5 Problems on Interest Group Seminar - Assignment 3 | L A 1, Assignments of Humanities

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PS 303 - HW 3

Due Thursday 02/23/

1. [10]: When measuring the length of a table, two people A and B have obtained the values lA =
   1. 52 ± 0. 01 m and lB = 1. 505 ± 0. 006 m, respectively.

(a) Do these measurements agree to within the uncertainty? (Explain your answer) Solution:

(b) Which measurement is more accurate? (Explain your answer) Solution:

2. [25]: Consider a golden bar of volume V = 2. 0 l. The density of gold is ρ = 19. 32 g/cm^3. Show work - not just the final answers.

(a) Write down the algebraic relationship expressing mass as a function of volume and density. Solution:

(b) What is the mass of the bar? (Be careful about the units!) Solution:

(c) How much does the bar cost if the price of the gold is $15 per gram?

(d) If you put aside $10 a day, how long would you have to save money in order to buy it? Solution:

(e) What is the weight of the bar on the Earth (where the gravity g = 9. 8 m/s^2 )? Solution:

3. [40] The Sun is approximately a sphere of radius R = 695000 km. The (average) density of the Sun is ρ = 1. 41 g/cm^3. The volume of any sphere of radius R is V = 43 πR^3. (Show work.)

(a) Find the volume of the Sun. Solution:

(b) What is the mass of the Sun? (Be careful about the units) Solution:

(c) If the radius of the Sun was 2 times larger than it is, how many times would its volume be larger? Explain your answer. Solution:

(d) Is volume proportional to radius?

(e) If the radius of the Sun were 2 times larger than it is and the density were the same, how many times would its mass be larger? Explain your answer. Solution:

(f) Is mass proportional to radius for a constant density? Solution:

(g) If the density of the Sun were 2 times larger than it is and the radius were the same, how many times would its mass be larger? Explain your answer. Solution:

(h) Is mass proportional to density for a constant radius? Solution:

4. [25] A neutron star is the cinder left when a giant star burns out and collapses on itself. A single teaspoonful of neutron star material weighs as much as an ordinary mountain. The density of the neutron star matter is about ρ = 3 × 1014 g/cm^3. The radius of a typical neutron star is R = 10 km. The volume of any sphere of radius R is V = 43 πR^3 .(Show work.) (a) Find the mass of a sugarcube of the neutron star matter with volume 1 cm^3. Solution:

(b) The mass of the total human population on the Earth is estimated to be 3 × 1011 kg. How many times (by what factor) is the mass of the neutron star sugarcube larger than the mass of all humanity?

Solution:

(c) What is the mass of a typical neutron star? Solution:

(d) What is the gravity on the surface of the neutron star, if the gravity is given by the formula g = γM/R^2 , where γ = 6. 67 × 10 −^11 N m^2 /kg^2 , M is the mass and R is the radius of the star. Solution:

(e) How many times is the gravity on the surface of the neutron star larger than on the Earth? The gravity on the Earth is gEarth = 9. 8 m/s^2.

5. [20]: Consider a square and a cube of side 1 mm.

(a) If the square’s side is increased by a factor of 5, by how many times does its area increase? (Calculate both areas first). Solution:

(b) Now consider an arbitrary surface with a boundary, say a butterfly. If you scale the butterfly by a factor of 5 (increase the distance between any two points by 5 times), by how many times does the area of the butterfly increase? (Hint: You can place a square grid on the butterfly and take into account your previous answer about a single square.) Explain your answer. Solution:

(c) If the cube’s side is increased by a factor of 5, by how many times does its volume increase? (Again, calculate both volumes first). Solution:

(d) Now consider a bear. If you scale the bear by a factor of 5, by how many times does the volume of the bear increase? (Hint: You can place a cube grid in the bear and take into account your previous answer about a single cube.)