Instructor: Hacker
Course: Physics 121
Sample Exam 2 Solutions (Mathematical Preliminaries)
Print your name neatly. If you forget to write your name, or if the grader can’t read your
writing, you can lose up to 100 points. Answer all the questions that you can.
This exam will consist of 21 multiple-choice problems. You may not use calculators or
other electronic devices on this exam. The use of such a device will be regarded as an
attempt to cheat, and will be pursued accordingly. All diagrams and figures on this exam
are rough sketches: they are not generally drawn to scale.
No partial credit will be given for these problems. However, you can miss one of the 21
problems without penalty. Your grade will be based on your best 20 problems. You will
not receive extra credit for getting all 21 right.
Your grade on the exam will be based entirely on the answers that you circle on this
sheet. If you have no answer or a wrong answer there, the grader will not look at the
page with the problem to see if the right answer appears there. Illegible or ambiguous
answers will be graded as wrong. You are responsible for copying your answers clearly,
correctly, and in the right place.
Although there is no partial credit on this exam, you must show your work in the space
provided on the exam. There is additional scratch paper at the end of the exam: do
not use it unless you have filled all the scratch space provided on the page with the
problem. If you answer a difficult problem without doing any written work, the grader
will assume that you got the answer by guessing or by copying from someone else, and
will not give you credit for the problem even though you’ve indicated the correct solution
on the answer sheet.
Circle your answers here. Do not detach this sheet from the test.
1.
a
b
a b
2. ○
c d
○
e
8.
a
b
d e
○
15.
a
b
c d
○
e
c
d
e
9.
a
b c
○
d
e
16.
a
b
c d
○
e
c d
○
e
c
3.
a
b c
○
d
e
10.
a
b
c
d e
○
17.
a
b
4.
a
b
c
d e
○
11.
a
b
c
d e
○
18.
a
b c
○
5.
a
b
c d
○
e
12.
a
b c
○
d
a b
19. ○
c
6.
a
b
c
d e
○
13.
a
b
c
d e
○
20.
7.
a
b
c d
○
14.
a
b
c
d
e
e
e
○
a
b
a b
21. ○
d
e
d
e
c d
○
e
c
e
d
Physics 121 sample exam 2 solns
Copyright ©Wayne Hacker 2010. All rights reserved. 2
Measurement: Dimensions, Units, and Significant Figures
Dimensional consistency
Problem 1. Consider the formulas (i) and (ii). Are the formulas dimensionally consistent? Here A = area, r = distance, x = distance, t = time, and v = velocity.
x
(i) A = 4πr2
(ii) v =
t
(a) (i) is consistent; (ii) is not
(b) (ii) is consistent; (i) is not
*(c) Both (i) and (ii) are consistent
(d) Neither (i) nor (ii) is consistent
(e) None of these
Solution: In (i), A has units of (length)2 ; 4 and π are dimensionless; and r has dimensions of (length). In units, the formula looks like
(length)2 = (length)2
Hence (i) is dimensionally consistent.
; x has units of (length); and t has units of (time). In units,
In (ii), v has units of length
time
the formula looks like
(length)
(length)
=
(time)
(time)
Hence (ii) is also dimensionally consistent.
Problem 2. Determine whether equations (i) and (ii) below are dimensionally consistent.
(i) v =
x − x0
t
(ii) x = a(t + v)2 ,
where x = distance, v = velocity, t = time, and a = acceleration.
*(a) (i) is dimensionally consistent; (ii) is dimensionally inconsistent
(b) (i) is dimensionally inconsistent; (ii) is dimensionally consistent
(c) (i) and (ii) are both dimensionally consistent
(d) (i) and (ii) are both dimensionally inconsistent
Solution: Can’t add velocity and time since they are dimensionally inconsistent.
Physics 121 sample exam 2 solns
Copyright ©Wayne Hacker 2010. All rights reserved. 3
Units
Problem 3. A German shepherd weighs 77 lb. Which is the correct conversion factor
for the relationship 2.2 lb = 1 kg to convert the dog’s mass into kilograms?
2.2 lb
1 kg
(a)
*(b)
1 kg
2.2 lb
kg
(c) 2.2 kg · lb
(d) 2.2
lb
(e) None of these
Solution: Use the conversion factor 2.2 lbm/kg. To make the units work, multiply by
the conversion factor:
1 kg
= kg
lbm ·
2.2 lbm
To cancel the pounds.
Comment: You might be wondering how weight, with dimensions of force, can equal
mass with dimensions of mass, actually the conversion relationship is 2.2 lbm = 1 kg,
where lbm is known as pound mass with units of mass. A mass of 10 lbm has a weight
of 10 lbf, where lbf is pound-force, which is what we call lb (i.e., lbm = lb).
Significant figures
Problem 4. How many significant figures are there in: m = 120.30 kg?
(a) 2
(c) 4
(e) None of these
(b) 3
*(d) 5
Solution: The first zero is significant because it is between nonzero digits (rule 2); the
second is significant because it is to the right of the decimal point (rule 4). Hence: 5
significant digits.
Problem 5. Calculate 120.51 + 3.2. Round your answer to the correct number of significant figures.
(a) 120
*(c) 123.7
(e) None of these
(b) 124
(d) 123.71
Solution: 120.51 + 3.2 = 123.71. The 3.2 is only accurate to the tenths place; so by
rule 10, the sum is also only accurate to the tenths place. Hence we round it to 123.7.
Physics 121 sample exam 2 solns
Copyright ©Wayne Hacker 2010. All rights reserved. 4
Arc length
Problem 6. An angle measures 1.06 radians. What is its measurement in degrees?
