Week 15 page 1 of 11
CT15-1. A bead of mass m slides on a circular wire
of radius R. The wire rotates about a vertical axis with
angular velocity . What is the kinetic energy of the
bead?
1
1
2 2
2 2
A) T  mR   mR 
2
2
1
1
2 2
2
2
2
B) T  mR   mR cos  
2
2
1
1
2 2
2
2
2
C) T  mR   mR sin  
2
2
1
1
2 2
2 2
D) T  mR   mR 
2
2
2
1
2
T

mR



sin

E)
2

2/6/2016


 University of Colorado at Boulder

Week 15 page 2 of 11
CT15-2. A bead of mass m slides on a circular wire
of radius R. The wire rotates about a vertical axis with
angular velocity . When the bead is at angle, how
high is the bead above the lowest point of the wire?
A) h  Rcos
B) h  Rsin
C) h  R  sin  cos 
D) h  R 1  sin 

E) h  R 1  cos 
2/6/2016
 University of Colorado at Boulder
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Week 15 page 3 of 11
CT15-3. A bead of mass m slides on a circular wire
of radius R. The wire rotates about a vertical axis with
angular velocity . When the bead is at angle, what
is the  component of the centripetal force on the
bead?
2 2
A) Fc  mR  cos
2
B) Fc  mR sin

2 2
C) Fc  mR  sin
2 2
D) Fc  mR  sin cos
2
E) Fc  mR sin cos
2/6/2016
 University of Colorado at Boulder
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Week 15 page 4 of 11

CT15-4. A bead of mass m slides on a circular wire
of radius R. The wire rotates about a vertical axis with
angular velocity . The equation of motion of the
bead is
g
  sin  2 sin cos  0
R

What are the equilibrium value(s) of ?
A)   0
1  g 

B)   cos 
 R2 

1  g


sin

C)
2 
 R 
1  g 
1  g 


0
and


cos


0
and


sin




D)
E)
 R2 
 R2 
2/6/2016
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Week 15 page 5 of 11
CT15-5. A bead of mass m slides on a circular wire of
radius R. The wire rotates about a vertical axis with
angular velocity . The equation of motion for small

1  g


cos

motions about the equilibrium 0
2  is
 R 
   2 sin 20   0
What is the oscillation frequency of the bead?
2
A)   
B)   
2
2
C)    sin0
D)    sin 0
E)    sin
2/6/2016
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
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Week 15 page 6 of 11
CT15-6. A block of mass m1 slides on an inclined plane of mass m2. The
Lagrangian of the system is
2
1
1
1
L  x , x , y, y   m1  y cos  x   m1y 2sin 2  m 2 x 2  m1gy sin
2
2
2
What is the Lagrangian equation of motion for
the variable x?
A) m1x  0
m1
B)  m1  m2  x  0
C) m2 x  m1 cos y  0
D) x  cos y  0
m2
E)  m1  m2  x  m1 cos y  0
y

x
2/6/2016
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Week 15 page 7 of 11
CT15-7. A block of mass m1 slides on an
inclined plane of mass m2. The equations of
motion are
m1
x
g sin cos
m1sin 2  m 2
m1  m 2
y
g sin
m1sin 2  m 2
What is a good approximation to these
equations in the limit of a very heavy plane?
y  g sin
A) x  0
m1
m1  m2
x

g
sin

cos

y

g sin
B)
m2
m2
y  g sin
C) x  g sin cos
cos
1
yg
D) x  g
sin
sin
cos
yg
E) x  g
sin
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m1
y
m2
 University of Colorado at Boulder

x
Week 15 page 8 of 11
CT15-8. A particle of mass m slides on the
outside of a cylinder of radius a. A good choice
a 
of generalized coordinates is (r,) What are the
kinetic and potential energy of the particle? 
1
T  mr 22
2
A)
U  mgr cos
1
T  mr 22
2
B)
U  mgr sin
1
T  m r 2  2
2
D)
U  mgr cos

2/6/2016





1
T  m r 2  r 2 2
2
C)
U  mgr sin
1
T  m r 2  r 2 2
2
E)
U  mgr cos
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Week 15 page 9 of 11
CT15-9. A particle of mass m slides on the
outside of a cylinder of radius a. What
condition must be satisfied by the force of
constraint at the point (angle where the
particle leaves the cylinder? 
A) Qr  mg
C) Qr  mg sin
E) Q r  0
2/6/2016
a 
B) Q r   mg
D) Qr  mg cos
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Week 15 page 10 of 11
CT15-10. A particle of mass m slides on the
outside of a cylinder of radius a. The particle

leaves the cylinder at angle and speed v,
where
2
cos0 
3
2ag
v
3
Once the particle leaves the cylinder, what is its height y(t) above the
ground?
1 2
1 2
A) y  a  v sin0 t  gt
B) y  a  v sin0 t  gt
2
2
2a
1 2
2a
1 2
C) y   v sin0 t  gt
D) y   v sin0 t  gt
3
2
3
2
2a
2
E) y   v sin0 t  gt
3
2/6/2016
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y
Week 15 page 11 of 11
CT15-11. A particle of mass m slides on the
outside of a cylinder of radius a. The particle
leaves the cylinder at angle and speed v,
where
2
cos0 
3
2ag
v
3
The particle then flies through the air for time


4ag
t f  v sin0  1  2 2  1
3v sin 0


How far from the center of the cylinder does the particle land?
A) x  vt f cos0  a sin0
B) x   vt f  a  sin0
C) x  vt f sin0  a cos0
D) x   vt f  a  cos0
E) x  vt f  a sin0
2/6/2016
 University of Colorado at Boulder
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x