Class 090923

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In this assignment you will be asked to

·         do a little more with your rubber band calibration graph, which you constructed previously

·         find or confirm the correct accelerations for the previously done Atwood and ball-down-ramp experiments

·         do a little more with the rubberband-chain force diagrams you sketched in class, and learn a little terminology

·         plug a = F_net / m into the second and fourth equation of motion, and do your best with a little algebra

Trapezoidal Graph of Rubber Band Calibration Data

When you calibrated your rubber band chain you observed the position of the end of the chain with various numbers of dominoes. 

Plot the information on a graph of y = number of dominoes vs. x = position of end.  Your graph will consist of five points.

Give the coordinates of each of your five points; be sure to include units.

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Now use your points to make a series of trapezoids:

• Connect your five points with line segments.  There will be four line segments, each with a slope you calculated in response to previous class notes.
• From each point sketch the 'vertical' line segment from that point to the horizontal axis.  You will sketch five line segments, each representing the number of dominoes suspended in the corresponding trial.  Having sketched these segments, you will have constructed a series of four trapezoids.

The graph you have constructed will be called a 'trapezoidal approximation graph'.

Briefly describe your graph in words

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• Find the area of each trapezoid.
• Label the slope of each trapezoid by placing it in a rectangular 'box' just above the 'slope segment' at the top of the trapezoid.
• Label the area of each trapezoid writing it inside the trapezoid and circling it.

What are the four slopes?

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What are the four areas?

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What does each slope represent (explain what the rise represents, what the run represents, and what the slope therefore represents; be sure to include units with each quantity):

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A rise represents a change in the number of dominoes supported, which is an indication of tension. 

A run represents a change in length, in cm. 

Therefore

• slope = rise / run = (change in number of dominoes) / (change in length), or (change in tension) / (change in length). 

The slope is therefore an average rate of change of tension with respect to length.  The units of the slope will be dominoes / cm, where each 'domino' represents a force equal to the weight of a domino.

What does each area represent (explain what the altitude of the equal-area rectangle represents, what the base represents, and what the area therefore represents).

'Graph altitudes' represent force, in units of domino weight.  The average graph altitude on an interval therefore represents the approximate average force on that interval. 

The width of the interval represents the change in rubber band length, in cm; this is the displacement of the end of the rubber band 'chain' during the interval. 

The area of the trapezoid therefore represents

area = average force * displacement,

which is the work done to stretch the rubber band for that interval.  

The units are dominos * cm, where again 'domino' refers to the weight of a domino.

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Your correct accelerations for lab situations:

In a number of lab situations acceleration is assumed constant, and we observe the time needed for an object to undergo a known change in position, either starting or ending at rest.

To solve the motion in such a situation we the following procedure will always work efficiently:

• Apply the definition of average velocity.
• Sketch the v vs. t trapezoid.
• Use the trapezoid to determine the initial and final velocity of the object on the interval.
• Apply the definition of average acceleration.

One very common error is to divide average velocity by time interval (vAve / `dt).  This is not a calculation that comes up when following the above procedure, and it isn't a calculation that tells us anything important about the motion.  Be sure you understand why this calculation doesn't happen in the given procedure.

Another common error is for students to double the average velocity to get the final velocity.  For the case where initial velocity is zero and acceleration is constant, this is actually done.  However this is a result of the fact that the v vs. t trapezoid is in this case a triangle, and is not something we generally do.  You shouldn't even think about doubling the average velocity, and certainly shouldn't get into the habit of doing so.  Draw the trapezoid and do as the drawing dictates.

