CSULA
I)
FINAL EXAM
(SOLUTION)
PHYSICS 211
Summer’12
Professor: Rafael Obregon
CONCEPTUAL QUESTIONS
1) A woman walks first at a constant speed of v1 along a straight line, a distance d, from point A to point B and then
back along the line d from B to A at constant speed of v2. Derive an expression for her average speed (in m/s) over
the entire trip in terms of both speeds (v1 and v2). (Recall: vavg = total distance/total time)
d represent the distance between A and B.
t1 is the time for which she walks from A to B at speed v1: t1  d / v1
t 2 is the time for the return trip from B to A at speed v2: t 2  d / v2
The average speed is: vavg = Total distance/Total time → v  d  d 
avg
t1  t 2
2v  v
2d
2d

 1 2
d / v1  d / v 2
 v  v1  v1  v 2

d  2
 v1  v 2 
2) In the figure, the coefficient of kinetic friction between the surface and the blocks is µk. If M = 1.0 kg, find an expression for
the magnitude of the acceleration (a) of either block (in terms of F, µk, and g).
From FBD for the right mass, equation of motions will be:
F  T  f k1  3Ma
f k1  k n1  k (3M ) g
&
From FBD for the left mass, equation of motions will be:
T  f k 2  Ma
f k 2  k n2  k (M ) g
&
Replacing M = 1 kg, the first equation of each mass will become:
F  T  3 k g  3a
T  k g  a
(1)
(2)
Adding equation (1) and (2):
F  4  k g  4a  a 
F  4 k g F
  k g
4
4
3) A certain spring that obeys Hooke’s law is stretched by an external agent. The work done in stretching the spring by 10 cm is
4J. How much additional work is required to stretch the spring an additional 10 cm?
1
2
4.00 J  k  0.100 m  . Therefore k  800 N m and to stretch the spring to 0.200 m requires extra work
W 
2
1
 800 0.2002  4.00 J  12.0 J
2
4) A glider of mass m is free to slide along a horizontal air track. It is pushed against a launcher at one end of the track. Think of
the launcher as a light spring of force constant k compressed by a distance x. The glider is released from rest.
From an equation that establishes the conservation of energy of the system spring-mass, show that the glider attains a final
speed of: v  x k / m .
K  U s i  K  U s  f
 0  12 kx2  12 mv 2  0  kx2  mv 2  v 
kx2
k
x
m
m
II) Solve
1) (10 pts). A projectile is fired in such way that its horizontal range is equal to four times its maximum height. Find the angle
of projection.
(You may use the trig expression: sin 2  sin  cos )
R
vi2  sin 2 i 
&
g
h
vi2 sin 2  i
2g
; Since R  4h
→
4vi2 sin 2  vi2 sin 2 vi2 2 sin  cos


 2 sin   2 cos
2g
g
g
Thus: tan   1    45 
2) (10 pts). The only force acting on a 2.0 kg body moving along the x axis is given by Fx = (8.0x) N, where x is in m. If the
velocity of the object at x = 0 is +3.0m/s, how fast is moving at x = 2.0 m?
(Hint: Use definition of Work as Area under the Curve and Work-Kinetic Energy Theorem)
Since: Wnet   2(8 x)dx  J  4 x 2 2  J  16 J
But: Wnet  K  K B  K A  12 mv 2f  12 mv i2


0


Therefore: 16J 
v 2f
0

 vi2  v 2f  (3m / s) 2  v f 
 16  9 m / s  5.0m / s
3) An object of mass m1 placed on a frictionless, horizontal table is connected to a string that passes over a pulley and then is
fastened to a hanging object of mass m2 (m2 > m1) as shown in the figure.
A) (2 pts). Draw a Free-Body Diagram (FBD) for each block showing all forces acting ON
each mass (label them all correctly).
B) (2 pts). Write Newton’s 2nd Law equations of motion for each mass
Mass 1
Mass 2
T  m2 g  m2 a
 F  ma → T  m1a
x
 F  ma → or
(1)
(2)
y
m2 g  T  m2 a
C) (3 pts). From part B, derive and expression for the magnitude of the acceleration of the system (in terms of both masses
and g).
m2 g
Adding (1) and (2): m2 g  m1  m2 a  a 
m1  m2
D) (3 pts). From equations on parts B and C, derive and expression for the tension of the string (in terms of both masses and
g).
m1m2
g
Using equation (1): T  m1a 
m1  m2
4) A puck of mass m1 is tied to a string and allowed to revolve in a circle of radius R on a frictionless, horizontal table. The
other end of the string passes through a small hole in the center of the table, and an object of mass m2 is tied to it (see figure).
The suspended object remains in equilibrium while the puck on the table revolves with constant speed v.
A) (2 pts). Write Newton’s 2nd Law equations of motion for mass 2.
Mass m2 is in equilibrium:
F  T  m g  0
y
2
or
T  m2 g
B) (2 pts). Identify and write an expression for the radial force (Fc) acting on the puck. (in terms of m2 and g).
The tension in the string provides the required centripetal acceleration of the puck.
Thus,
Fc  T  m2 g
C) (6 pts). From part B derive and expression for the speed (v) of the puck (in function of m1, m2, R and g).
From: Fc 
m1 v 2
we have v  RFc 
R
m1
 m2 

