hphys unit 04 ubfpm packet 2012

Name: ________________________________

Unbalanced Force ModelPhysics Diagrams

Here you can keep track of all of the diagrams that you have learned to draw. You can come back and update it after future models, if you’d like. This list might come in handy when you start tackling complex “goal-less” problems.

Name of Diagram Model Example Notes

Honors Physics / Unit 04 / UBFPM

– 1 – from Modeling Workshop Project © 2006

Interlude: Vector Practice!Always use a protractor, ruler, and pencil when working with graphical vector constructions!

1. An object moving in a circular path with a constant speed of 2.0 m/s changes direction by 30º in 3.0 s.

a. What is the magnitude of the change in velocity over the 3.0 s interval? Remember that velocity is a vector. Draw a vector diagram and perform the graphical construction to find

€

Δv .

b. What is the magnitude of the average acceleration over the 3.0 s interval?


from Modeling Workshop Project © 2006 ! – 2 –

2. The vectors shown in the figure below show three positions occupied by a dog moving across a field in a general upper-left to lower-right direction. A coordinate system superimposed on the field provides a frame of reference. The time interval between these positions is 2.0 s.

71

(c) What is the magnitude and direction of �F1 + �F2?

x

y

F!

1

F!

250 N

100 N

11. The vectors shown in the figure below show three positions occupied by a dog movingacross a field in a general upper left to lower right direction. A coordinate systemsuperimposed on the field provides a frame of reference. The time interval betweenthese positions is 2.0 s.

5 10 15 20 25 30x !m"

5

10

15

20

y !m"

R!

1

R!

2

R!

3

(a) What is the magnitude and direction of the dog’s displacement from position�R1 to position �R2? Use the Pythagorean theorem and an appropriate inversetrigonometric function to determine these. You may report the direction as anegative angle measured clockwise from the positive x-axis.

(b) What is the magnitude and direction of the dog’s average velocity over the timeinterval of the displacement in part (a)? Measure the angle in the same manneras in (a).

(c) Estimate the dog’s instantaneous velocity at the instant it is at position �R2. Usethe double interval method. Give both its magnitude and direction.

(d) Is this motion feasible for a dog? Explain.

Section 4.5

a. Draw and label the displacement vector from 1 to 2. What is the magnitude and direction of the dog’s displacement from position 1 to position 2? Remember to use the scale to convert your answer into meters.

b. What is the magnitude and direction of the dog’s average velocity over the time interval of the displacement in part a?

c. Estimate the dog’s instantaneous velocity at the instant it is at position 2. Use the double interval method. Give both the magnitude and the direction.

d. Is this motion feasible for a dog? Explain.



3. Topanga likes to swing on a tire tied to a tree branch in her yard, as in the figure. If Topanga and the tire have a combined mass of 82.5 kg, and Cory pulls Topanga straight back far enough for her to make an angle of 30º with the vertical, what is the tension in the rope supporting Topanga and the tire?

a. Solve this problem with a graphical vector construction.

b. Solve this problem using vector components (trigonometry replaces the protractor).



Experiment: Pulling Carts with Springs

Sketch and label the experiment setup:

What could we measure? How could we measure it?

Whiteboarding and Post-Lab Notes



Reading: Acceleration is Directly Proportional to Net Force and Inversely Proportional to Mass



Our experiments in the lab showed us that an object experiencing a

constant force will have a constant acceleration, just as the object in the

strobe photograph of a dry ice puck below.

Figure 1: The dry ice puck is being pulled to the right by a small, white elastic loop.

You can see that the loop is stretched the same amount throughout the motion,

indicating that the force to the right exerted by the loop on the puck is constant.

We know from our studies of this photo in the previous chapter that the puck’s

acceleration is constant.

The velocity vs. time graph for the above motion is shown in the mar-

gin in figure 2. We see a constant slope of the velocity vs. time graph,

which confirms that the acceleration of the dry ice puck is constant. Now

let’s see what happens if we attach two rubber loops identical to the first

to the dry ice puck.

