Download Kinematics Multiples

Springs
From Princeton Review Book
1. A spring of force constant 800 N/m is hung from a ceiling. A block of mass 4.0 kg is
hung from its lower end and allowed to come to rest. How far will the block stretch the
spring?
a. 0.49 cm.
b. 0.98 cm
c. 3.2 cm.
d. 4.9 cm.
e. 9.8 cm
*D. When the block comes to rest, it is in equilibrium. This means that the downward
pull of the Earth is exactly balanced by the upward force of the spring (watch units!)
F  0
mg  Kx  0
x
mg (4kg)(10m / s 2 )

 .05meters  5cm
K
800 N / m
2. A block of mass m begins at rest at point A. As it moves back toward the wall due to
the force exerted by the stretched spring, it is also acted upon by a frictional force whose
strength is given by the expression bx, where b is a positive constant. What is the block’s
speed when it first passes through the equilibrium position (x = 0)?
a. A
( K  b)
m
b. A
( K  b)
m
c. A
( 1 2 K  b)
m
d. A
( 12 K  b)
m
e. A
1
2
( K  b)
m
*B. Because of the frictional force, energy is NOT conserved. First, calculate the work
done by friction—since the force is not constant, we must INTEGRATE!!!
A
A
W   F cos dx   (bx)( 1)dx 
0
0
bx 2
2
A
0

 bA 2
2
Now apply work/change in TME:
TME A  Wnc  TME B
1
2
KA 2  (-
bA 2
)  12 mv 2
2
Solve for v 2 
A 2 ( K  b)
m
So v  A
(K - b)
m
From Old AP’s
3. An ideal spring obeys Hooke’s law: F=-Kx. A mass of 0.50 kg when hung vertically
from this spring stretches the spring 0.075 meters. The value of the force constant for the
spring is most nearly: (most nearly means g = 10 m/s2)
a. 0.33 N/m.
b. 0.66 N/m.
c. 6.6 N/m.
d. 33 N/m.
e. 66 N/m
*E. Although it does not say so, whenever a mass is hung vertically—assume it is a static
problem—that it is not dropped and allowed to bob up and down. Assume it is placed
gently and allowed to come to rest. This means equilibrium:
F  0
mg  Kx  0
(.5kg)(10m / s 2 )  K (.075m)  0
K  66.6 N / m
Round down to 66 becuase we rounded g UP from 9.8 to 10, so we have an overestima te.
You would get 6.6 N/m if you forgot to multiply by g.
You would get .66 if you divided by g.
4. (1993) Two identical massless springs are hung from a horizontal support. A block of
mass 1.2 kg is suspended from the pair of springs, as shown above. When the block is in
equilibrium, each spring is stretched an additional 0.15 meters. The force constant of
each spring is most nearly:
a. 40 N/m
b. 48 N/m.
c. 60 N/m.
d. 80 N/m.
e. 96 N/m.
*A. Draw a FBD. There are TWO upward forces = Kx countering the downward weight.
Setting them equal for equilibrium:
2 Kx  mg
K
mg (1.2kg)(10m / s 2 )

 40 N / m
2x
2(.15m)
5. (1974) A spring which does NOT obey Hooke’s law supplies a force of magnitude
Ax2, where x is measured downward from the equilibrium position of the unloaded spring
and A is a constant. A mass m is attached to the end of the spring and is released from
rest at x = 0. What is the maximum downward displacement?
a.
mg
A
c.
mg
A
e.
3mg
A
b.
d.
3mg
A
2mg
A
*B. Since the mass is changing height, you should conserve energy. Define sea level as
X=0 labeled in diagram. HOWEVER, since the spring is nonlinear, you will have to
derive the potential energy function for it yourself.
6. (1993) A block on a horizontal frictionless plane is attached to a spring, as shown
above. The block oscillates along the X-axis with turnaround points of plus and minus A.
Which of the following statements about the block is correct?
a. At x = 0, its velocity is zero.
b. At x = 0, its acceleration is at a maximum.
c. At x = A, its displacement is a maximum.
d. At x = A, its velocity is at a maximum.
e. At x = A, its acceleration is zero.
*C. Since this is the turnaround point, this is the greatest displacement.
7. (1993) A block on a horizontal frictionless plane is attached to a spring, as shown
above. The block oscillates along the X-axis with turnaround points of plus and minus A.
Which of the following statements about energy is correct?
a. The potential energy of the spring is at a minimum at x = 0.
b. The potential energy of the spring is at a minimum at x = A.
c. The kinetic energy of the block is at a minimum at x = 0.
d. The kinetic energy of the block is at a maximum at x = A.
e. The kinetic energy of the block is always equal to the potential energy of the
spring.
* A. Since the spring is unstretched at x = 0, there is no stored energy.
8. (1984) Which of the following is true for a system consisting of a mass oscillating on
the end of an ideal spring?
a. The kinetic and potential energies are equal at all times.
b. The kinetic and potential energies are both constant.
c. The maximum potential energy is achieved when the mass passes through its
equilibrium position.
d. The maximum kinetic energy and maximum potential energy are equal, but
occur at different times.
e. The maximum kinetic energy occurs at maximum displacement of the mass
from its equilibrium position.
* D. Since energy is conserved, the max. KE at equilbrium when there is no stored
energy equals the Elastic Potential Energy at a turnaround point when there is no KE.
9. A block oscillates without friction on the end of a spring as shown above. The
minimum and maximum lengths of the spring as it oscillates are, respectively, Xmin and
Xmax. The graphs below can represent quantities associated with the oscillation as
functions of the length X of the spring. Which graph can represent the total mechanical
energy of the block-spring system as a function of X?
* E. Trick question. The TME is constant.
10. A block oscillates without friction on the end of a spring as shown above. The
minimum and maximum lengths of the spring as it oscillates are, respectively, Xmin and
Xmax. The graphs below can represent quantities associated with the oscillation as
functions of the length X of the spring. Which graph can represent the kinetic energy of
the block as a function of X?
* D.
The KE must be zero at the turnaround positions, which limits the choice to C or D.
To find the velocity as a function of position, look at TME:
1
2
mV 2  12 Kx 2  TME
So V 
2(TME - 12 Kx 2 )
m
11. When an mass on a spring that is moving back and forth between its two turnaround
positions is at its maximum displacement from the equilibrium position, which of the
following is true of the values of its speed and the magnitude of the restoring force?
Speed
Restoring Force
a.
Zero
Maximum
b.
Zero
Zero
c.
½ Maximum
½ Maximum
d.
Maximum
½ Maximum
e.
Maximum
Zero
*A
Speed is zero by definition of turnaround point.
Restoring force is maximum since spring is maximally stretched
(equilibrium is no net restoring force)
12. (1974) An object on the end of a spring vibrates along a vertical line with simple
harmonic motion of amplitude D as shown above. The kinetic energy is a maximum at:
a. position X only.
b. position Y only.
c. position Z only.
d. both positions X and Z.
e. all positions since the kinetic energy is constant.
* B.
Even though the equilibrium position is not unstretched for vertical springs, the KE is
still a maximum at equilibrium (because by definition of equilibrium, the potential energy
function is a minimum).