Midterms: Place, Rules, How to study
PLACE: This Monday we have the first midterm, covering Units 1-5. All midterms will be
during regular class hours but will take place in the FMAB auditorium (Film and Media Arts
Building) which is located south of the Marriott Library - check it out on the campus map. If
possible, please be there at least 15 minutes before 3:00 pm, so you can get seated, get
your exam, fill out all the required information and be ready to start working on problems
at 3:00 pm.
IMPORTANT-PLEASE NOTE THE EXAM RULES:
1. You must bring your University ID to take the exam.
2. You need to bring a pen or two. If you write with a pencil you will not be able to request
a regrade.
3. Bring a scientific/graphing calculator. Devices that can store text are not allowed.
4. Bring a 4"x6" card with formulas or anything you want. You can write on both sides.
HOW TO PREPARE FOR THE EXAM:
1. Go back and review the Homework assignments and make sure you know how to do
them from scratch on your own.
2. Study this Review for Midterm 1 which includes most of the examples from the lectures.
3. Look at old exams and solutions on the main course website: click on "Old exam
reviews".
Mechanics Review 1, Slide 1
Classical Mechanics
Review 1: Units 1-5
Mechanics Review 1, Slide 2
Example: Moving Particle
The velocity of a particle moving along the x axis varies
according to the expression
v(t) = (40 – 5t2) m/s,
where t is in seconds.
(A) Find the average acceleration in the time interval t =0
to t =2.0 s.
A: aavg = -10 m/s2
(B) Determine the acceleration at t = 2.0 s.
A: a = -20 m/s2
Dv
aavg =
Dt
dv
a=
dt
Mechanics Unit 1, Slide 3
Example: Ski Jump
A ski jumper leaves the ski track moving in a horizontal
direction with a speed of 25.0 m/s. The landing incline below
her falls off with a slope of 35.0o. Where does she land on the
incline? What is her speed when she lands?
1 2
y(t) = - gt
2
x(t) = vx0t
y(t) = -tan f × x(t)
vy (t) = -gt vx (t) = vx0
Answer:
xf = 89.3 m, yf = -62.5 m
d = 109 m, t = 3.57 s
Mechanics Unit 2, Slide 4
Example: Ferris Wheel
A typical carnival Ferris wheel has a radius of 9.0 m and
rotates 4.0 times per minute.
(a) What is the period T of the Ferris wheel? A: T =15 s
(b) What is the speed of the riders? A: v = 6π/5 m/s
(c) What magnitude acceleration do the riders
experience?
T =1/ f
w = 2p / T
v = wr
2
v
ac 
r
Mechanics Unit 3, Slide 5
Example: Hockey Puck
A hockey puck of mass 0.30 kg, slides on the horizontal
frictionless surface of an ice rink. Two hockey sticks strike
it at the same time as shown. Determine the magnitude
and the direction of the puck’s acceleration.
åF
= max
åF
= may
x
y
a = a 2x + a 2y
q = tan-1 (ay / ax )
a = 34 m/s2
θ =300
Mechanics Unit 4, Slide 6
Example: Two Blocks
Two blocks of masses m1 and m2 , are placed in contact with
each other on a frictionless horizontal surface. A constant
horizontal force F is applied to m1.
A) Find the acceleration of the blocks.
B) Determine the magnitude of the contact force between the
two blocks.
åF
åF
y,1
=0
åF
åF
y,2
=0
= m1ax
F - P = m1a
x,1
= m2 ax
P = m2 a
x,2
m2
®a=F
m1 + m2
Mechanics Unit 4, Slide 7
Example: Car on a Slope
A car of mass m is on an icy driveway inclined at an
angle θ. Find the acceleration of the car assuming
the driveway is frictionless.
åF
= max
åF
=0
x
y
ax = gsinq
Mechanics Unit 4, Slide 8
Example: Three Boxes
Three boxes each of mass 14 kg are on a frictionless table,
connected by massless strings. A force T1 pulls on the
rightmost box (A) such that the three boxes accelerate at a
rate of a = 0.7 m/s2 .
1.What is the magnitude of T1? T1 = (mA + mB + mC )a
FA = mA a
2. What is the net horizontal force on A?
3. What is the force that box B exerts on A?
T1 - FA,B = mA a
4. What is the net force on box B?
FB = mB a
5. What is the force box C exerts on B?
å
å
FB,C = FC,B = mC a
Mechanics Review 1, Slide 9
Example: Connected Objects
A block of mass m1 on a frictionless horizontal surface is
connected to a ball of mass m2 by a cord over a frictionless
pulley. A force of magnitude F at an angle θ with the
horizontal is applied to the block and the block slides to the
right. Determine the magnitude of the acceleration of the two
objects. Find the normal force on m1.
