Transcript Problem 1: Hunter in the Forest (25 points)

PHYSICS 218 Final Exam
Fall, 2006 STEPS
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For grader use only:
Problem 1 (20) ___________
Problem 2 (20) ___________
Problem 3 (10) ___________
Problem 4 (20)___________
Problem 5 (15) ___________
Problem 6 (20)___________
Total (105) ___________
Problem 1: (20 points)
Two blocks, with masses m 1 and m 2 , are stacked as shown and placed on a frictionless horizontal
surface. There is friction (coefficient of friction  ) between the two blocks. An external force is
applied to the top block at an angle  below the horizontal. a) Draw a free body diagram for each
of these blocks. b) What is the maximum force F that can be applied for the two blocks move
together.
F

m1
m2
N1
y
N1
N1
N2
m1
x
N1
F
m1 g
m2
m2 g
b) Fx  m1a x
Fx  m2 a x
F cos   N1  m1a x (1)
N1  m2 a x (3)
Fy  m1a y
Fy  m2 a y
N1  m1 g  F sin   0 (2)
N 2  N1  m2 g  0
N1  m1 g  F sin 
F cos   m1 g  F sin   m1a x
 (mg  F sin  )  m2 a x
F cos   m1 g  F sin  
m1 g (1 
F
m1
)
m2
cos    sin  (1 
m1
)
m2
m1
 (m1 g  F sin  )
m2
Problem 2: (20 points)
2
A spring with negligible mass exerts a restoring force Fx ( x)  x  x , if it is
stretched or compressed (  and  are known constants). A block of mass m is pushed
against the spring so that the spring is compressed by amount of A. When the block is
released, it moves along a frictionless, horizontal surface and then up the incline that has
coefficient of friction  .
Find the potential energy of the system before the block is released.
How far does the block travel up the incline before starting to slide back down?
1)
2)
N
2
1
F fr
mg h

1) U ( x)    Fx ( x)dx
x2
x3
U ( x)    (x  x )dx  
   Const
2
3
2
2) W
nonconservative

 mV12
mV22
 U (r2 ) 
 U (r1 ) 
2
2
A2
A3
U (1)  

2
3
U (2)  mgh
h
W fr 
sin
  mg cosdx  mg cos
0
h
sin 
h
A2
A3
 mg cos 
 mgh  

sin 
2
3
A2
A3


2
3
h
mg (1   cot  )
Problem 3: (10 points)

An object of mass m is at rest in equilibrium at the origin. At t=0 a new force F (t ) is applied that has
components
F x ( t )  c1 t
F y (t )  c 2  c 3 x,
where
c1 , c 2 , c 3
are known constants. Calculate the velocity vector as a function of time.
Fx  max
a x (t ) 
c1t
m
c1t
c1t 2
Vx (t )   a x dt   dt 
 Const
m
2m
Vx (0)  0  Const  0
c1t 2
c t3
dt  1  Const1
2m
6m
x(0)  0  Const1  0
x(t )   Vx dt  
x(t ) 
c1t 3
6m
Fy  ma y
c1t 3
c2  c3
6m
a y (t ) 
m
c2t c1c3t 4
V y (t ) 

 Const 3
m 24m 2
V y (0)  0  Const 3  0

c1t 2   c2t c1c3t 4  
j
V (t ) 
i  

2 
2m
 m 24m 
 c2t c1c3t 4 
c2t 2 c1c3t 5


y(t )   Vy (t )dt   

dt 

 Const 4
2 
2m 120m2
 m 24m 
y (0)  0  Const 4  0
c2t 2 c1c3t 5
y (t ) 

2m 120m 2

c1t 3   c2t 2 c1c3t 5  
j
r (t ) 
i  

2 
6m
 2m 120m 
Problem 4: (20 points)
A car in an amusement park rides without friction around the track. It starts with
velocity V0 at point A at height H.
Find the velocity of the car at point B. Denote it as V1 .
What is the radius R that the car moves around the loop without falling off.
1)
2)

ir

i
A
C
H
R
B
1) Conservation of energy:
mV02
mV12
 mgH 
2
2
 V02

V1  2
 gH 
 2

2) To find the maximum radius, we need to find the magnitude of critical
velocity for the car not to fall off the track. From the second law at point C:
Fr  mar
V2
 mg  m
R
V 2  gR
From conservation of mechanical energy at points A and C:
mV02
mgR
mgH 
 2 Rmg 
2
2
V02
gH 
2
R
5
g
2
Problem 5: (15 points)
Two masses, m1 and m2 , are attached by a massless, unstretchable string which passes over a pulley
with radius R and moment of inertia about its axis I. The horizontal surface is frictionless. The rope
is assumed NOT to slip as the pulley turns. Find the acceleration of mass m1 .
x
N2
T2
T2
m2
T1
m2 g
T1
m1
y
m1 g
Fy  m1a1 y
m1 g  T1  m1a1 y (1)
a x  a y  a (4)
a  R
(5)
I
R
m1 g  (T1  T2 )  (m1  m2 )a
T1  T2 
m1 g 
a
Ia
 (m1  m2 )a
R2
m1 g
m1  m2 
I
R2

 ext  I (rhr )
T2  m2 a2 x (2) RT1  RT2  I (3)
Fx  m2 a2 x
Problem 6: (20 points)
A bullet of mass m is fired with velocity of magnitude Vm into a block of mass M. The block is
connected to a spring constant k and rests on a frictionless surface. Find the velocity of the block
as a function of time. (Assume the bullet comes to rest infinitely quickly in the block, i.e. during
the collision the spring doesn’t get compressed.)
x
k
Vm
M
Px (before)  Px (after)
 mVm  (m  M )Vx
mVm
Vx  
mM
d 2x
Fx  ( M  m)a x  ( M  m) 2
dt
d 2x
 kx  ( M  m) 2
dt
2
d x
k

x0
2
dt
mM
x(t )  A cos  t  B sin  t;  
k
mM
dx
  A sin  t  B cos  t
dt
mVm
x(0)  A  0; V (0)  B  
mM
V (t ) 
V (t )  
mVm
cos  t
mM