PHYS 211 Recitation Review Problems: Solutions

Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions
PHYS 211 Recitation Review Problems: Solutions
1) Sled on Ice
The first set of questions involve a sled being pulled on ice (where friction is negligibly small and should
be ignored). A person wearing spiked shoes can apply a force to the sled in one of six different ways:
A) To the right with increasing magnitude D) To the left with increasing magnitude
B) To the right with constant magnitude E) To the left with constant magnitude
C) To the right with decreasing magnitude F) To the left with decreasing magnitude
They can also G) not push the sled (zero force).
For each of the below descriptions of motion, choose the one force (A-G) which would keep the sled
moving as described (or respond NONE if none of the forces could do so).
The sled…
__B__ is moving toward the right and speeding up at a steady rate (constant acceleration)
__G_ is moving toward the right at a steady (constant) velocity
__E__ is moving toward the right and slowing down at a steady rate (constant acceleration)
__E__ is moving toward the left and speeding up at a steady rate (constant acceleration)
__B__ is moving toward the left and slowing down at a steady rate (constant acceleration)
__B__ is slowing down at a steady rate and has a constant acceleration to the right
Could any of the forces bring a sled initially at rest to a constant non-zero velocity to the left at some later
time? If so, which, if not, why?
F) To the left with decreasing magnitude will have this effect. Initially with a non-zero force to the left
the sled will accelerate to the left. As the magnitude of the force decreases, the rate at which the sled
speeds up decreases, until the magnitude of the force falls to zero (there is no net force) at which point the
sled will move at constant non-zero velocity to the left.
2) Race
Two blocks (masses m and 4m) are lined up (at rest) on a starting line. Starting at the same time, they will
be pushed with forces of constant, equal magnitude F across an icy (frictionless) surface until they reach a
finish line some distance away. Circle the correct choices and fill in the blanks to accurately complete the
story:
Compared to mass 4m , mass m will have a (larger, equal, smaller) acceleration. Mass m will finish the
race in time T. Mass 4m will finish it (faster, slower), in time ___2T____. As it crosses the finish line,
mass m will have a velocity of v. Mass 4m will have a velocity of _ ½ v________.
Notes:
F ma 4ma a 4 a ; x a t a t t a a t 2t
1 2 1 2
m 4m m 4m 2 m m 2 4m 4m 4m m 4m m m
v a t a 2t v
1 1
f ,4m 4m 4 m 4 m m 2 f , m
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Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions
3) Cart
For a lecture demo in front of class, a little cart (mass 0.5 kg) travels
on a long, straight, level air track. Shown at right is a graph of
velocity vs. time of the cart. Let's define "right" to be the positive
direction.
1. What is the net force (with correct sign) on the cart at t=3 s?
A) 0 N
m
B) +3 N
v
6
a s
4
m
2
C) -3 N
s
t
1.5 s
D) +2 N
m
F ma 0.5kg 4 2 2 N
s
E) -2 N
+4 m/s
+2 m/s
-2 m/s
+velocity
time (s)
1 2 3 4 5
2. At t=2.5 seconds, how would you describe what the car is doing?
A) It is moving to the right with steady speed.
B) It is moving to the right with decreasing speed
C) It is moving to the left with steady speed
D) It is moving to the left with decreasing speed
E) None of the above/not enough information
3. At t=5 seconds, the air for the air track is suddenly turned off, so kinetic friction (alone) causes
the cart to grind to a halt in another 0.5 sec. During that half-second when the cart is stopping,
call the force of friction on the cart by the track f CT .
What can you conclude about the frictional force f' TC on the track by the cart during this time?
(assume the mass of the track itself is 5.0 kg, i.e. ten times the mass of the little cart)
A) f' TC = -10 f CT
B) f' TC = +10 f CT
C) f' TC = -f CT /10
D) f' TC = -f CT
E) f' TC = +f CT
Newton’s 3 rd Law: Action-reaction pairs are equal and opposite
4. In the previous question, (during the half-second when the car stops due to friction) what is the
coefficient of kinetic friction between the cart and the track?
A) 0.8
m
B) 0.5
v
2
a s
4 m
2
C) 0.4
s
t
0.5 s
D) 0.2
m
4 2
a s
E) 0.1
F ma mg
m 0.4
g 10
2
s
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Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions
4) Rocket
The questions on this page all refer to this (highly simplified) scenario.