Being very sure you analyze the motion correctly, please report the following.  If you've done some or all of these correctly, you can insert a copy (or copies) from your previous document(s):

The data and accelerations of the three trials for the Atwood machine:

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The slope is therefore

The two slopes obtained when you graph acceleration vs. number of added rubber bands for the Atwood machine:

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The coordinates of the three points you graphed:

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The data and accelerations for the three trials (domino flat, on long side, on short side) for the ball-down-ramp experiment:

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The two 'graph slopes' obtained when you graph acceleration vs. ramp slope:

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The coordinates of the three points you graphed:

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Give a detailed description of how you proceeded from raw data to acceleration for one of the Atwood Machine trials:

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Give a detailed description of how you proceeded from raw data to acceleration for one of the ball-down-ramp trials:

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Newton’s Second Law

Here are a few fairly obvious statements. 

·         It takes more force to accelerate lots of mass than just a little mass.

·         The more acceleration you want the more force you need.

·         More force implies greater acceleration.

All these statements are a bit imprecise.  More precise statements require more words, with a potential for confusion.  So we’re going to work from the imprecise statements that first form our ideas, to the precise statements on which we base the actual science.

More precise and accurate versions of the above statements might look like the following:

• To give two different masses the same acceleration, the greater mass requires the greater force.
• To give a certain mass a greater acceleration, you need to apply a greater force.
• If you apply the same force to two different masses, the lesser mass will have the greater acceleration.

The last statement 'If you apply the same force to two different masses, the lesser mass will have the greater acceleration' is not really so. 

• For example you can apply all sorts of force to a merry-go-round and it never goes anywhere, it just rotates, with the parts near the rim moving faster than the parts near the center.
• So for now we're going to remember that real objects can rotate, but we're going to confine our attention to situations that don’t involve rotation.
• It an object doesn't rotate when a force is applied to it, we say that it is acting like a particle.
• Particles don’t rotate.  They only translate.

We can (and will) verify by experiment that the following statement holds.  This is Newton's Second Law:

The net force exerted on a particle is the product of its mass and its acceleration:

• F_net = m a.

You need to memorize this entire statement, along with the definitions of average rate, average velocity, average acceleration, the equations of uniformly accelerated motion and the interpretation of a v vs. t trapezoid.

The net force on a particle is what you get if you combine all the forces acting on it.

A couple of important consequences of Newton's Second Law:

• If its acceleration is zero then the net force on it is zero.
• If it’s sitting still then its acceleration is zero. 
• If it’s moving at constant speed in a constant direction then its acceleration is zero. 
• Otherwise its acceleration isn’t zero and the net force on it is not zero.

Substituting a = F_net / m into the second and fourth equations of motion:

If we solve Newton’s Second Law for acceleration we get

• a = F_net / m

If substitute F_net / m for a in the second equation of uniformly accelerated motion, vf = v0 + a `dt, what equation do we get?

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We can rearrange your equation to get

·         F_net * `dt = m vf – m v0

Show, as best you can at this point, the steps needed to get from your previous answer to this form:

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If substitute F_net / m for a in the fourth equation of uniformly accelerated motion, vf^2 = v0^2 + 2 a `ds, what equation do we get?

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We can rearrange your equation to get

·         F_net * `ds = ½ m vf^2 – ½ m v0^2.

Show, as best you can at this point, the steps needed to get from your previous answer to this form:

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Impulse, momentum, energy, work

The quantities F_net * `ds, F_net * `dt, 1/2 m v^2 and m v arise naturally when a = F_net / m is substituted into the second and fourth equations of motion.

We give these quantities names, which you should learn immediately:

• F_net * `ds is the work done by the net force F_net on an interval for which the displacement is `ds
• F_net * `dt is the impulse of the net force F_net on an interval of duration `dt.
• 1/2 m v^2 is the kinetic energy of a particle of mass m moving at velocity v.
• m v is the momentum of a particle of mass m moving at velocity v.

Our substitutions give us the two equations

·         F_net * `dt = m vf - m v0

and

·         F_net * `ds = ½ m vf^2 – ½ m v0^2.

Both of these equations apply to the behavior of a particle of mass m on an interval.

