Physics I Class 111031

Your data for the first two experiments should be submitted promptly.  Your analysis should be submitted within a week, as should the rest of the assignment.

Balancing dominoes experiment:

Report all relevant data from the experiment, being sure to clearly identify the quantities you report and what they mean.

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Find the torque produced by the weight of each domino on the first beam, about the point of rotation.  You may assume domino weights of 15 grams each.

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Candy bar experiment: 

How many oscillations did the candy bar complete in a minute?

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What was the length of the rubber band chain when supporting 4 dominoes?

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What was the length of the rubber band chain when supporting 8 dominoes?

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Through how many radians did the reference point move during the 1-minute timing (it moved through a complete circle for every cycle you counted)?

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What therefore was the angular velocity omega of the reference point?

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How far was it, on the average, from the lowest point to the highest point in the candy bar's oscillation (you didn't measure this; just visualize the motion and make an estimate)?

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The diameter of the reference circle is equal to the estimate you made for the preceding question.  How fast was that reference point moving around the arc of that circle?

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What is the slope of the force vs. length graph for your rubber band chain?  If you measured the domino stack then you can use the fact that for every millimeter of height the stack has mass 1.9 grams.  If not just assume a mass of 15 grams.

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Your last answer is your estimate of k, and your count resulted in your previous answer for omega.  What therefore is m?

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Experiment with 3 rubber bands

The 'initial point' of each of your rubber band chains will be the end nearest the central paper clip, and the 'terminal point' will be the end further away from that clip.  The length vector for each chain is the vector from its initial point to its terminal point.

What are the x and y components of each of the three length vectors?

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What are the magnitudes of each of the three length vectors?

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Divide the x and y components of each of these vectors by its length.  What are your results?  The vectors you get here are called 'unit vectors', because if you divide a vector by its length you get a vector of length 1 (i.e., a vector of unit length).

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According to your calibration of the colored calibrating chain, what was the tension in each of the rubber bands?

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Multiply the tension by the x and y components of the corresponding unit vector.  Your result for each rubber band chain will be the components of its tension vector.

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Add up the x components of your three tension vectors.  What do you get?

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Add up the y components of your three tension vectors.  What do you get?

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The sum of your x components is the x component of the resultant vector; the sum of the y components is the y component of the resultant vector.  What therefore is the magnitude of your resultant vector?

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The resultant vector represents the total effect of the three forces acting on the paper clip in the middle.  The paper clip has zero acceleration, so the net force is actually zero.  One measure of how accurate this experiment might be is the comparison between your resultant and the ideal.  A reasonable way to make the comparison is to compare the magnitude of your calculated resultant with the largest of the three tension forces. 

What is the magnitude of your resultant, as a percent of the magnitude of the largest of the three tension forces?

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`q001.  The mass of the Earth is about 6 * 10^24 kg.  The mass of the Moon is about 8 * 10^22 kg.  The two are separated by about 400 000 kilometers.  G = 6.67 * 10^-11 N m^2 / kg^2. 

What is the gravitational force between the Earth and the Moon?

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What is the acceleration of the Moon toward the Earth?

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What is the acceleration of the Earth toward the Moon?

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Assuming that the Moon's path in its orbit is a circle (which is pretty much the case) of radius about 400 000 kilometers, and that it takes 28 days to complete one orbit (again pretty close to the actual time required), then how fast is it moving?

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What therefore its its centripetal acceleration?

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`q002.  Summary of SHM:

• If the force vs. length graph of an elastic object is linear then its slope is called its force constant. 
• If the tension force exerted this object is  (in a certain way not yet specified) responsible for the net force on an object then that force will be of the form F_net = - k x, where x is the position of the object relative to its equilibrium position. 
• If the net force on a mass m at position x is - k x, then the mass will either remain in its equilibrium position (x = 0), or will oscillate in a manner modeled by the projection on a line through the origin of a reference point moving around a circle with angular velocity omega = sqrqt(k/m)..  You should understand how to use the formula omega = sqrt(k / m), and you should understand motion around a circle which is characterized by a constant angular velocity omega.  You can expect that it will take a few examples and more explanation for you to understand the part about the projection.. 
• If A is the radius of the circle then the amplitude of the motion is A, the speed of the point about the reference circle is v = A * omega, the maximum KE of the object occurs at the equilibrium point and is equal to 1/2 m v^2, the total mechanical energy of the oscillation is 1/2 m v^2, and the potential energy at position x relative to the x = 0 position is 1/2 k x^2.

An object of mass 50 grams is suspended from a rubber band chain.  When it supports a hanging mass weighing .5 Newtons the chain is 70 cm long.  When it supports a hanging mass weighing .9 Newtons the chain is 90 cm long. 

What is the average slope of the force vs. length graph for the given length interval?

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Assuming that the force vs. length graph is linear (not really so for a rubber band chain, but close enough not to make a big difference in the oscillation of our object), what is the value of k for this chain?

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What therefore should be the angular velocity of our reference point?

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How long should it take the reference point to complete one circuit around the circle?

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`q003.  A vector `R of constant length 1 has its initial point at the origin.  Its terminal point moves at a constant speed around the circle, so that the vector moves like a spinning dial.   It should be clear that the circle has radius 1.