A rocket is launched and travels straight up (defined to be positive here).
The net force on the rocket as a function of time is shown in this graph.
1. Between t=1 hr and 3 hrs, what happens to the speed of the rocket?
A) The rocket is speeding up the whole time
B) The rocket is slowing down the whole time.
C) The rocket is moving with a constant speed the whole time.
D) The rocket is speeding up for a while, and then it slows down.
E) There is not enough information given to decide.
+4
+2
Net Force (MegaNewtons)
1
2 3 4 5
time (hrs)
Force is always positive,
in direction of motion
Inside the rocket, there is some payload, as shown. On the left is
Rover I (RI). On the right is Rover II (RII), which is identical to RI.
Sitting on top of RII is a sensor package (S) which weighs much less
than the Rovers do.
Rover I (RI)
up
Sensor (S)
Rover II (RII)
2. At t=1 hr, how does the acceleration of sensor (S) compare to the
Floor
acceleration of Rover II?
A) a(S) = a(RII) B) a(S) < a(RII) C) a(S) > a(RII)
D) Without knowing more numbers, we can’t tell how the accelerations of S and RII compare.
They are attached same acceleration
3. At t=1 hr, how does the normal force of the ground on Rover I (RI) compare to that of the
ground on Rover II?
A) N(RI) = N(RII) B) N(RI) < N(RII) C) N(RI) > N(RII)
D) Without knowing more numbers, we can’t tell how the normal forces on RI and RII compare.
RII is effectively “heavier” because of the sensor package sitting on top of it.
4. A student is asked to sketch a force diagram for Rover II only, as viewed from the ground just
after launch ("N" represents "normal", W represents "Weight") Note that none of the diagrams
are complete, because the student has not properly labeled forces "ON object BY object", but
which is best?
m a
N
N
N'
N
N'
W W W W
A)
B) C) D) E)
The normal force from the ground is up, from the sensor package is down
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Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions
5. In class we watched a video of a hammer and feather dropped on the moon. Both stared from
the same height and landed on the lunar surface. Assume that the mass of the hammer is 100
times greater than the mass of the feather.
During the time that they fall freely, consider the following two statements:
i) The net force on the hammer is equal to the net force on the feather.
ii) The net acceleration of the hammer is equal to the net acceleration on the feather.
A) Without knowing more numbers, we cannot decide
B) Both statements are true
C) i is true, but ii is false.
D) i is false, but ii is true
E) Both statements are false.
The forces (mg) are DIFFERENT because of the different masses. However the acceleration of
both is the same, namely g moon .
6. A rover is moving about on a flat Martian surface. Its position as a function of time is given
m 2
m
x t 3 t , y t 5 t, z t 0
extremely precisely by 2
s
(You should assume there is rolling friction for the rover on Mars!)
What can you conclude about the net force on the rover?
s
A) There is not enough information given to conclude anything precise about the net force.
B) The magnitude of the net force is a constant in time, and always points in the + ˆ i direction.
C) The magnitude of the net force is not constant in time, but always points in the + ˆ i direction.
D) The magnitude of the net force is a constant in time, but the direction changes with time.
E) The magnitude of the net force is not constant in time, and the direction also changes with
time.
Find the acceleration by taking 2 time derivatives. Only a
x (t) = 6 m/s 2 is non-zero. Because the
acceleration is constant in the +i direction, so is the net force.
7. A penny sits on an old fashioned (flat, horizontal) circular record turntable,
rotating at a constant speed as illustrated in the "top-down" diagram at the right.
1¢
Which of the following sets of vectors best describes the directions of the velocity,
acceleration, and net force acting on the penny at the point indicated in the
diagram? (Note: this is not a force diagram, I'm just asking for the direction of
these three different vectors)
F
v
a
F
a=0
v
F
a=0
v
F
a
v
F
a
v
A)
B) C)
D) E)
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Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions
8) Car
A car (mass M) is going around a banked curve of radius R
and bank angle at constant speed v. Sitting on its very well
polished (frictionless) hood is a car top carrier (mass m)
which the occupants forgot to secure. Assume that the curve
is very icy (frictionless between road and car) but that the
car is traveling at the correct velocity to make it through the
turn.
