Name ____________________
Mr. Willie
Hastings AP Physics C
Midterm Exam Review-3
A
1. The cart shown above is made of a block of mass m and four solid rubber tires each of mass
m/4 and radius r. Each tire may be considered to be a disk. (A disk has rotational inertia ½ ML2 ,
where M is the mass and L is the radius of the disk.) The cart is released from rest and rolls
without slipping from the top of an inclined plane of height h. Express all algebraic answers in
terms of the given quantities and fundamental constants.
(a) Determine the total rotational inertia of all four tires.
(b) Determine the speed of the cart when it reaches the bottom of the incline.
Free Response 1. (continued)
(c) After rolling down the incline and across the horizontal surface, the cart collides with a
bumper of negligible mass attached to an ideal spring, which has a spring constant k. Determine
the distance xm the spring is compressed before the cart and bumper come to rest.
(d) Now assume that the bumper has a non-negligible mass. After the collision with the bumper,
the spring is compressed to a maximum distance of about 90% of the value of xm in part (c). Give
a reasonable explanation for this decrease.
2. You are to perform an experiment investigating the conservation of mechanical energy
involving a transformation from initial gravitational potential energy to translational kinetic
energy.
(a) You are given the equipment listed below, all the supports required to hold the
equipment, and a lab table. On the list below, indicate each piece of equipment you
would use by checking the line next to each item.
____ Track
____ Cart
____ String
____ Meterstick
____ Electronic balance
____ Stopwatch
____ Set of objects of different masses
____ Lightweight low-friction pulley
(b) Outline a procedure for performing the experiment. Include a diagram of your
experimental setup. Label the equipment in your diagram. Also include a description of
the measurements you would make and a symbol for each measurement.
(c) Give a detailed account of the calculations of gravitational potential energy and
translational kinetic energy both before and after the transformation, in terms of the
quantities measured in part (b).
2. (continued)
(d) After your first trial, your calculations show that the energy increased during the
experiment. Assuming you made no mathematical errors, give a reasonable explanation
for this result.
(e) On all other trials, your calculations show that the energy decreased during the
experiment. Assuming you made no mathematical errors, give a reasonable physical
explanation for the fact that the average energy you determined decreased. Include
references to conservative and nonconservative forces, as appropriate.
3. A skier of mass m will be pulled up a hill by a rope, as shown above. The magnitude of the
acceleration of the skier as a function of time t can be modeled by the equations
a = amaxsinπt/T
=0
(0 < t < T)
(𝑡 ≥ 𝑇)
where amax and T are constants. The hill is inclined at an angle ɵ above the horizontal, and
friction between the skis and the snow is negligible. Express your answers in terms of given
quantities and fundamental constants.
(a) Derive an expression for the velocity of the skier as a function of time during the
acceleration. Assume the skier starts from rest.
(b) Derive an expression for the work done by the net force on the skier from rest until
terminal speed is reached.
3. continued
(c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed.
(d) Derive an expression for the total impulse imparted to the skier during the acceleration.
(e) Suppose that the magnitude of the acceleration is instead modeled as a = amaxe–πt/2T for all
t > 0 , where amax and T are the same as in the original model. On the axes below, sketch
the graphs of the force exerted by the rope on the skier for the two models, from t = 0 to a
time t > T . Label the original model F1 and the new model F2 .
mgsinɵ
T
Question 1
15 points total
Distribution
of Points
(a) 2 points
For determining the rotational inertia of each tire
1
1𝑚 2 1
𝐼 = 𝑀𝐿2 =
𝑟 = 𝑚𝑟 2
2
24
8
For the correct total rotational inertia for all 4 tires
1
𝐼𝑡𝑜𝑡 = 4𝐼 = 𝑚𝑟 2
2
1 point
1 point
(b) 7 points
For an indication of the conservation of mechanical energy
Etop = Ebottom ; ∆U=-∆K ; or equivalent
1 point
For correct expressions for energies at the top
Ktop = 0; Utop _ mgh + 4(mgh/4)= 2mgh
1 point
For a correct expression for potential energy at the bottom and
that kinetic energy at the bottom is the sum of
translational and rotational kinetic energies
Ubottom = 0; Kbottom = Ktrans + Krot
1 point
For a correct expression for translational kinetic energy at the bottom
Ktrans = 1/2 (2m)υ2 = mυ2
1 point
For a correct expression for rotational kinetic energy at the bottom
Krot = Iω2/2
1 point
For recognition of the relationship between translational and rotational velocity
ω=v/r
1 point
Substituting these expressions to determine total kinetic energy at the bottom
1 1
𝑣2 5
𝐾𝑏𝑜𝑡𝑡𝑜𝑚 = 𝑚𝑣 2 + ( 𝑚𝑟 2 ) 2 = 𝑚𝑣 2
2 2
𝑟
4
Setting potential energy at the top equal to the kinetic energy at the bottom
5mυ2/4 = mgh
For the correct solution for v
8
𝑣 = √ 𝑔ℎ
5
1 point
(c) 4 point
Distribution of
points
For recognition that energy is conserved (Although it is an inelastic collision,
the mass of the bumper is negligibly small, thus its kinetic energy is negligible
and there is no loss of energy.)
