Download Midterm 1

Midterm 1
Physics 14 Fall 2001
J. Morris
Name: ___________________
1: ________
2: ________
3: ________
4: ________
5: ________
Total: __________
Notes: This is a closed book and closed note exam. Be sure to read each question carefully. Try to be neat and
concise. In order receive full credit you must show your work and justify your answers. The points allocated to
each problem are shown in parenthesis following each question.
1. Provide short answers to the questions (a) through (f). To receive full credit you must justify your reasoning;
yes and no answers without supporting explanation are not sufficient.
a) Briefly define (in words) what is meant by velocity and acceleration. What distinguishes velocity from
speed? Can an object have constant speed and be accelerating? If so, provide an example. (5)
Velocity is the time rate of change of displacement. Essentially this is speed with direction included.
Acceleration is the time rate of change of the velocity. Because velocity is a vector an acceleration will arise if
an object changes its speed, direction, or both.
Velocity has both magnitude (speed) and direction. Thus, the distinguishing feature between velocity and speed
is that velocity also includes the direction the object is moving.
An object can have constant speed and at the same time be accelerating provided it changes its direction of
heading – for example by moving along a curved path at constant speed.
b) Distinguish between vector and scalar quantities and provide a few examples of each. (5)
A vector quantity has both a magnitude and direction whereas a scalar has only a magnitude.
Some examples of each include:
Vectors: Force, acceleration, displacement, velocity.
Scalars: Distance, speed, temperature, mass.
c) Can a body be increasing in speed as its acceleration decreases? Justify your answer with an example. (5)
Yes, a body can be increasing in speed as its acceleration decreases. Recall that acceleration is the rate of
change of velocity. Though the acceleration is decreasing there generally still is an acceleration, it’s just not as
large. However, if there is an acceleration then the speed will continue to increase.
As an example from daily life imagine you are getting onto the freeway. At first you may have the accelerator
pressed nearly to the floor – in such case your speed is increasing at a rapid rate. Once you get onto the
freeway you let up on the pedal but still keep it somewhat depressed until you are moving with the flow of
traffic. Even though you have let up on the pedal you are still speeding up until you reach the speed of traffic
flow.
d) While in a car rounding a corner one feels a force pulling them outward. Similarly, when one speeds up or
slows down they feel forces pulling them backward and forward. What is the cause of these forces? (5)
The cause of these forces is one’s tendency to move in a straight line at constant speed (inertia) coupled with
the fact that they are in a car that is accelerating.
In the case of the car rounding a corner the car is accelerating because its velocity is changing direction (it is
also possible that it is speeding up or slowing down but need not be). At each instant while rounding the corner
your natural state of motion would be that of a straight-line tangent to your path. However, as the car makes
the turn it does not take this path – consequently you feel pulled outward – this is because the car is
accelerating inward (toward the center of the circle).
While in a car speeding up you feel pulled backward because your natural state of motion is that of constant
velocity. Consequently, as the car speeds up you are left behind – your tendency is to move at the same speed
and the car is continually increasing its speed. Similar reasoning applies to when the car slows down.
e) A ball rolls down a ramp from rest. For which of the three ramps shown will the ball go down the ramp with
increasing speed and increasing acceleration? Assuming that the ball starts out at the same speed in each case
for which ramp will it reach the bottom in the least amount of time? Explain your reasoning. (5)
In all three ramps the ball increases it speed. In order for it to increase its acceleration (the rate its speed
increases) the ramp must get steeper as the ball rolls down. This is the case for the first ramp.
The ball in the second ramp acquires most of speed early on since the ramp starts off steeply. Consequently, the
ball covers most the distance at a larger speed than the other two ramps. In particular, the flat section of the
ramp is covered very rapidly, whereas this is not the case for the first ramp.
f) Is an astronaut seated aboard an orbiting space station accelerating? Does the astronaut have weight? Do they
have mass? What is the cause of the astronaut’s perception of their weight? (5)
Since the astronaut is in orbit (which is not a straight line) they are accelerating, in fact they are in free-fall.
