Physics 121, Final Exam

**Do not turn the pages of the exam until
you are instructed to do so.**

**You are responsible for reading the following rules
carefully before beginning.**

**Exam rules:** You may
use *only* a writing instrument while
taking this test.

**You may not**

**Procedure:**

1. Answer
the multiple-choice questions (problems 1 – 10) by marking your answer on
the scantron form. For each
multiple-choice question (problems 1 – 10), select only one answer. **Questions with more than one answer
selected will be considered incorrect.**

2.
The analytical problems (11 – 17) must be answered in
the blue exam booklets. You ** must** answer problems 11, 12, 13, and 14 in one of the
blue booklets, and problems 15, 16, and 17 in the other blue booklet.

3.
The answer to each analytical problem must be well motivated
and expressed in terms of the variables used in the problem. You will receive partial credit where
appropriate, but only when we can read your solution. **Answers that are not motivated will not receive any
credit, even if correct.**

4. At the end of the exam, you must hand in the blue exam
booklets, the scantron form, the exam, and the formula sheets. All items must be clearly labeled with
your name and student ID number. **If
any of these items is missing, we will not grade your exam, and you will
receive a score of 0 points.**

**Note:** You are not
allowed to use a cheat sheet on this exam. Please refer to the formula sheet at the end of this package
for important equations.

**Note:** If you do not
answer a question in terms of the variables provided, you will not receive
credit for that question.

**Note: **You will get 2.5 extra
points if you put your student ID correctly on your scantron form and answer
the analytical questions in the correct exam booklet.

A rocket is fired vertically upward with a
constant acceleration greater than *g*. The
rocket engine runs for several seconds and then stops. Which of the following statements is
true if we plot the velocity of the rocket as a function of time?

c The velocity versus time graph will be parabolic.

c The velocity versus time graph will be a straight line.

c The velocity versus time graph will consist of two straight-line segments, both with a positive (but different) slope.

c The velocity versus time graph will consist of two straight-line segments with slopes of opposite sign.

c Ns

c N/s

c Nm

c N/m

__Problem 3__ (1.25 points)

Suppose NewtonÕs Law of Universal Gravitation
were modified to read: *F* = *GmM*/*r*^{3},
rather than the observed inverse-square law force. KeplerÕs third law would then read

c (*T*_{1}/*T*_{2})^{2} = (*r*_{1}/*r*_{2})

c (*T*_{1}/*T*_{2})^{2} = (*r*_{1}/*r*_{2})^{3}

c (*T*_{1}/*T*_{2})^{2} = (*r*_{1}/*r*_{2})^{4}

c (*T*_{1}/*T*_{2})^{2} = (*r*_{1}/*r*_{2})^{2}

__Problem 4__ (1.25 points)

Two masses *m*_{1} and *m*_{2} sit on a table connected by a rope. A second rope is attached to the
opposite side of *m*_{2}. Both masses are pulled along the table
with the tension in the second rope equal to *T*_{2}.
Let *T*_{1}
denote the tension in the first rope connecting the two masses. Which of the following statements is
true?

c *T*_{1} = *T*_{2}

c *T*_{1} > *T*_{2}

c *T*_{1} < *T*_{2}

c We
need to know the relative values of *m*_{1} and *m*_{2} to answer this question.

__Problem 5__ (1.25 points)

The linear density of a long thin rod, of
length *L*, decreases
from a value of *d* at the left
end to zero at the right end. How
far from the left end is the rodÕs center-of-mass located?

c *L*/5

c *L*/3

c (2/3)
*L*

c (4/5)
*L*

__Problem 6__ (1.25 points)

A sphere rolling on a horizontal flat surface slows down because of

c the friction force.

c the deformation of the surface.

c the ball and the surface are essentially rigid.

c the gravitational force.

__Problem 7__ (1.25 points)

Three balls start at the same vertical position
but follow different frictionless paths as they descent from a height *h*. Which
of the following statements is true?

c The balls all reach the lower level at the same time.

c The balls all reach the lower level with the same speed but at possibly different times.

c The ball that takes the longer path reaches the bottom with the lowest velocity.

c The balls all reach the lower level with the same speed and at the same time.

__Problem 8__ (1.25 points)

Suppose you are holding a bicycle wheel by a handle connected to the axle in front of you. The axle points horizontally away from you and the wheel is spinning clockwise from your perspective. Now try to tilt the axle to your left (center of mass moves leftward). The wheel will swerve

c upward.

c downward.

c to your left.

c to your right.

__Problem 9__ (1.25 points)

An ideal gas undergoing a Òfree expansionÓ

c does positive work.

c increases its internal energy.

c decreases its internal energy.

c does not change its internal energy.

__Problem 10__ (1.25 points)

The coefficient of performance of a Carnot engine operated as a heat pump is

c 1
– *T*_{L}/*T*_{H}

c (1
- *T*_{L}/*T*_{H})^{-1}

c (*T*_{H}/*T*_{L} - 1)^{-1}

c *T*_{H}/*T*_{L} – 1

__Problem 11__ (12.5 points)

The operation of an automobile internal combustion engine
can be approximated by a reversible cycle known as the Otto cycle, whose *PV* diagram is shown in the Figure below. The gas in cylinder at point a is
compressed adiabatically to point b.
Between point b and point c, heat is added to the gas, and the pressure
increases at constant volume.
During the power stroke, between point c and point d, the gas expands
adiabatically. Between point d and
point a, heat is removed from the system, and the pressure decreases at
constant volume. Assume the gas is
an ideal monatomic gas.

(a)
Assuming there are *n*
moles of gas in system, what are the heats |*Q*_{H}| and |*Q*_{L}|?
Express your answer in terms of *n*, *R*, *T*_{a}, *T*_{b}*, T*_{c},
and *T*_{d}.

(b)
What is the efficiency of the Otto cycle? Express your answer in terms of *T*_{a}, *T*_{b}*, T*_{c},
and *T*_{d}.

(c)
Express the efficiency of the Otto cycle in terms of just the
compression ratio *V*_{a}/*V*_{b} and g. Hint: use the
fact that during an adiabatic process *PV*^{g} = constant.

(d) How does the efficiency change when we replace the monatomic gas with a diatomic gas?

__Problem 12__ (12.5 points)