 gR
 m1 
5) A block of mass m is released from rest at the top of a frictionless incline that makes an angle θ with the horizontal (see
figure). The block slides a distance d down the incline from the point of release. While sliding, it contacts an unstressed
spring of negligible mass with constant k. It is observed that the mass is brought momentarily to rest after compressing the
spring an additional distance x.
(Hint: Choose the zero point of gravitational potential energy of the object-spring-Earth system as the configuration in which
the object comes to rest. Since the incline is frictionless, for the total mechanical energy, we have
KB  U gB  U sB  K A  U gA  U sA )
A). (5pts) Write an equation that establishes the Principle of Conservation of Energy (in terms of m, g, h and x).
K B  U gB  U sB  K A  U gA  U sA 
0  mgh  0  0  0  12 kx2  mgh  12 kx2
B). (5pts) Show that the initial separation d between object and spring is given by: d 
kx2
x
2mg sin 
From the graph: s  (d  x)  h  h  (d  x) sin 
sin 
( d  x) 
Replacing h into equation from part B and solving for d:
kx2
kx2
d 
x
2mg sin 
2mg sin 
6) Two objects are connected by a light string passing over a light, frictionless pulley as shown in the figure. The object of mass
m1 is released from rest at height h above the table. Using the isolated system model:
A). (5pts) Write an equation that establishes the Principle of Conservation of Energy.
2
2
U i  Ki  U f  K f → 0  0  0  m1 gh  12 m1v  12 m2 v  0  m2 gh
B). (5pts) Determine the speed of m2 just as m1 hits the table
m1 gh 
1
 m1  m2  v2  m2 gh
2
→
v
2  m1  m2  gh
 m1  m2 
7) A bullet of mass m is fired into a block of mass M initially at rest at the edge of a frictionless table of height h (see figure).
The bullet remains in the block, and after impact the block lands a distance d from the bottom of the table.
A). (2pts) Using conservation of momentum from just before to just after the impact of the bullet with the block, find an
expression for the initial speed (vi) of the bullet (in terms of m, M and vf).
M m
mv i  M  m v f → v i  
v (1).
 m  f
B). (2pts) Taking the origin at the point where the block-embedded bullet leaves the table, write expressions for d and h at
any time t.
dvft
1
h  gt 2
2
&
C). (2pts) Using part B, find an expression for the time t of impact, when the block-bullet reaches the floor (in terms of h and
g)
2h
t
g
D). (2pts) Using equations from parts B and C, derive an expression for the final speed (vf) of block-bullet just after the
impact (in terms of h, g and d)
vf 
g
d
d

t
2h
gd 2
2h
E). (2pts) Finally, determine the initial speed (vi) of the bullet (in terms of M, m, g, d, and h).
Substituting vf into (1) from part A, gives:
2
 M  m  gd
vi  

 m  2h
8) (10pts) The free-fall acceleration on the surface of the Moon is about one-sixth that on the surface of the Earth ( g M  16 g E ).
The radius of the Moon is about 0ne-fourth the radius of the Earth ( RM  14 RE ).
Find the ratio of their average densities,  M /  E .
Recall that:   M / V
g
&
Vsphere  43 R 3
3
GM G  4 R / 3 4

  G R
2
2
R
R
3
→
gM 1 4 G M RM / 3
 
gE 6 4 GERE / 3
→
 M  gM  RE   1 
2


     4 
E  gE   RM   6 
3