Class 111024

Some video notes are in the process of being translated, and should be linked by the afternoon of 111025.

'seed' questions ... another resource

Physics 201 in-class experiment:

Report your data. For this experiment data consist of the x and y coordinates of the three rubber bands, and the lengths of your colored calibrating rubber bands when attached to the end of each of the three:

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Explain what your data mean:

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This data report should be copied and submitted separately, as soon as possible.

The next part should be submitted by Monday. We will take a little time Wednesday to talk about these instructions.

Your next step will be to sketch a y vs. x graph depicting the six points you observed, with line segments between each pair of points to represent the three rubber band chains.

Having sketched the graph you should find the 'rise' and 'run' of each rubber band chain, then use the Pythagorean Theorem to find the length of each.

You will then be asked to report the length of each rubber band chain vs. the length of your colored calibrating rubber bands.

You can report this at any time between now and next Monday.

First give a table of chain length vs. length of colored chain:

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Briefly explain how you got the length of one of the three chains, based on its x and y coordinates. Pick a chain which has nonzero x and y components.

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Physics 241/231 In-class experiment

Explain what you did and report your data, in such a way that a non-English-speaking supervisor could put it into his language translator and tell what you did, what you observed and how you observed it.

Report this as soon as possible:

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You don't need to report anything else until Monday. We will probably talk Wednesday about this situation, and you will take additional data. However you should start thinking about the following, and plan how you will obtain additional data to fulfill the stated goals.

I note that one group obtained a good bit of data for a car, while the other obtained a good bit of data for the strap. One thing you can expect is that each group will take at least a few observations to corroborate the results of the other. We might then trade off some data for the ultimate analysis. This should be valid, to the extent that the magnets are similar in strength; I expect the uncertainties due to the inconsistent nature of friction to be greater than uncertainties due to variations between magnets.

Your first goal in this experiment is to get a graph of energy dissipated vs. magnet proximity. All the energy is dissipated, presumably by friction. So measurements which, in one way or another, can be used to figure out either frictional force vs. position or frictional torque vs. position will serve this purpose.

Since the dissipated energy all presumably came from the magnetic force, your graph could be relabeled magnetic PE vs. proximity.

This graph can be used to estimate a graph of magnetic force vs. proximity. Your challenge is to figure out how. (Hint: start with a trapezoid on a general graph of PE vs. proximity, and a trapezoid a general graph of magnetic force vs. proximity. Analyze and interpret both trapezoids. Apply you interpretation to this situation).

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A number of students have submitted answers to the questions posed in the 111010 document. This is proving to be a valuable exercise, and it is recommended (but not required) for everyone. If your score on the Major Quiz was less than 93, you should seriously consider submitting your responses. The questions are for the most part relatively short.

You should by this time know the following and be able to apply them to answer questions, set up and analyze lab situations, and solve problems.

How to calculate the units of a product or quotient of quantities having known units.
Definition of average rate
Definition of average velocity
Definition of average acceleration
How to interpret the slope and area of a 'graph trapezoid'.
How to identify an interval
How to declare a positive direction
How to identify the quantities v0, vf, a, `ds, `dt, `dv and vAve for a situation involving uniformly accelerated motion.
The two equations of uniformly accelerated motion that follow from the definitions of average velocity and average acceleration, and how these equations also arise by considering the v0, vf, `dt trapezoid.
How to combine appropriate pairs of the quantities v0, vf, a, `ds, `dt, `dv and vAve using the definitions of average velocity and acceleration and where possible to reason out, over a series of steps, the values of all seven quantities.
How to obtain the third and fourth equations of uniformly accelerated motion from the first two. (Lacking this knowledge, you should at least be able to write down the four equations; however University Physics students should not lack this knowledge).
University Physics students: How to derive the four equations of uniformly accelerated motion using calculus, how to find acceleration, velocity and position functions given one of these functions and, where necessary, intial conditions; how to integrate the force function to find work; how interpretation of a single trapezoid on a graph leads to the interpretation of the derivative and integral of the function represented by that graph.
How to solve any of the four equations of uniformly accelerated motion for any given variable.
How to identify which equation(s) of uniformly accelerated motion can be used with known information to obtain the values of the five quantities v0, vf, `dt, a and `ds.
How to use the equations of uniformly accelerated motoin to find the values of v0, vf, a, `ds and `dt when three of the quantities are known.
How to use Newton's Second Law F_net = m a along with the second and fourth equations of unifomrly accelerated motion to obtain the work-kinetic energy theorem and the impulse-momentum theorem, and the definitions of work, kinetic energy, impulse and momentum.
How to sketch the projection of a vector on a line.
How to calculate the components of a vector on a coordinate plane, given its magnitude and its angle with the positive x axis.
What information is required to add two or more vectors, and how to do so.
How to represent the forces present in a variety of situations, and their components parallel to the coordinate axes.
How to calculate the components of the forces present in a variety of situations, and what information is necessary to do so.
The definition of potential energy.
The difference between work being done on an object and work being done by the object.
The work-energy theorem in terms of `dW_NC, `dKE and `dPE.

