1
HW1
1. Two spheres are cut from a certain uniform rock. One has radius r=5.0 cm. The mass of the other is eight times greater. Find its radius R .
Since mass ∝ radius 3
R = r · 8 1 / 3 = 10 cm
2. The mass of a copper atom is m = 2 .
27 · 10
ρ = 8920 kg/m 3
−
25 kg , and the density of copper is
. (a) Determine the number of atoms in V = 1 cm 3 of copper.
N = ρV /m = 3 .
93 10 22
(b) Visualize the one cubic centimeter as formed by stacking up identical cubes, with one copper atom at the center of each. Determine the volume of each cube.
V /N = 2 .
54 10
−
29 m 3
(c) Find the edge dimension of each cube, which represents an estimate for the spacing between atoms.
V
N
1 / 3
= 2 .
94 10
−
10 m
3. Newton’s law of universal gravitation is represented by F = GM m/r 2 where F is the magnitude of the gravitational force exerted by one small object on another,
M and m are the masses of the objects, and r is a distance. Force has the SI units kg · m/s 2 . What are the SI units of the proportionality constant G?
[ G ] = [ F ] m 2 /kg 2 = m 3 /kg/s 2
4. Kinetic energy KE (Chapter 5) has dimensions kg m 2 /s 2 . It can be written in terms of the momentum p (Chapter 6) and mass m as KE = p 2 / 2 m (a) Determine the proper units for momentum using dimensional analysis. (Use the following as necessary: kg, m, and s.)
[ p ] = kg m/s
(b) Given the units of force, write a simple equation relating a constant force F exerted on an object, an interval of time t during which the force is applied, and the resulting momentum of the object, p.
Look for F ∼ t α p β with yet unknown α, β . Compare the dimensions: kg m/s 2 ∼ s α ( kg m/s ) β
Thus, α = − 1 , β = 1 and
F t ∼ p

2
5. a) Assume the equation x = At 3 + Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B. (Use the following as necessary:
L and T, where L is the unit of length and T is the unit of time.)
[ A ] = L/T 3 , [ B ] = L/T
(b) Determine the dimensions of the derivative dx/dt = 3 At 2 + B .(Use the following as necessary: L and T, where L is the unit of length and T is the unit of time.)
[ dx/dt ] = L/T
6. A rectangular building lot has a width of 72.5 ft and a length of 110 ft. Determine the area of this lot in square meters
72 .
5 × 110 f t 2 = 72 .
5 × 110 (0 .
305 m ) 2 = 740 .
9 m 2
7. A solid piece of lead has a mass of 31.14 g and a volume of 2.72 cm3. From these data, calculate the density of lead in SI units (kilograms per cubic meter).
1 kg/m3
31 .
14 (0 .
001 kg )
= 11 .
45 10 3
2 .
72 (0 .
01 m ) 3 kg/m 3
8. Suppose your hair grows at the rate 1/31 in. per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of 0.1 nm, your answer suggests how rapidly layers of atoms are assembled in this protein synthesis
1 in
31 day
=
1
31
2 .
54 10
−
2 nm × 10 9
24 · 3600 s
= 9 .
48 nm/s
9. Find the order of magnitude of the number of table-tennis balls that would fit into a typical-size room (without being crushed). (Assume that the dimensions of the room are 4 m by 4 m by 3 m.)
10 6
10. a) Compute the order of magnitude of the mass of a bathtub half full of water.
(Assume the tub measures 1.3 m by 0.5 m by 0.3 m.) m = ρV ∼ 10 2 kg
(b) Compute the order of magnitude of the mass of a bathtub half full of pennies.
(Assume the pennies are made entirely of copper.) m = ρ
Cu
V ∼ 10 3 kg

3
11. A surveyor measures the distance across a straight river by the following method.
Starting directly across from a tree on the opposite bank, she walks d = 92 m along the riverbank to establish a baseline. Then she sights across to the tree.
The angle from her baseline to the tree is θ = 31.0. How wide is the river?
d tan θ = 55 .
28 m
12. The displacement vectors and shown in the figure below both have magnitudes of L=1.58 m.
y
The direction of vector A is θ =38.8.
B
A
x a) Find
~
=
~
+
First, find components
~
= ( L cos θ , L sin θ ) B = (0 , L )
Then,
C x
= A x
+ B x
= L cos θ = 1 .
23 , C y
= A y
+ B y
= L sin θ + L = 2 .
57 magnitude p C x
2 + C y
2 = 2 .
85 direction arctan( C y
/C x
) = 64 .
4 o
(b) Find
~
=
~
−
C x
= A x
− B x
= L cos θ = 1 .
23 > 0 , C y
= A y
− B y
= L sin θ − L = − 0 .
59 < 0 magnitude p C x
2 + C y
2 = 1 .
37 direction arctan( C y
/C x
) + 360 0 = 334 .
4 0
(c) Find
~
= −
~
+
C x
= − A x
+ B x
= − L cos θ < 0 , C y
= − A y
+ B y
= − L sin θ + L > 0

4 magnitude p C 2 x
+ C 2 y
= 1 .
37 direction arctan( C y
/C x
) + 180 0 = 154 .
4 o
Note: arctan is confined between ± 90 o while angle is measured from 0 to 360.
Hence the above corrections - need to look at the picture.
13. The polar coordinates of a point are r = 6.00 m and θ = 250. What are the
Cartesian coordinates of this point?
~r = ( x, y ) , x = r cos θ = − 2 .
05 m , y = r sin θ = − 5 .
64 m
14. Vector
~ has a magnitude of A=26 units and points in the positive y-direction.
When vector B is added to A, the resultant vector
~
=
~
+
~ points in the negative y-direction with a magnitude of C=16 units. Find the magnitude of B?
First, write in components
~
= (0 , A ) C = (0 , − C )
Then
~
=
~
−
~
= (0 − 0 , − C − A ) = (0 , − C − A ) and
B = A + C = 42
15. A vector ~r has an x component of -24.5 units and a y component of 39.8 units.
Find the magnitude and direction of this vector.
magnitude r = p x 2 + y 2 = 46 .
73 direction θ = arctan( y/x ) + 180 o = 121 .
6 o counterclockwise from the +x axis
16. Use the component method to add the vectors and shown in the figure. The length of B is 3.55 m and the angle θ = 33 .
0. Length of A is 3.0 m. Express the resultant in unit-vector notation.
y
B
A
x

One has
~
= A cos θ~i + A sin θ~j = 2 .
516 ~i + 1 .
63 B = B~j = 3 .
55 ~j
Then
~
+
~
= A cos θ~i + A sin θ~j + B~j = A cos θ~i + ( A sin θ + B ) ~j = 2 .
51 ~i + 5 .
18 ~j
17. Consider the two vectors
~
= ~i − 3 ~j and
(a) Calculate
~
= −
~
+
~
= 0 ~i − 7 ~j
~i − 4 ~j .
(b) Calculate
~
−
~
= 2 ~i + ~j
(c) Calculate
|
~
+
~
| = 7
(d) Calculate
|
~
−
~
| =
√
5
(e) Calculate the directions of + and - .
~
+
~
: 270 o
(counterclockwise from the +x axis)
~
−
~
: 26 .
6 o
(counterclockwise from the +x axis)
5