The astronaut does have weight – this being the force of attraction between them and the earth. It is this force
that keeps them in orbit – otherwise they would continue in a straight line.
The astronaut does have mass – since this is the case of all bodies.
The astronaut does not have the perception of any weight since they are in a continual state of free-fall.
2. A person throws a rock at 37 above the horizontal with speed 50m/s toward a cliff as shown in the figure.
The rock hits the cliff when the rock attains its highest point.
a) Generally, what can be said about the horizontal and vertical components of the rock’s motion? On the figure
illustrate the rock’s vertical and horizontal velocity components at several instants after it is thrown. Assume
these instants occur at regular time intervals (e.g. 1sec, 2sec….) (10).
The horizontal and vertical components of the rock’s motion evolve independently of one another. There is no
acceleration in the horizontal direction and thus the horizontal component of the velocity remains constant –
that is, the rock covers equal horizontal distances in equal times.
In the vertical direction there is an acceleration of g downward and
the component of the rock’s velocity in this direction behaves
just like a rock thrown straight upward.
37
b) It is a geometrical fact that a right triangle with longest side 50 units and angle 37 has sides of 30 and 40
units as shown in the figure. Explain how this information can be used to determine the rock’s initial horizontal
and vertical velocity components. What is the rock’s minimum speed? (5)
Vectors add just like displacements – one can also resolve them into components
in exactly the same way one would resolve displacements – even though in this
case the vector in question is a velocity.
By considering the triangle at right it is clear that the horizontal component
of velocity is 40m/s and the vertical component is 30m/s.
50
30
37
40
The rock attains its minimum speed just before it hits the cliff. At this instant the rock is traveling horizontally –
there is no vertical component of velocity. Hence, the rock’s minimum speed is 40m/s.
c) Suppose the person throws another rock straight upward. How fast would they have to throw this rock so that
it attains the same height as the rock in part (a)? How do the times for each rock to attain this height
compare?(5)
Recall that the vertical and horizontal motions of the rock evolve independently of one another.
Thus, in throwing this other rock straight up all one need do is duplicate the first rock’s vertical
motion. Hence, one should throw this other rock straight up at 30m/s.
Since both rock’s vertical motions are identical the times required to reach their maximum
height are identical.
d) How far away is the person from the cliff? What height does each rock attain? (5)
Since the rock is moving upward at 30m/s it takes 3seconds for it to reach its highest point. During this time it is
traveling over at 40m/s. Thus the horizontal distance it moves (the distance to the cliff) is (40)(3) = 120 meters.
3. Consider a tug-a-war between four people pulling on a bag. The horizontal forces each person exerts on the
crate are shown in the figure (top view).
a) Given the forces shown in the figure determine the total force (vector) that the four people exert on the bag.
Make sure to illustrate how you arrived at this vector! (10)
F1
F2
F = F1 + F2 + F3 + F4
F4
F
F3
Recall that vectors add just like displacements. Thus, the net vector is determined
by connecting each vector head-to-toe and connecting the tail of the first to the
head of the last.
b) Can you think of a justification for the addition of vectors? In particular is there some fundamental physical
quantity that is characterized by an arrow that “adds” in a particular way? (5)
Vectors are represented by arrows. Thus, if one seeks to find a rule for adding arrows it suffices to consider a
particular physical quantity that one is familiar with and is conveniently represented by an arrow. One such
quantity is the displacement from one point to another. It is natural to “define” the sum of several
displacements to be that single displacement that one could make that would carry them from the very
beginning point to the final point. This is precisely how vectors are added, and whenever there is any doubt
about how a vector should behave in a particular situation one need only consider how a displacement would
behave in that situation.
4. Provide short answers to the following questions pertaining to measurements.
a) Is it possible to measure the length of something exactly? If not, how might one in practice estimate the error
of their measurement(s)? (5)
Generally it is not possible to measure the length of something exactly. Furthermore, it is also not even possible
to define the length of something exactly – to take an extreme example the surface of most substances is very
bumpy on an atomic scale.