To learn to apply these concepts and definitions you need to perform the required lab tasks, answer the questions and problems posed in homework assignments, submit your work on any question or problem you do not completely understand and/or can solve with confidence, and review instructor responses as posted at your access page. A good review of instructor comments will often be accompanied by resubmission of one or more problems.

You should read all of the following problems, make notes on what definitions and concepts you plan to use to solve them, and leave yourself room to later complete the solutions:

General College Physics:

You should read and think about how you would solve all the exercises in Chapters 1-7 which are labeled (I). You should make notes on any of these exercises you are not sure how to do.

Chapter 2 problems 28, 42, 52

Chapter 4 Problems 12, 24, 30, 52

Chapter 6 Problems 6, 12, 22, 42, 52

Chapter 7 Problems 4, 12, 16, 20; 50

Chapter 3, Problems 8, 16, 20, 30

Chapter 5, Problems 8, 14; 30, 36, 48

University Physics:

If you have a lot of trouble with the problems in a chapter then you should work through at least every other odd-numbered exercise (note exercises come before problems). If you have done the questions previously assigned in this course, and read through the text, you should be prepared to do the exercises. If you haven't done the questions previously assigned and can't do most of the exercises, then you really need to work through the previously assigned questions.

Chapter 2, Problems 56, 66, 72, 90

Chapter 3, Problems 44, 48, 56, 64, 68

Chapter 4, Problems 34, 46, 58

Chapter 5, Problems 62, 76, 86, 92, 102, 116

Chapter 6, Problems 56, 58, 64, 74, 82, 84, 90

Chapter 7 Problems 42, 56, 68, 74, 84

Chapter 8 Problems 64, 74, 84, 96, 100, 106

The following definitions and concepts will take us through the rest of the course. These definitions are given along with questions, and answers should be submitted in the usual manner.

Centripetal Acceleration: A point moving on an arc of a circle of radius r with speed v experiences centripetal acceleration a_cent = v^2 / r directed toward the center of the circle.

`q001. What is the centripetal acceleration of a 30 kg mass moving at 10 m/s around a circle of radius 8 m?

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What centripetal force is required to keep the object moving on its circular path?

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Radian: A radian is the central angle for which the corresponding arc distance on a circle is equal to the radius of that circle.

`q002: The end of a strap of length 30 cm, rotating about its center, travels 25 cm. Through how many radians has the strap rotated? What is the circumference of the circular path and through how many radians does the strap rotate as it completes one full rotation?

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Universal Gravitation: Two particles with masses m_1 and m_2, separated by distance r, will each experience a force of attraction to the other. The magnitude of the force of attraction is G m_1 m_2 / r^2 and the direction of the force on one particle is toward the other.

`q003: Recall that steel has density around 7.5 grams / cm^3.