When the vector makes an angle of 15 degrees with the x axis, what are its x and y components?

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Answer the same for angle 30 degrees.

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Answer for angle 45 degrees.

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What are the x components for 60, 75, 90, 105, 120, 135, 150, 165 and 180 degrees?

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Is there a pattern to your results?

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What are the y components for the same angles, and what is the pattern of these results?

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Why do the same numbers keep appearing for both components?

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By how many degrees does the y component appear to be ahead of the x component?

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Sketch and describe a graph of the x component vs. the angle.

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If the `R vector is moving around the circle at 30 degrees per second, then what is the x component after 1/2, 1, 2, 3 and 4 seconds?

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Assuming that the `R vector is moving around the circle at 30 degrees per second, graph the x component vs. time.  Describe your graph.

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`q004.  What were the lengths of your 'colored' set of rubber bands when supporting 2, 6 and 10 dominoes?

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What were the weights of those domino stacks, and what therefore were the tension forces at the three lengths you reported?

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What therefore is your best estimate of the average tension force as the rubber band stretches from the 2-domino length to the 6-domino length?  If you didn't measure your domino stacks, just assume a mass of 15 grams per domino.

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How much work do you conclude the tension force therefore does between these two lengths?

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How much work do you estimate the tension force does between the 6-domino length and the 10-domino length?

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If the tension force was path-independent, then the chain would exert the same forces on return from the 10-domino length to the 4-domino length as were exerted as it stretched from the 4-domino length to the 10-domino length.  If that were the case, then how much work would the chain do on an object as it snaps back from the 10-domino length to the 4-domino length?

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`q005.  Formulas for moments of inertia of a few important system:

Hoop (a thin ring with all the mass effectively concentrated at distance R from the axis of rotation), with axis of rotation through the center of the bounded disk and perpendicular to the plane of that disk:

I = M R^2

Uniform disk rotating about an axis through its center and perpendicular to its plane:

I = 1/2 M R^2, where M is the mass of the disk and R its radius.

Uniform cylinder rotating about its central axis:

I = 1/2 M R^2, where M is the mass of the cylinder and R its radius.

Uniform sphere rotating about an axis through its center:

I = 2/5 M R^2

Thin rod rotating about an axis through its midpoint and perpendicular to the rod:

I = 1/12 M R^2

This rod rotating about one of its ends:

I = 1/3 M R^2.

Find the moment of inertia of each of the following, its angular acceleration due to the given torque, and its kinetic energy at the given angular velocity?

A 12-gram uniform foam disk of diameter 16 centimeters, subject to a torque of 100 000 dyne centimeters (a dyne is a gram cm / s^2)., rotating at 4 revolutions per second.

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A 2-kilogram 2 x 4 board 3 meters long, rotated about an axis through its center; and the same board rotated about one of its ends.  In both cases consider the system subject to a torque of 50 meter * Newtons and to be rotating about its axis at 2 radians / second.

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The Earth, whose radius is about 6400 kilometers and whose mass is about 6 * 10^24 kg, as it spins about its axis.  Find the torque that would be required to increase the period of the Earth's rotation from 24 hours to 25 hours in 100 million years.

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Required of University Physics only:

A 15-gram domino rests on a rotating strap, extending from position 10 cm to position 15 cm relative to the axis of rotation. 

If we consider all the mass of the domino to be concentrated at its center, then what is its moment of inertia relative to the axis?

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If we cut the domino in half, so that one half extends from the 10 cm position to the 12.5 cm position and the other from the 12.5 cm position to the 15 cm position, and if we assume that the mass of each half is concentrated at its center, what is the total moment of inertia of the two pieces?

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The total moment of inertia of the two pieces must be equal to the moment of inertia of the whole domino.  So why were the two results different?

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What do you conclude about the accuracy of our assumption that the mass of a domino, or a piece of the domino, is located at its center?

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`q006.  A horizontal light foam 'beam' of length 25 cm supports masses of 10 g, located at one end, 20 g located 10 cm from that end, and 15 g at the other end.

What is the torque of each of the weights, relative to the center of the 'beam'?

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What is the moment of inertia of the system relative to the center of the 'beam'?

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How much additional torque is required to hold the 'beam' stationary in the horizontal position?

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If this torque is produced by an upward force at one end of the 'beam', which end is it and how much force is required?

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What therefore is the angular acceleration of the system that supporting force is removed?

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How long will it take the beam, once released, to accelerate through an angular displacement of .1 radian?

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How much work will the net torque do during this displacement?

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What will be the angular velocity of the system after the .1 radian displacement?

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Has the gravitational potential energy of the system changed during this rotation?  If so, has it increased or decreased?  In how many ways can you tell that this is the case?

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What is the torque produced by each of the weights, relative to the end on which the 10 gram mass rests, and how much force would be required at the other end to keep the system at rest?

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University Physics only:

Suppose the system has rotated pi/6 radians (30 degrees) from horizontal. 

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How far is the line of action of a weight on the end from the axis of rotation?

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What is the net torque on the system at this position?

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What is the angular acceleration of the system?

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