In the lab, you performed experiments using identical springs and low

friction carts with the objective of measuring, quantitatively, the relation-

ship between force and acceleration. Figure 3 is a stroboscopic photo of

the same dry ice puck shown in figure 1, this time being pulled by two

�

�

�

�

�

�

�

0 1 2 3 40.0

0.1

0.2

0.3

0.4

0.5

time �s�

velocity�m�s�

Figure 2: The velocity vs. time graph for

the puck in figure 1.

elastic loops. Thus, the force exerted on the puck to the right is twice what

it was in figure 1.

Figure 3: The dry ice puck is now being pulled to the right by two elastic loops.

You can see that the loops are stretched the same amount as the one loop in figure

1, meaning the puck is experiencing twice the force as in figure 1.

Taking data from this photo, we can create a table with time, position

and velocity columns. Note that we used the double interval method for

determining the velocity of the puck at each clock reading.

Clock

Reading Position Velocity

(s) (m) (m/s)

0.00 .085 —

0.42 .170.300−.085

.84−0= .256

0.84 .300 .367

1.26 .478 .478

1.68 .702 .592

2.10 .975 —

Plotting the velocity and time data from the table gives us the velocity

vs. time graph in figure 4. Included in the graph is a dotted line that corre-

sponds to the velocity vs. time data for the puck when it was pulled with

only one elastic loop. Note that the slope of the velocity vs. time graph

is steeper when a greater force is applied to the puck (which we already

knew would be the case). Let’s now calculate the acceleration of the puck

when it is pulled with two elastic loops by taking the slope of our velocity

vs. time curve. Using the data from the graph:



aavg =∆v∆t

=0.53 m/s

2.0 s= 0.265 m/s

2

Just as you saw with your own experiments, the acceleration of an object

�

�

�

�

�v

�t

�v��.68�.15�m�s � 0.53 m�s�t��2.0�0�s � 2.0 s

one loop, slope�0.133 m�s2

two loops

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

time �s�

velocity�m�s�

Figure 4: The velocity vs. time graphs for

the dry ice puck being pulled with one loop

(dotted line without data points) and two

loops. Note that the slope of the curve (that

is, the puck’s acceleration) is doubles when

the force is doubled.

appears to be directly proportional to the net force applied to that object.

In fact, you took data for several different forces (integer multiples of the

force exerted by a single spring) to determine that this direct proportion-

ality was a fact. Figure 5 is a generic graph of acceleration as a function

of applied net force, similar to what you measured in the lab. It might,

at first sight, seem puzzling that we say the relationship between accel-

eration and force is directly proportional, since the graph does not pass

through the origin. But we determined that the curve’s offset from the ori-

gin could be easily explained in terms of the small amount of friction the

cart was experiencing.

�

�

�

�

�

force �springs�

acceleration�m�s�

s�

Figure 5: Experiments show that the accel-

eration of an object is directly proportional

to the total net force on that object. Our

graph shows a non-zero intercept only be-

cause we did not take into account the fric-

tion force.

In fact, elaborate experiments designed to reduce friction forces to neg-

ligible values confirm our conclusion that an object’s acceleration is di-

rectly proportional to the net force exerted on the object. These experi-

ments, along with our experiments, allow the following mathematical rep-

resentation:

a ∝ Fnet or a = kFnet (1)

Our experiments included the study of how acceleration depends on

the mass of the object. We kept the net force on the object constant through-

out our experiments, increasing the mass of our carts by adding bricks.

After using units of kilograms to measure the mass of the cart and bricks,

we found a relationship between acceleration and mass that looks like it

might be an inverse relationship. We linearized our data by plotting the

reciprocal of mass (that is, one over the mass) as our independent variable

and acceleration as our dependent variable. The result looked something

like the generic plot shown in figure 6.