There are always errors inherent to one’s measurement of a physical quantity. To determine such errors what
one should do is take several independent measurements of the quantity. Inevitably what one will find is that the
measurements cluster around some average value. The degree to which the separate measurements differ from
the average is a reasonable estimate of the uncertainty in the measured value of that quantity.
b) Suppose four people take measurements of someone’s wrist and arrive at: 16cm, 15cm, 15cm, and 17cm.
What value might you quote for the person’s wrist? Roughly what uncertainty is associated with these
measurements? (5)
The average of all the measurements is 15.75cm. As it stands this number has too many figures – considering
that the data only have two and are differing by order 1cm. If one looks at the deviations of each measurement
from the average and averages those deviations one arrives at 0.75cm. Putting these facts together one might
quote the person’s wrist as 15.8cm with an uncertainty of 0.8cm.
5. A ball is attached to a rope of length l = 4m and released from rest at point A. The ball swings through its
lowest point B and then comes to rest at point C as shown in the figure. Assume the time it takes for the ball to
move from point A to C is t = 2sec.
a) How do the times for the ball to swing down and up compare? (5)
C
A
Just as with a ball tossed up in the air (this is a controlled fall)
the times for the ball to swing down and up are equal – taking
one second for each.
B
b) What is the ball’s average speed as it travels from point A to point B? How does this average speed compare
with the instantaneous speed at point B? (5)
Recall that the average speed is given by distance traveled divided by the elapsed time. The distance the ball
travels from point A to B is (l)/2 = 6.3m. Hence, the average velocity is 6.3m/s. Since the ball is always
increasing its speed as it swings downward the instantaneous speed at point B is greater than the average
speed.
c) What is the direction of the ball’s average velocity as it swings from point A to C? Justify your answer! (5)
Recall that the average velocity can be interpreted as the average of all the velocity vectors that make up the
motion. For every point on the left side of the diagram where the ball is moving downward there is a
corresponding point on the right side that is moving up. It’s not hard to see that the up and down motions
cancel but the left to right motion does not. The average velocity is parallel to an arrow directed from point A
to C.
d) A lighter ball is attached to the rope and swings from point A to C. How does the time required for this
lighter ball to move from point A to C compare with that of the heavier ball? (5)
All objects accelerate toward the earth at the same rate. Though the ball is attached to a cord what is
determining the rate of its swinging motion is gravity. Thus, the time required to swing from one point to
another is independent of the mass of the ball.
6. Consider a book of mass m = 0.5kg being pushed by someone’s hand on a horizontal table as shown in the
figure. Assume the hand pushes the book with a force of magnitude F = 6N and the frictional force between the
book and the surface has magnitude f = 2N.
a) Draw a free-body diagram for the book. (5)
F
m
N
F
m
f
W
b) What is the weight of the book? (5)
W = mg = (0.5)(9.8) = 4.9 N
c) What is the acceleration of the book? (5)
Using Newton’s second law the net force acting on the book in the horizontal direction is 6 – 4 = 2N. Since a =
F/m , one finds
a = 8m/s2.
d) Suppose the book is initially at rest and then pushed for 1.5 seconds. Determine the final speed of the book,
its average speed, and how far it travels in the process. (5)
Because the acceleration is 8m/s2 the book speeds up 8m/s every second, or equivalently 4m/s every half
second. Since the book starts from rest and accelerates for 1.5 seconds the speed after this amount of time is 8
+ 4 = 12m/s.
The average speed is just the average of the speed at the start of the motion (0) and the speed at the end of the
motion (12m/s). Hence, the average speed is 6m/s.
Recall that the distance traveled is the product of the average speed and the time. Hence, d = (6)(1.5) = 9m.
e) For each force acting on the mass determine its respective reaction force. Make sure to state clearly on what
body this reaction force acts. (5)
The reaction to F (force pushing of the hand on the book) is: F(b on h) (force of the book pushing back on the
hand)
The reaction to W (gravitational force of the earth on the book) is: W(b on e) (gravitational force of the book on
the earth).
The reaction to f (frictional force of the table on the book) is: f(b on t) (frictional force of the book on the table)
The reaction to N (normal contact force the table exerts on the book) is: N(b on t) (normal contact force that the
book exerts on the table).