Download Midterms: Place, Rules, How to study

Midterms: Place, Rules, How to study
PLACE AND TIME: Friday, September 16 we have the first midterm, covering Units 1-6. All
midterms will be during regular class hours but will take place in the S Beh auditorium which
is located south of the Marriott Library - check it out on the campus map. If possible, please
be there at least 5 minutes before the exam starts, so you can get seated, get your exam, fill
out all the required information and be ready to start working on problems.
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 if you think there was an error in grading.
3. Bring a scientific/graphing calculator. Devices that can store text are not allowed.
4. Bring a 4"x 6" 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 the Review for Midterm 1 which will be posted on the main course website and
also on webAssign and includes many 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 1-6
Mechanics Review 1, Slide 2
Example: Stone thrown from a building
A stone is thrown from the top of a building 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 to reach the ground?
What is the direction of motion (velocity) of the stone just before it
hits the ground?
vx (t) = vx0 = v0 cosq
x(t) = x0 + vx0t = x0 + v0 cosq t
vy (t) = vy0 - gt = v0 sinq - gt
1
y(t) = y0 + vy0t - gt 2
2
t = 4.22 s
θ = tan-1(vyf / vxf) = tan-1(-31.4/17.3)
Mechanics Unit 2, 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
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 5
Example: Three Boxes
Three boxes each of mass m 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.
1.What is the magnitude of T1? T1 = (mA + mB + mC )a
2. What is the net horizontal force on A?
FA = mA 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 6
Example: Book and Coffee Cup
A book of mass mb is connected by a light string to a coffee
cup of mass mc. The book is launched up the frictionless
slope of angle θ with an initial speed vi.
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
For part B : Use kinematics
v 2f - vi2 = 2a(x f - xi )
Mechanics Review 1 , Slide 7
Example: Two Blocks and a Pulley
A block of mass m2 on a horizontal surface with coefficient of
kinetic friction µk, is connected to a ball of mass m1 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.
å Fx,2 = F cosq - T - mk n = m2 a,
åF
åF
y,2
= n + F sin q - m2 g = 0
y,1
= T - m1g = m1a
n = -F sinq + m2 g
æ F cosq - m1g - mk n ö
a =ç
è
m1 + m2
÷
ø
Mechanics Unit 6 , Slide 8
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 = 310
Mechanics Unit 4, Slide 9
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
F
®a=
m1 + m2
Mechanics Unit 4, Slide 10
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
= may
x
y
ax = gsinq
Mechanics Unit 4, 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: Hockey Puck sliding on Ice
A hockey puck on a frozen pond is given an initial speed of v0 =
20.0 m/s. If the puck always remains on the ice and slides a
distance d =115 m before coming to rest, determine the
coefficient of kinetic friction between the puck and ice. A: 0.117
åF
= max
- fk = ma
åF
= may
N - mg = 0
x
y
fk = mk N
v - v = 2da
2
2
0
Mechanics Unit 6, Slide 13
Example: Block on Incline Plane
Suppose a block is placed on a rough surface inclined relative
to the horizontal. The incline angle is increased until the block
starts to move. Show that you can obtain μs by measuring the
critical angle θc at which this slipping just occurs.
mgsinq - fs = 0
n - mgcosq = 0
fs,max = ms n = ms mgcosqc
ms = tanqc
Mechanics Unit 6, Slide 14
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
a
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
m1
m1g
a=
m1 + m2
Mechanics Unit 5, Slide 15
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 x mg
B. Determine the force exerted by the seat on the pilot at the
top of the loop. A: n = 0.91 x mg
Bottom : n - mg = mac
Top :
mg + n = mac
v2
ac 
r
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
2


v
banked? A: 20.1o
1
  tan 
  v/r
2
v
a
r

 rg 
å F = N sinq = ma
å F = N cosq - mg = 0
x
c
y
Mechanics Review 1, Slide 17