Review Final Exam (final version)
Final: 75% from Ch9, 10,11,13 (review below) and 25% from Exam 2 review sheet
Proof of important theorems (DO NOT FORGET TO JUSTIFY STEPS TO GET FULL CREDIT) You will be asked
to answer 18 points worth):
1.
Starting from Newton’s Law for a system of masses prove that
d pTotal
 F netExternal (LN A16
dt
Ch9 s1) and then prove that the total momentum of a system is conserved if the net external
force is zero 5points (LN A16 Ch9 s2) [6points]
2. Prove that for an elastic collision (K is conserved) v2 final  v1 final  (v2initial  v1initial )
(LN A16 Ch9s5)
3. Prove, starting from
F
net i
i
  mi ai that the Center of Mass motion obeys
i
M aCM   F netExternal i provided that we define rCM as M rCM   mi ri . (LN A17 Ch9 s8) –
i
i
[6points] .
4. Starting from the definition of torque   r  F prove that for a Rigid Body rotating around an
axis we get   I where I 
m r
i i
2
(LN Ch10 s5) [6points]
i
5. Starting from the definition of angular momentum, L / o  r  p , prove that
d L/ O
  /O
dt
[6points]
6. Theorem 1 of rotational dynamics: State the restrictions on the motion of CM, if any, of the law
 / CM  I / CM  / CM [3points]
7. Theorem 2 of rotational dynamics: Starting from the definition of the total Kinetic energy for a
1
2
system of masses: K   mi vi 2 , prove that for a rigid body: K 
1
1
Mv 2CM  I / CM / CM 2
2
2
{Hint: rewrite ri as ri  RCM  i }. [10 points]
8. Bernoulli’s Equation: 1. Derive the work of the Normal forces on a section streamline tube in a
steady state flow in terms of the pressures at either end PA and PB. 2. What theorem did we use
to derive Bernoulli’s equation? [6 points]
9. State and prove Archimedes’ principle
Short exercises. (3 problems graded on an 18 points total)
1. Compute the cross product of V  2i with the vector
W  (3, 2, 0) [5points]
2. What is the torque exerted by the force F at right around O,
around O’ and around P, respectively? [6points]
3. State the center of mass location of an arbitrary triangle and
then prove that it is so. [6points]
4. A mass M=5kg is located 2m from a mass m=1kg. Compute the
center of mass of the system and determine its distance from M. [6points]
5. A spacecraft approaches Jupiter at 20km/s while Jupiter itself is moving at 13km/s. The
spacecraft does ½ a turn around the planet and is then leaving Jupiter’s gravity in a direction just
opposite to its initial approach of the planet. What is the final velocity of the spacecraft. Justify
the equation you use and the approximation you make. [8points]
6. A car of mass m=750kg travelling a 20m/s has a head on inelastic collision with a truck of mass
M=30,000kg travelling at 15m/s the opposite way. What is the final velocity of the two,
assuming the positive axis in the initial direction of the car? [6points]
7. A skater of body inertia I=3kgm2 is spinning at 30rd/s with arms extended each at 1m from her
axis of rotation. All of a sudden she gets two 3kg weights dropped in her hands. What is her
final angular velocity after grabbing the 3kg weights? [8points]
8. Derive the maximum height at which a pump can pump water up from a stream, if the pump is
located at the top of the pipe. (water=1000kg.m3) [8points]
9. A hydraulic car lift uses a cylinder of radius R. The manual pump that operates the car lift uses a
piston of radius r=0.005m. The maximum force used to operate the pump is 200N. What must
be the radius of the lift cylinder to be able to lift cars of mass m=1200kg? [8points]
10. A see-saw is in equilibrium with an m=20kg little girl located at 2.5m from the fulcrum. The
chubby boy on the other side is 1.2m from the fulcrum. What is the chubby boy’s weight?
[6points]
11. An eraser is thrown upward with an initial velocity of 25m/s at 45 degrees from the horizontal
and with an initial angular velocity around its CM of 10 rd/s. What is the angular velocity of the
eraser at the top of its trajectory? [6points]
12. A bullet of mass m=0.015kg and traveling a 800m/s strikes a uniform rod of mass M=2kg and
length l=0.8m rotating horizontally with an angular velocity of =2 rd/s around a pivot point at
its end ( I=1/3 Ml2 ). At what distance away from the pivot point should the path of the bullet
be in order to stop the rod (assume the bullet gets embedded in the rod). [8points]
Typical Problems. ( 36 points assigned on final)
1. Ballistic pendulum: A 150g bullet moving horizontally imbeds itself in a 5kg block of wood
hanging by a 2m rope from the ceiling. After impact the block reaches a maximum height when
the rope makes an angle of 25o with the vertical. What was the speed of the bullet before
impact? State which laws you are applying. [12 points]
2. A m=2kg ball of play-dough is thrown at v=10m/s at the edge of a door of inertia I=2.5 kgm2 and
width d=0.75m, opened by an angle of 30o. What is the minimum time it will take the door to
close after the ball has stuck to it? [12 points]
3. The massless string of a m=0.05kg yoyo is wound on it at r=0.02m. When released, the yoyo is
accelerating downward at 2m/s2. What is the inertia of the yoyo? Assuming the mass of the
yoyo is concentrated on its outer rim what is its radius? [12 points]
4. Derive the thrust Ma on a rocket in terms of the exhaust velocity ve of the burnt fuel and the
rate =dM/dt at which the burnt fuel is expelled. (LN Ch9s10) [12 points]
5. A mass m hangs from a massless rope wound around a pulley of mass M concentrated on its
outer rim where the rope is wound. Find the downward acceleration a of the mass m. [12
points]
6. Same set up as previous problem, but now find the velocity of mass m when it has dropped by a
height h [12 points]
7. An Atwood machine has a pulley with inertia I, a mass m1 on the left resting on the floor and a
mass m2 on the right a height h above the floor. (m2>m1). What is the angular velocity of the
pulley when m2 hits the floor? [12 points]
8. A heavy pulley with a sheave of radius R=0.15m is rotating under the influence of a mass m=0.7
kg hanging from a rope wound around the sheave. The mass m is accelerating downward at
2m/s2. What is the inertia I of the pulley? [12 points]
9. A hollow cylinder of mass M rolls without slipping down an incline at  from the horizontal.
What is the velocity of the cylinder’s CM after it has travelled a distance l along the incline after
starting from rest. Justify all steps! [12 points]
10. A cylindrical tank of radius R filled with water to a height H leaks through a hole of radius r at a
height h (h<H) above the bottom. At what rate (volume/time) is the tank loosing water? [12
points]
11. An airplane weighs 8 104 N and has a total wing area of A=40m2 (includes both wings). The air
path over the wings is on average 20% longer than under the wings. Derive a working estimate
of how fast it should be traveling in order to be airborne. (Hint: Assume that vbot=Air speed, and
that it should take the same time for an air molecule to travel top and bottom in order to
recombine behind the wing [12 points]
Vectors: V  (Vx , Vy )  (V cos  , V sin  )  Vx iˆ  V y ˆj
V  Vx 2  V y 2
tan  
Vy
Vx
Kinematics quantities:
r  ( x  r cos  , y  r sin  )
dx
dy
v  (vx  , v y  )
dt
dt
dv y
dv
a  (ax  x , a y 
)
dt
dt
Uniformly Accelerated Motion (along x-axis for example):
1 2


