Goal: Problem solving
Source: UMPERG
A quantity of gas is confined to a cylinder. The cylinder is vertical
and capped by a moveable piston of mass 2 kg and area 0.1 m2.
The gas is heated until the piston rises 20 cm. The amount of work done
by the gas is most nearly
Goal: Interrelate and contrast the concepts of work, kinetic energy and impulse.
Source: UMPERG-ctqpe96
Compare two collisions that are perfectly inelastic. In case (A) a car
traveling with velocity V collides head-on with a sports car having half
the mass and traveling in the opposite direction with twice the speed.
In case (B) a car traveling with velocity V collides head-on with a
light truck having twice the mass and traveling in the opposite
direction with half the speed. In which case is the work done on the
car during the collision the greatest?
(4) The total momentum of both systems is zero, so after the collision
there is no KE in either system. System (A) has more kinetic energy
initially. There is no way, however, to determine how much of the
kinetic energy in the combined system of the two vehicles is dissipated
in the automobile as opposed to the other vehicle.
This question serves only to provoke a discussion of the dissipation of
energy in a collision. Students are tempted to assume that each
vehicle must absorb its own initial KE.
How do the forces acting on the car in the two cases compare?
Which collision takes longer?
Which vehicle do you think will suffer the greatest damage?
Promote a discussion of auto safety.
Goal: Hone the scalar nature of work and distinguish work from impulse.
Source: UMPERG-ctqpe74
A block having mass M travels along a horizontal frictionless surface
with speed v. What is the LEAST amount of work that must be done on
the mass to reverse its direction?
(3) Zero work must be done. Students will likely become entangled in
the sign of the work as well as the interpretation of the requirement to
“reverse its direction”. The most defensible answer after (3) is (2).
Some students may confuse the sign of the work, Students who choose (2)
or (4) have career potential as a lawyer.
This is an excellent problem for engaging students in a discussion of
work and energy. A mass traveling in the opposite direction with the
same speed would have the same kinetic energy. The work-kinetic energy
theorem then states that no net work need be done on the mass. The
work-kinetic energy theorem also resolves any ambiguity in the sign of
the work if the mass is just brought to rest.
Draw a diagram indicating the direction of motion and the direction of
the force acting on the mass. What is the direction of the
displacement?
If the surface had friction and the mass just slid until it stopped, how
much work would the friction force do?
It is easy to demonstrate several situations for which an object
reverses its direction and no new work is done. All it requires is a
conservative force. For example, let a ball roll up an incline and then
back down. Or, allow a mass to encounter a spring. Or, have a marble
roll around a semicircular track. This latter case is interesting
because the force acting on the mass (Normal) does no work.
Goal: Recognize physical conditions under which conservation principles hold.
Source: UMPERG-ctqpe134
A
child is standing at the rim of a rotating disk holding a rock. The
disk rotates without friction. The rock is dropped at the instant
shown. What quantities are conserved during this process.
(3) is the correct response if the rock is simply dropped. Some
students may fail to include the rock as part of the system after it is
dropped.
Objects traveling in a straight line do have angular momentum with
respect to any origin that is not on the path of the object. The rock
does not cease to have angular momentum with respect to the center of
the disk when it is dropped. Although the angular momentum and energy of
the rock will change as the rock falls, its angular momentum and energy
just after it is dropped are the same as just before.
Does the rock have angular momentum (or energy) just before it is
dropped? just after it is dropped?
If energy (angular momentum) is lost, what happens to it?
Changes in angular momentum are caused by a net torque. What torques
act on the system?
Have students relate their answer to this question to the previous one.
Goal: Interlink several dynamical concepts and associate with a physical process.
Source: UMPERG-ctqpe98
Mass M1 is traveling along a smooth horizontal surface and
collides with a mass M2 (stationary) which has a spring
attached as shown below.
The spring between the blocks is most compressed when
5: When the spring is maximally compressed, both masses have the same
velocity which is the velocity of the center of mass.
The total energy of an isolated system can be decomposed into three
categories; kinetic energy associated with center of mass motion,
kinetic energy of bodies in the center of mass coordinate frame and
potential energy associated with the interaction of bodies comprising
the system.
Is M2 ever traveling faster than M1?
Do the masses ever have the same velocity?
How would you find Pcm, the momentum of the center of
mass?
How is the kinetic energy of the center of mass related to its
momentum?
A
sketch of the velocities of the two masses over time would look
something like the graph at the right. [Note that the velocity of
M1 can be negative after the collision if it is less massive
than M2.] Such a graph helps make the relationships between
Vcm, the relative velocity of the masses and the spring
compression clear.
Goal: Link energy and kinematic quantities.
Source: UMPERG
Two masses, m and M, are released from rest at a height H above the
ground. Mass m slides down a curved surface while M slides down an
incline as shown. Both surfaces are frictionless and M > m.
Which of the following statements is true?
(4); even though both blocks arrive at the bottom with the same speed, m
has a larger initial acceleration and attains a larger speed faster than
M, despite having to travel a slightly longer distance. This item helps
to focus attention on identifying those salient characteristics of the
problem that relate to the time it takes the blocks to slide down the
ramps. Some students will cue on the distance traveled, some on the
differing masses of the blocks, some on m picking up speed faster than
M.
The curved surface makes it impossible for students to use either
kinematics or Newton’s Second Law to determine the exact time it takes m
to reach the bottom. Some students may correctly conclude that both
blocks arrive at the bottom with the same speed, and thereby erroneously
conclude that this must mean they arrive at the same time as well.
The curved track case also offers an opportunity to explore whether
students realize that the total work done by the gravitational force
goes into changing the kinetic energy of the block, even with a normal
force present since this normal force does no work on the block.
What features of the problem determine the time it takes the masses to
reach the bottom?
What’s the same about both blocks if they are released from the same
height? What’s different?
Does traveling a shorter distance always mean less time?
For those who answered (1), ask what would happen to the time it would
take M to reach the bottom if the 45° angle were made more, or less
steep (think of the top vertex of the triangle being on a hinge).
Clearly in the limit where M would drop vertically a distance
SQRT(H2+L2), the time it would take to reach the
other vertex of the hypoteneuse would be shorter than for any angle less
than 90°.
Commentary:
Answer
(1) This problem helps interrelate concepts from mechanics and
thermodynamics. The work can be determined from the work done against
the gravitational force.