(a)
(c)
(e)
1.06π
degrees
180
1.06
degrees
2π
None of these
(b)
*(d)
(1.06)(90)
degrees
π
(1.06)(180)
degrees
π
Problem 7. An angle measures 87◦ . Find its exact measurement in radians.
87
180
(a)
rad
(b)
rad
180π
87
87
87π
rad
(d)
rad
*(c)
180
180
(e) None of these
Solution:
87π
rad
180
Problem 8. In the figure at right, what is the
measure of the angle θ?
√
(a)
14 rad
(b) 5π/7 rad
(c) 5/7 rad
*(d) 7/5 rad
(e) None of these
7
θ
5
Problem 9. A circle has a radius of 5 cm. A central angle has measure θ = π/3 rad.
What is the length of the arc intercepted by the angle?
(a) 2/15 cm
(c) 5/3 cm
(e) None of these
*(b) 5π/3 cm
(d) 10π/3 cm
Solution: (5 cm)(π/3) = 5π/3 cm
Problem 10. A bicycle wheel has a radius of 27 inches. It turns so that a point on the
rim travels 11 inches. What angle has the wheel turned through? Round your answer to
the nearest 0.01 rad.
(a) 27/11 rad
(b) 11 rad
(c) 1 rad
*(d) 11/27 rad
(e) None of these
Solution: θ = s/r = (11 in)/(27 in) = 11/27 rad
Physics 121 sample exam 2 solns
Copyright ©Wayne Hacker 2010. All rights reserved. 5
Trigonometry
Problem 11. In the figure at right, v/u = ?
(a) 1/ sin α
(c) 1/ tan α
(e) None of these
u
(b) tan α
*(d) sin α
β
v
α
w
Problem 12. Use a drawing of a special triangle, or from memory, determine sin 30◦ .
√
(a) √
1/ 2
*(b) 1/2
(c)
3/2
(d) 1
(e) None of these
Problem 13. A highway runs directly east
and west. An airplane flies across the highway
in a direction θ = 34◦ north of east. What
is the total distance that the airplane will fly
before it is 21 miles north of the highway?
(a) tan 34◦ /21 miles
(c) sin 34◦ /21 miles
(e) None of these
21 cos 34◦ miles
21/ sin 34◦ miles
(b)
*(d)
Solution: Let d be the total distance flown by the airplane. Then 21/d = sin 34◦ ; so
d = 21/ sin 34◦ miles.
Problem 14. Let θ be an acute angle such that cos θ = 2, determine the value of θ in
terms of the fundamental angles 30◦ , 45◦ , 60◦ , and 90◦ .
(a) 30◦
(c) 60◦
*(e) No such angle
(b)
(d)
45◦
90◦
Solution: No such angle; cos θ > 1 is impossible.
Problem 15. In the figure at right find the exact
value of φ. (The figure is not necessarily drawn to
scale.)
−1
(a) φ = sin (8/7)
*(c) φ = tan−1 (8/7)
(e) None of these
−1
(b) φ = tan (7/8)
(d) φ = cos−1 (7/8)
−1
Solution: tan φ = 8/7; so φ = tan (8/7)
8
φ
7
Physics 121 sample exam 2 solns
Copyright ©Wayne Hacker 2010. All rights reserved. 6
Problem 16. An angle with measure θ degrees is in the third quadrant. What is sin θ?
(a) sin(θ − 180◦ )
*(c) − sin(θ − 180◦ )
(e) None of these
(b) sin(360◦ − θ)
(d) − sin(θ − 360◦ )
Solution: Since θ is in the third quadrant, the reference angle is θref = θ − 180◦ ; and
sin θ ≤ 0. Hence sin θ = − sin θref = − sin(θ − 180◦ ).
Vectors
Problem 17. Which of the vectors in the
graph corresponds to −10ˆı + 5ˆ?
~
~
(a) A
(b) B
~
~
*(c) C
(d) D
(e) None of these
~ and C
~ have a negSolution: Only B
ative x-component and a positive ycomponent. In −10ˆı + 5ˆ, the magnitude
of the x-component (10) is greater than
the magnitude of the y-component (5).
~ but not of B.
~
This is true of C
~
B
AK
y
6
A
A
~
C
~
A
A
A
A
iP
P
PP A
PP
A
-
x
~
D
~ Y~ , and Z
~
Problem 18. The vectors X,
are labelled on the figure at right. Which
of the following equations is true?
~ =X
~ − Y~
~ = Y~ − X
~
(a) Z
*(b) Z
~ =X
~ + Y~
~ = −X
~ − Y~
(c) Z
(d) Z
(e) None of these
y
6
~
Z
Y~
]
JJ
J
J
J
J
-
x
?
~
X
~ = h−1, 5i and B
~ = h−2, −3i. Find A
~ + B.
~
Problem 19. A
*(a) h−3, 2i
(c)
h3, 8i
(e)
None of these
(b) h4, −5i
(d) h2, −15i
Solution: h−1, 5i + h−2, −3i = h−1 + (−2), 5 + (−3)i = h−3, 2i
Physics 121 sample exam 2 solns
Copyright ©Wayne Hacker 2010. All rights reserved. 7
~ = h−3, 4i, what is the magnitude of A?
~
Problem 20. If A
(a) 25
(b) 7√
*(c) 5
(d)
7
(e) None of these
q
p
Solution: kAk = A2x + A2y = (−3)2 + (4)2 = 5
Problem 21. An airplane is flying at a speed of 190 mph in a direction 24◦ north of
east. What is the northward component of the plane’s velocity? Round your answer to
the nearest mph.
*(a) 190 sin 24◦ mph
(c)
190 sin 66◦ mph
(e)
None of these
(b)
(d)
190 cos 24◦ mph
190 tan 66◦ mph
Solution: vy = v sin θ = (190 mph) sin 24◦