Goal: Problem solving with rotational dynamics
Source: UMPERG-ctqpe148
A hoop
of mass 4 kg and radius 10 cm rolls without slipping down an incline
30° to the horizontal. The acceleration of the center of the hoop
is most nearly
Goal: Problem solving with dynamics
Source: UMPERG-ctqpe135.3
A
disk, with radius 0.25 m and mass 4 kg, lies on a smooth
horizontal table. A string wound about the disk is pulled with a
force of 8N. What is the acceleration of the disk?
(4) Students find it difficult to grasp that the angular dynamic
relationship does not replace, but rather augments, the 2nd law.
Goal: Problem solving
Source: UMPERG-ctqpe135.1
A
disk, having radius R and mass M, is free to rotate about an axis
through its center. A massless string is wound around disk and attached
to mass m. The moment of inertia for a disk given by is
1/2(MR2). If M=4m what is the acceleration of mass m?
(5) Students answering #2 are likely making the common mistake of
thinking that the tension in the string is mg.
Goal: Link acceleration to the slope of a velocity/time graph
Source: CT151.2-6
An
object’s motion is described by the graph above. The instantaneous
acceleration at t=10 sec is most nearly…
(1) Useful follow-up questions include; when does the object have
positive acceleration, when negative acceleration; does the object ever
stop?; when is it farthest from the origin?
Goal: Interpreting graphs
Source: CT151.2-5
An
object’s motion is described by the graph above. The average
acceleration during the first 10 s is most nearly…
(3) Students may have difficulty understanding what they are
asked. Recasting the problem in terms of areas helps. The only
contenders should be #2 or #3. Counting blocks should make it clear that
the result is much closer to #3.
Goal: Problem solving and developing strategic knowledge
Source: UMPERG-ctqpe101
You are given this problem:
A
block sits on a frictionless incline. Given the angle of incline, the
distance along incline, and the mass of block, find the acceleration
after traveling a distance d.
What principle would you use to solve the problem MOST EFFICIENTLY?
(2) The 2nd law is needed to find the acceleration. Students who
answer that only kinematics is needed are relying on memory.
Goal: Reasoning with dynamics
Source: UMPERG-ctqpe43
Block
m1 sits on block m2 and both sit on the floor of
an elevator at rest. When the elevator starts to move down, the normal
force on the upper block will …
(3) As it starts the elevator must accelerate downward and so
will the upper block. The only forces on the block are gravity and the
normal force. The normal force must diminish so gravity can provide the
downward acceleration.
Students answering #2 may have interpreted the question to mean ‘as the
elevator moves’ and think that the elevator moves with constant velocity.
Goal: Reasoning with circular motion
Source: UMPERG-ctqpe37
A small ball is released from rest at position A and rolls down a
vertical circular track under the influence of gravity.
When
the ball reaches position B, which of the indicated directions most
nearly corresponds to the direction of the ball’s acceleration?
Enter (9) if the direction cannot be determined.
(2) At position B the acceleration has a tangential component and
a radial component. Both components can be determined at position B.
Worked out carefully one gets 18 degrees above position #2. It is common
for students to neglect one component or the other.
Goal: Hone angular kinematic quantities and distinguish them from linear kinematic quantities.
Source: UMPERG
A mass moves in a circle with uniformly increasing anglular velocity.
As the angular velocity ω increases, the linear acceleration of
the mass has…
(4) This requires exploration. Some students may think that the
direction is changing because the acceleration points toward the center
of the circle. They may be unaware that there is also a component of the
acceleration in the tangential direction.
Some students may answer (3) thinking only of the radial acceleration
and that ‘towards the center’ is a direction.
Goal: Hone angular kinematic quantities and distinguish them from linear kinematic quantities.
Source: UMPERG
A mass moves in a circle with uniformly increasing angle.
As the angle θ increases, the linear acceleration of the mass has
…
(2) Students have a lot of difficulty reconciling linear kinematics with
angular kinematics. Unless shown how to take derivatives in polar
coordinates, or shown how to represent rotational kinematic quantities
as vectors, students can only memorize specific relationships.
Some students may answer (1) thinking that ‘towards the center’ is a
direction.
Commentary:
Answer
(4) Students should realize that the acceleration must be less
than a sliding mass on a frictionless surface would have which is #2.
Engage the students in a discussion of why the acceleration cannot
depend upon the radius.