A) Draw FBDs for both the car and the carrier.
Car
Carrier
a
B) Given that the car makes it around the frictionless bank, does the carrier? HINT: After you write
your two equations (two force components) divide them – this will get rid of things in which you
are uninterested.
It will. The equations are quite similar. The vertical component of the normal/interaction forces must
cancel gravity. The horizontal component must provide centripetal acceleration:
2
2
F x,car N FCarrier,Car sin
Ma v
cent.
M
R v
tan
Fy,car : N FCarrier,Car cos
F Rg
G,Car
Mg
2
2
F x,car FCarrier,Car sin
ma v
cent.
m
R v
tan
Fy,car : FCarrier,Car cos
F Rg
G,Carrier
mg
C) How does the mass of the car top carrier affect the velocity at which the car must travel to make it
around the frictionless banked curve?
It doesn’t – note that the interaction force divides out along with the normal force from the curve
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Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions
9) Towing a Sled
A mother tows her daughter on a sled on level ice. The
friction between the sled and the ice is negligible, and
the tow rope makes an angle of to the horizontal. The
combined mass of the sled and the child is M . The sled
has an acceleration in the horizontal direction of
magnitude a.
A) What is the tension T in the rope?
We draw the free body diagram and acceleration diagram from the sled and note that
only the horizontal component of the tension is responsible for the acceleration,
allowing us to solve for the acceleration:
a
T cos
Ma T
Ma
cos
B) What is the magnitude of the normal force N exerted by the ice on the sled?
The normal force we get from the vertical component:
N T sin Mg N Mg T sin Mg Ma tan
N
Now the mother starts up an incline (angle ).
She continues to pull at the same relative angle
and at the same acceleration a. What new
tension and normal force must exist?
Work in a tilted coordinate
system where the acceleration
vector is along the “x-axis”
and the normal to the plane it
the “y-axis.” Then we find:
a
Ma Mg sin
T cos
Mg sin Ma T
cos
N T sin Mg cos N Mg cos T sin Mg cos Ma Mg sin tan
N
Check: These reduce to our previous expressions in the limit that goes to zero.
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Physics 211 Fall 2012 Midterm 2 Recitation Review Solutions
10) Blocks
A block of mass m 1 is sitting on top of a more massive
block of mass m 2 that rests on a horizontal table.
There is a horizontal force of magnitude F pulling the
top block to the right. There is friction between the
two blocks and between the bottom block and the
table, both with kinetic coefficient of friction μ . The
blocks are connected by a massless, ideal rope that
goes around a fixed massless and frictionless pulley.
A) Draw FBDs and acceleration diagrams for both blocks 1 & 2
1 2
There are two action-reaction pairs – the friction and normal forces
between the blocks F f .
12
and
12
B) What constraint relationship is there between the
acceleration of block 1 and block 2 given the coordinate
system(s) that you defined in your acceleration diagrams?
a
a
We define the coordinate systems of the two blocks backward from each other so that a 1 = a 2 = a. If we
chose to make “to the right” positive then the accelerations would have opposite signs.
C) From your FBDs & acceleration diagrams write four equations of motion using Newton’s 2 nd law
F F T f m a F : F F m g
x,1 12 1 y,1 12 G,1 1
F T f f m a F : N F F m m g
x,2 2G 12 2 y,2 12 G2 1 2
D) How many unknowns appear in your equations? Are any of your equations irrelevant, i.e. contain
unknown(s) that appear nowhere else and whose value(s) you don’t care about? If this number is
bigger than 4, do you have any other equations relating these unknowns?
We have six unknowns: T, f12, f2G, F12
, N,
a
None of the equations are irrelevant since we need the normal forces to think about friction.
We have two other equations relating friction and normal forces
E) What are the frictional forces on the blocks (in terms of known quantities)?
;
f F m g
12 12 1
f N m m g
2G
1 2
F) Solve for the acceleration a of the system. Note that to eliminate the tension T you might just add
(or subtract) two of your equations from part C.
We can add the two horizontal equations to cancel the tension:
F T f T f f m a m a m m a
12 2G
12 1 2 1 2
1 1
1 2
a F 2f12 f2G
F 2m1g m1 m2
g a
m1 m2 m1 m2
m1 m2
F g 3m m
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