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?
9. A mass of 2.7 kg is connected to a horizontal spring whose stiffness is 8.5 N/m. When the
spring is relaxed, x = 0. The spring is stretched so that the initial value of x = +0.16 m. The mass
is released from rest at time t = 0. Remember that when the argument of a trigonometric function
is in radians, on a calculator you have to switch the calculator to radians or convert the radians to
degrees. Predict the position x when t = 1.06 s