åF
åF
åF
x,1
= F cosq - T = m1a,
y,1
= n + F sin q - m1g = 0
y,2
= T - m2 g = m2 a
 F cos  m2 g
a  
m1  m2




n  F sin   m1 g
Mechanics Review 1, Slide 10
Example: Loop the Loop
A pilot of mass m in a jet aircraft executes a loop-the- loop. In
this maneuver the aircraft moves in a vertical circle of radius
2.70 km at a constant speed of 225 m/s.
A. Determine the force exerted by the seat on the pilot at the
bottom of the loop. A: n = 2.91 mg
B. Determine the force exerted by the seat on the pilot at the
top of the loop. A: n = 0.913 mg
Bottom : n - mg = mac
Top :
mg - n = mac
v2
ac 
r
Mechanics Review 1 , Slide 11
Example: Atwood Machine
Two objects of unequal mass are hung vertically over a
frictionless pulley of negligible mass. Determine the
magnitude of the acceleration of the two objects and the
tension in the string.
åF
y,1
 m1  m2 
 g
a  
 m1  m2 
= m1g -T = m1a
T
a
m2
T
åF
y,2
= T - m2 g = m2 a
m1
a
m1g
 2m1m2 
 g
T  
 m1  m2 
m2g
Mechanics Review 1 , Slide 12
Example: Field Goal
A field goal kicker can kick the ball 30 m/s at an angle of 30 degrees w.r.t. the
ground. If the crossbar of the goal post is 3m off the ground, from how far
away can he kick a field goal?
y
x
3m
D
y-direction
x-direction
voy = vo sin(30o) = 15 m/s
vox = vo cos(30o) = 26 m/s
y = yo + voyt + ½ at 2
D = xo + vox t + ½ at 2
3 m = 0 m + (15 m/s) t – ½ (9.8 m/s2) t 2
= 0 m + (26 m/s)(2.8 s) + 0 m/s2 (2.8 s )2
t = 2.8 s or t = 0.22 s.
= 72.8 m
Mechanics Review 1, Slide 13
Example: Two Masses and a Pulley
A cart with mass m2 is connected to a mass m1 using a
string that passes over a frictionless pulley, as shown
below. Initially, the cart is held motionless, but is then
released and starts to accelerate.
a
m2
Find the acceleration of the masses and the
tension in string.
åF
åF
x,2
= T = m2 a, å Fy,2 = 0
y,1
= m1g - T = m1a
g
m1
Mechanics Unit 5, Slide 14
Example: Stone thrown from a building
A stone is thrown from the top of a building upward at an
angle of 30.0o to the horizontal with an initial speed of 20.0
m/s. The height of the building is 45.0 m. How long does it
take the stone to reach the ground? What is the direction of
motion of the stone (direction of velocity) just before it strikes
the ground?
1 2
y(t) = y0 + vy0t - gt
2
x(t) = x0 + vx0t
vy (t) = vy0 - gt
vx (t) = vx 0
t = 4.22 s
θ = tan-1(-31.4/17.3)
Mechanics Unit 2, Slide 15
Example: Book and Coffee Cup
The 2.0 kg book is connected by a light string to a 300 g
coffee cup. The book is launched up the frictionless 20o slope
with an initial speed of 5.0 m/s.
A. Calculate the acceleration of the book.
B. How far does the book go up the slope before it stops?
C. Calculate the Tension in the string
åF
åF
x,b
= -mb gsin q - T = mb a,
y,c
= T - mc g = mc a
Use kinematics
v 2f - vi2 = 2a(x f - xi )
Mechanics Review 1 , Slide 16
Example: Banked Curve
A car moving at the designated speed of 13.4 m/s can
negotiate a curve even when the road is covered with ice, if
the ramp is banked (meaning that the roadway is tilted toward
the inside of the curve). The radius of the curve is 50.0 m.
A. What is the angular speed of the car?
B. What is the acceleration of the car?
C. At what angle should the curve be
banked? A: 20.1o
  v/r
v2
a
r
2


v
1
  tan  
 rg 
Mechanics Review 1, Slide 17
Example: Three forces
Three forces are exerted on an object placed on an inclined
plane. The three forces are directed as shown in the figure.
The forces have magnitudes F1 = 3.00 N, F2 = 8.00 N and
F3 = 6.00 N.
(a) What is the component of the
net force parallel to the incline?
(b) What is the component of the net
force perpendicular to the incline?
(a) F3 sin30 - F1
(b) F2 - F3 cos30
Mechanics Review 1, Slide 18
Example
A particle starts from the origin at t = 0 with an initial
velocity having an x component of 20 m/s and a y
component of -15 m/s. The particle moves in the x-y
plane with only an x component of acceleration, given by
ax = 4.0 m/s2.
(a) Determine the velocity at any time t.
(b) Calculate the speed and direction of motion of the
particle at t = 5.0 s. A: v = 43 m/s, θ = - 21o
(c) Determine the x and y coordinates of the particle at
any time t.
vx (t) = vx0 + axt
vy (t) = vy0 + ayt
1 2
x(t) = x0 + vx0t + ax t
2
1 2
y(t) = y0 + vy0t + ayt
2
Mechanics Unit 2, Slide 19