lengths. What is the stiffness of one of these half-length springs?