�

�

�

�

�

1

mass�kg�1�

acceleration�m�s�

s�

Figure 6: The acceleration of an object is in-

versely proportional to its mass.

Our experimental results (which are confirmed by many other experi-

ments over the ages) allow us to write the following mathematical repre-

sentation:

a ∝1

mor a = k

1

m(2)

Combining the results of our two experiments, we arrive at the onedimensional version of Newton’s Second Law.

a ∝Fnetm

(3)

where Fnet is the net force acting on an object whose inertial mass m can be

treated as being concentrated at a single point.

Worksheet 1: “Goal-less” Problems! (Super cool.)

4. The 80 kg box travels to the bottom of the 2 m long ramp with a constant speed of 1.4 m/s.

a. What models apply to this situation and why? (See the italics + bold + underline? That part is way important!)

b. Draw at least four diagrams/graphs to illustrate the situation. Choose the diagrams and graphs that you find most useful. Draw these BIG enough to be useful and annotate them like crazy. The diagrams are key.

c. Using the models you have chosen, solve for any unknown quantities. Use more than one method to find the same answers whenever possible. Show your work and use units. Always start with a variable expression (that is, your first line should always start with symbols only, and you should plug in things that you know in a later step).

!



5. The 80 kg box starts from rest and travels to the bottom of the 2 m long ramp, but its motion is opposed by a 150 N frictional force.Follow the same general procedure as in the previous problem: tell which models apply, draw and annotate diagrams, solve for unknowns.

!



6. The 80 kg box starts from rest and travels to the bottom of the 2 m long, frictionless ramp. Again, follow the same general procedure.

!



7. In the preceding three problems, was the contact normal force that the incline exerted greater than, less than or equal to the weight of the box? Explain.

8. In problem 6, suppose the incline were tilted at a greater angle. How would your answers change? Use a side-by-side comparison of two vector addition diagrams as part of your explanation.

9. In problem 4, determine the unbalanced force on the cart at the instant its velocity is 1.4 m/s.

In problem 6, determine the unbalanced force on the cart at the instant its velocity is 1.4 m/s.

Are your answers to the above two questions the same or different? If they are the same, explain how this is so even though there is no friction in problem 3. If they are different, explain how this is so even though the cart has the same instantaneous velocity of 1.4 m/s.

10. Suppose the cart in problem 6 were given a shove so that it started sliding up the ramp with a velocity of 3.0 m/s. How far up the ramp would it go?



Worksheet 2: Multiple Model Problem Solving, cont.Solve the following problems with the same approach as you did for the “goal-less” problems on Worksheet 1. List the models that apply (and why!), draw and annotate a lot of diagrams (at least four, but probably more), and then use the models to solve for as many unknown quantities as you can.

11. Three blocks are in contact with each other on a frictionless horizontal surface. A person applies a horizontal force of 18 N to the smallest block. Hint: there are multiple ways to define your system here. You could draw several FBDs for the same instant (snapshot).



12. A 70.0 kg box is pushed by a 400 N force at an angle of 30° to the horizontal. Starting from rest, the box travels 15.0 m in 2.78 seconds.



13. A 70 kg box is pulled by a 400 N force at an angle of 30° to the horizontal.Starting from rest, the box travels 15 m in 2.47 s.

!



14. Compare the contact normal force that the ground exerts on the box in problems 12 and 13. In which case does the box experience a greater force?

15. In problem 13, suppose that the person pulled at a greater angle from the horizontal (meaning the angle in the drawing would be greater than 30º). How would your answers change? Use a side-by-side comparison of two vector addition diagrams as part of your explanation.

16. In problem 12, suppose that the person pushed with a greater angle from the horizontal. How would your answers change? Use a side-by-side comparison of two vector addition diagrams as part of your explanation.



UBFPM Model Summary

My First Concept Map (Aww…)



hphys unit 04 ubfpm packet 2012

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