1 point
For a correct expression for potential energy of the spring at maximum
compression
UK = kxm2/2
1 point
For applying conservation of energy by equating the potential energy of
1 point
the spring at maximum compression EITHER to the gravitational potential
energy of the cart and wheels at the top of the inclined plane OR to the
kinetic energy of the cart and wheels at the bottom of the inclined plane
1
1
5
2
2
𝑘𝑥
=
2𝑚𝑔ℎ
OR
𝑘𝑥
=
𝑀𝑣 2
𝑚
𝑚
2
2
4
For a correct solution of either of these equations for xm (including a correct
substitution for v from part (b) for the second equation) or an answer consistent
with work done in (b)
𝑥𝑚 = 2√
1 point
𝑚𝑔ℎ
𝑘
(d) 2 points
For an explanation that discusses the inelastic collision with a loss of
mechanical energy or a reduced velocity resulting in a smaller compression.
The discussion should have been correctly stated. If there were any incorrect
statements, then 1 point was subtracted. Points were neither added or
subtracted for irrelevant statements, such as references to friction.
2 points
Question 2
15 points total
Distribution
of points
(a) 1 point
For choosing the meterstick and stopwatch, regardless of what else is checked
1 point
(b) 4 points
For a procedure that indicates the height needed to calculate gravitational potential
Energy
1 point
For a procedure that indicates distance and time measurements to calculate velocity
For a diagram and a clear indication of the height measurement
For a diagram and a clear indication of the distance measurement
1 point
1 point
1 point
Example #1
mc
mb
H
h






Use the electronic balance to determine the mass mc of the cart and the mass mb of one
object.
Attach the object to the cart using the string.
Place the cart on the track and hang the object so that the string passes through the pulley.
Allow the object to fall a distance h from its initial position to the floor, using the
meterstick to measure the distance fallen.
Use the stopwatch to measure the time t it takes the object to fall the distance h.
Measure the height H of the table.
Question 2 (continued)
Distribution
of points
(b) continued
Example #2
d
h





Use the electronic balance to determine the mass m of the cart.
Set the track at an incline, and measure the height h of the incline.
Place the cart at the top of the incline, and release from rest.
Using the stopwatch, measure the time t it takes for the cart to move down the
incline.
Measure the distance d that the cart moves down the incline.
(c) 6 points
For a clear indication of the initial potential energy of the system
For a clear indication of the final potential energy of the system
For a clear indication of the initial kinetic energy of the system
For a clear indication of the final kinetic energy of the system
For a correct calculation of the instantaneous velocity of the system
Example #1
Initial gravitational potential energy: Ug0 = mcgH + mbgh
Final gravitational potential energy: Ugf = mcgH
Initial kinetic energy: K0 = 0
Final kinetic energy: Kf = ½(mc+mb)vf 2
Acceleration is constant, so d = 1/2(vo+vf)t, where d is the distance along the track.
vf = 2h/t
1 point
1 point
1 point
1 point
2 points
Question 2 (continued)
Distribution
of points
(c) continued
Example #2
Initial gravitational potential energy: Ug0 = mgh
Final gravitational potential energy Ugf = 0
Initial kinetic energy K0 = 0
Final kinetic energy K0 =1/2mv2
Acceleration is constant, so d = ½(vo+vf)t
vf = 2d/t
(d) 2 points
For identifying a reasonable cause for the increase in energy
1 point
For a reasonable explanation related to the cause identified
1 point
Example
An unintentional push was applied to the cart, thus increasing the initial energy.
(e) 2 points
For identifying a reasonable cause for the decrease in energy related to the
nonconservative forces acting on the system
1 point
For a reasonable explanation related to the cause identified
1 point
Example
Friction acting on the object decreases the speed, thereby decreasing the energy.