Download Problem 1 (20 points)

PHYS 310: Exam1 September 22, 2010
Name:
Problem 1 (10 points). Express the vector 2i + 3j –k in the primed triad i’j’k’ in which the x’y’axes are rotated about the z-axis (which coincides with the z’-axis) through an angle of 30.
3
iˆ  iˆ  cos 30 
2
ˆj  iˆ  sin 30  1
2
kˆ  iˆ  0
1
iˆ  ˆj    sin 30  
2
ˆj  ˆj   cos 30  3
2
kˆ  ˆj   0
iˆ  kˆ  0
ˆj  kˆ  0
kˆ  kˆ  1
 3 1

3

0

 3 2 
2
2
 2  
 Ax  


   3
3
A    1

 y   2 2 0   3    2 3  1
  1
 Az  
0 1     1 
 0






A  3.232iˆ  1.598 ˆj   kˆ
PHYS 310: Exam1 September 22, 2010
Name:
Problem 2 A 9.00-kg object starting from rest falls through a viscous medium and experiences a



resistive force R  bv , where v is the velocity of the object. The object reaches one-half its
terminal speed in 5.54 s.
(a) (10 points) Determine the terminal speed.
(b) (10 points) At what time is the speed of the object three-fourths the terminal speed?
(c) (10 points) How far has the object traveled in the first 5.54 s of motion?
PHYS 310: Exam1 September 22, 2010
Name:
Problem 3 (30 points). A 50.0-g object connected to a spring with a force constant of 35.0 N/m
oscillates on a horizontal, frictionless surface with an amplitude of 4.00 cm. Find
(a) (10 points) the total energy of the system
(b) (10 points) the speed of the object when the position is 1.00 cm.
(c) (10 points) the kinetic energy and the potential energy when the position is 3.00 cm.
PHYS 310: Exam1 September 22, 2010
Name:
Problem 4 (30 points)
A mass m moves along the x-axis subject to an attractive force given by 17β2mx/2 and a
retarding force given by 3mx , where x is its distance from the origin and β is a constant. A
driving force given by mAcos t , where A is constant, is applied to the particle along the x-axis.
(a) (10 points) What is the equation describing the motion of the mass?
(b) (10 points) What value of  results in steady-state oscillations about the origin with
maximum amplitude?
(c) (10 points) What is the maximum amplitude?
(a) mx  3 mx 
17 2
 mx  0mAcos t
2
(b)
3
17 2

2
2
 r 2   2  2 2  4  2
   and  2 
(c)
Amax
derivation of Amax
25
F
 d2   2   2   2

4
2m d
2A

15  2
r  2
 d 
5

2