What is the gravitational force between a steel ball of diameter 5 cm and another steel ball of diameter 2.5 cm, if their surfaces are separated by a distance of 1 micron? (Helpful note: As long as they are far enough apart not to touch, they may be regarded as particles, with all the mass of each concentrated at its center. The 1 micron between their surfaces can be neglected for the purpose of finding the force.)

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If the gravitational force between these object is the only force they experience, and if they are initially stationary, then what will be the acceleration of each ball? If they were particles where would they meet?

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How long would it take the spheres, provided they are initially stationary, to move through the 1 micron distance and meet?

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Moment of Inertia: The moment of inertia of a particle of mass m constrained to rotate about a given axis is m r^2, where r is its distance from the axis. The moment of inertia of any rigid object constrained to rotate about a fixed axis is the sum of the m r^2 contributions from all the particles which constitute that object.

Formula: I = sum (m r^2)

Formulas for certain rigid objects with uniform mass distributions (specific conditions and clarifications to be specified later):

hoop: I = M R^2

disk: I = 1/2 M R^2

sphere: I = 2/5 M R^2

rod: I = 1/12 M R^2, I = 1/3 M R^2

`q004: A nut and bolt in the foam disk has mass about 12 grams.

Relative to an axis through the center of the disk, what is the moment of inertia of a bolt located 10 cm from the axis? Answer the same for similar bolts located 4 cm and 7 cm from the axis.

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If there are four of each type of bolt, what is their total moment of inertia?

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What is their total mass M?

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If the disk has radius R = 12 cm, then what is the ratio of M R^2 for this disk to the moment of inertia you calculated?

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Change in angular position: We understand this as the angle through which a rigid object rotates. But motions can be complicated, and rotation of a rigid object is understood relative to an axis, so we often have to be careful about just what we mean by 'the angle through which an object rotates'.

`q005: What is the change in the angular position of a 30 cm strap, rotating about its center, if a point on its end moves 2 cm? Give your answer in radians.

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Also reason out,without looking up conversion factors, the number of degrees in this rotation as well as the fraction of a complete rotation. (Hint: If you first figure out the fraction of a complete rotation then it should be easy to figure out the number of degrees. Another hint: How far does that point travel during a complete revolution?) At the very least give this question your best start.

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More specific definition of change in angular position: The change in the angular position of a rigid object rotating about a fixed axis is equal to the distance moved by one of its particles, divided by the distance of that point from the axis. Alternatively if `r is the vector originating at and perpendicular to the axis and which terminates at a particle not on the axis, then the change in angular position

Angular velocity: The angular velocity omega of a rigid object rotating about a fixed axis is the rate of change of its angular position about that axis, with respect to clock time.

`q006. What is the average angular velocity of a 30 cm strap, rotating about its center, if a point on its end moves 2 cm in .4 second? The units of your answer will include radians. Answer also in units that include degrees, as well as in units that include rotations.

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Angular acceleration: The angular acceleration alpha of a rigid object rotating about a fixed axis is the rate of change of its angular velocity about that axis, with respect to clock time.

`q007: If the strap in the preceding problems slows from angular velocity omega_1 = 5 radians / second to angular velocity omega_2 = 2 radians / second in .6 seconds, what is its average angular acceleration during this interval?

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Line of action of a force: The line of action of a force is the straight line which is parallel to the force, and passes through the point at which the force is applied.

Moment arm: The moment arm of a force about a point is the distance of the line of force from that point. (Refinement: Actually this isn't so; it's the vector from the point to the line, at the closest approach of the line to the point. The distance from the point to the line is the magnitude of the moment arm.)

`q008. One end of a horizontal 8-foot 2 x 4 board rests on a block, the other end in my hand. At a point on the 2 x 4 rests the coupler of a trailer, which exerts a downward force of 500 pounds.

If that point is 1.5 ft. from the end on the block, then what is the magnitude of the moment arm of that force relative to the end of the board on the block?

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What is the magnitude of its moment relative to my hand?

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Often we consider the moment arm of a force applied to an object which is constrained to rotate about a fixed axis. The moment arm of a force exerted on a rigid object constrained to rotate about a fixed axis is the unique vector originating on the axis and terminating on the line of action of that force, subject to the condition that the vector is perpendicular to both the force and the axis.