• F_net * `dt is the impulse of F_net on the interval.
• m vf is the momentum of the particle at the end of the interval
• m v0 is the momentum of the particle at the beginning of the interval
• so m vf - m v0 is the change in the particle's momentum on that interval

F_net * `dt = m vf - m v0 therefore states that, on the interval in question,

• the impulse of the net force acting on the particle is equal to the change in its momentum.

This is called the impulse momentum theorem.

• F_net * `ds is the work done by the force F_net on the interval.
• 1/2 m vf^2 is the kinetic energy of the particle at the end of the interval
• 1/2 m v0^2 is the kinetic energy of the particle at the beginning of the interval
• so m vf - m v0 is the change in the particle's kinetic energy on that interval

F_net * `ds = ½ m vf^2 – ½ m v0^2 therefore states that, on the interval in question,

• the work done by the net force acting on the particle is equal to the change in its kinetic energy.

This is called the work-kinetic energy theorem.

Briefly explain in your own words how we get the definitions of impulse, momentum and the impulse-momentum theorem.

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Briefly explain in your own words how we get the definitions of work, kinetic energy and the work-kinetic energy theorem.

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Force vectors

The three arrows you drew to depict the forces exerted by the rubber bands on the common paperclip are called force vectors. 

If you put the arrows end-to-end they should ideally form a ‘closed’ triangle (i.e., the last arrow should end where the first one began).  This would indicate that when the forces are combined, they add up to zero.

If two of the rubber bands are perpendicular, then you can define an x-y coordinate plane so that one force acts along the x axis and the other along the y axis. 

• The force in the direction you chose as the x direction can point either in the positive or negative x direction.  The same is so for the force in the y direction.
• The third force will not be directed along either of these axes; its direction will be partially in the x and partially in the y direction.

• Label the force in the x direction A, the force in the y direction B and the third force C.
• Each arrow represents the direction of one of the forces, and the relative lengths of the arrows represent the relative 'strengths' of the three forces (i.e., the longest vector represents the greatest of the three forces, the shortest vector represents the least of the three forces, with the lengths in proportion to the forces).
• We will refer to the arrows as the force vectors A, B and C.

Look at the sketch you made of the three forces. 

• Does force A (the one in the x direction) act in the positive or negative x direction?
• Does force B (the one in the y direction) act in the positive or negative y direction?
• Into what quadrant does force C (the third force) point? (if the x-y plane is in the standard vertical vs. horizontal orientation, the first quadrant corresponds to the upper right, the second to upper left, the third to lower left, the fourth to lower right)
• Which is greater, force A or force B?
• Which is longer, the A arrow or the B arrow?
• Which is the longest of the three arrows?

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Now draw the projection lines from the tip of arrow C to the x and y axes:

• The x projection line is a dotted line from the tip of the C arrow to the x axis, and is perpendicular to the x axis.
• The y projection line is a dotted line from the tip of the C arrow to the y axis, and is perpendicular to the x axis.
• Is one of your projection lines parallel to the x axis?  If so, which one is it?
• Is one of your projection lines parallel to the y axis?  If so, which one is it?

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Finally draw arrows from the origin to the ends of your projection lines:

• Draw an arrow starting at the origin and running along the x axis ,ending at the point where the x projection line meets the x axis.
• This arrow represents what we call the x projection of the vector C.
• Label this arrow Cx.

• Draw an arrow starting at the origin and running along the y axis to the point where the y projection line meets the y axis.
• This arrow represents what we call the y projection of the vector C.
• Label this arrow Cy.

• Which is longer, the x projection of C or the y projection of C?
• Which is longer, the x projection of C or the vector A?
• Which is longer, the y projection of C or the vector B?
• Place in order of length, from shortest to longest:  C, Cx, Cy, A, B.

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Homework:

Your label for this assignment: 

ic_class_090923

Copy and paste this label into the form.

Report your results from today's class using the Submit Work Form.  Answer the questions posed above.

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