 x  ax t  vox t  xo 
2


vx  ax t  vox

we can also use: v 2  vo 2  2a ( x  xo )
to replace one of the 2 equations above.
Useful integrals and derivatives:
1 n 1
d (ax n )
x C
 ax n 1
n 1
dx
d ( Sin(t   ))
d (Cos (t   ))
 Cos (t   )
  Sin(t   )
dt
dt
Dynamics:
n
 x dx 
FNet  ma
mM
r2
FS   kx
FG  G
K
v2
 r 2
r
  s FN
Ffk  k FN
a
FfsMax
1 2
mv
2
W  K
W  U
r2
W1 2   F dr
E U  K
r1
M rCM   mi ri   r  dm
d pCM
 M aCM  Fnet ext .
dt
Rotational Dynamics:
p  mv
  r  F =I 
I SolidCylinder 
KTotal 
L  r  mv  I 
1
Mr 2
2

1
I Rod / End  Mr 2
3
dL
I   mi ri 2   dmr 2
dt
1
1
I Rod / CM  Mr 2 I / O  I / CM  Md 2
12
2
1
1
MvCM 2 + I / CM  2
2
2
Fluids:
P  Fn / A
PB  PA   gh
 A S A v A  B SB vB
Physical Constant and values:
G  6.57 *1011 SI
REARTH  6400km
M EARTH  6  1024 kg
g=9.81m/s 2
1
1
 v A 2 + ghA  PA =  v B 2 + ghB  PB
2
2