Torque: The torque of a force exerted at a point, measured relative to a reference point, is a vector with magnitude and direction. The magnitude of the torque is the product of the magnitudes of the force and the moment arm. The direction is that of the cross product of the moment arm with the force; this direction is perpendicular to both the moment arm and the force, and is visualized using the right-hand rule.

Intuitively the torque exerted by a lever is the product of a force exerted perpendicular to the lever and the distance from the fulcrum at which the force is applied. The distance from the fulcrum is the moment arm of the force.

`q009. One end of an 8-foot 2 x 4 board rests on a block, the other end in my hand. At a point on the 2 x 4 rests the coupler of a trailer, which exerts a downward force of 500 pounds. If that point is 1.5 ft. from the end on the block, then what torque does the force exert about that end?

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How much force would I have to exert on my end to equal the magnitude of that torque?

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The torque resulting from a force on a rigid object constrained to rotate about a fixed axis is the cross product of the moment arm of the force, with the force. ... maybe better: A torque is a combination of forces exerted at one or more points of a rigid object which has the effect of changing its angular velocity.

Equilibrium: A system is in equilibrium if the acceleration of its center of mass, and its angular acceleration about any axis, are both zero.

`q010. Which of the following describe equilibrium situations and which do not:?

A block resting on an incline, with the parallel component of its weight equal to 140 Newtons while static friction is capable of producing a force of up to 170 Newtons.

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A block sliding along an incline, with the parallel component of its weight equal to 140 Newtons while static friction is capable of producing a force of up to 170 Newtons and kinetic friction a force of up to 140 Newtons.

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A 5 kg mass and a 6 kg mass suspended from opposite sides of a light frictionless pulley.

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A ball coasting at a constant velocity of 4 m/s.

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A system is therefore in equilibrium when the net force and net torque on it are both zero.

Formulas: sum(F) = 0, sum(tau) = 0.

sum(F_x) = 0, sum(F_y) = 0, sum(F_z) = 0, sum(tau) = 0

Newton's Second Law for rotating objects: An object whose moment of inertia about an axis is I and which has angular acceleration alpha about that point is experiencing a net torque of I * alpha.

Formula: tau = I * alpha

`q011. What is the net torque on an object whose moment of inertia about a certain point is 4 kg m^2 if it is accelerating at 7 radians / second^2?

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work\energy in angular motion: The work on a rigid object rotating about a fixed axis, resulting from a torque tau, is `tau dot `d`Theta.

`q012. How much work does a torque of 8 kg m^2 / sec^2 do as the object on which the torque is applied rotates through 6 complete revolutions?

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`q013. What is the work done by a torque of 30 kg m^2 / s^2 when applied for two seconds, during which its angular displacement is 6 radians?

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Effect of torque applied through angular displacement

`dW = tau * `dTheta

`q014. If the net torque on a rotating object does 60 Joules of work while the object rotates through 10 radians, what is the average net torque?

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Effect of torque applied for time interval

tau `dt = `d( I * omega)

for fixed I tau `dt = I * `dOmega

Work-kinetic energy theorem: same as before

`dW_net = `dKE

`q015. By how much will the kinetic energy of a rotating object change as a result of a net torque of 200 kg m^2 / s^2 applied through a rotation of 4 radians?

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Effect of a torque: angular impulse

angular impulse = tau_net * `dt

`q016. What is the angular impulse of a torque of 30 kg m^2 / s^2 when applied through an angular displacement of 6 radians, which requires 2 sseconds?

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angular momentum

angular momentum = I * omega

`q017. What is the angular momentum relative to a point of an object whose moment of inertia relative to that point is 1200 kg m^2 and whose angular velocity is 20 rad / sec?

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force constant: If the F vs. length graph for the force exerted by an elastic object has constant slope, then that slope is the force constant for that object.

`q018. A graph of the force, in Newtons, exerted by a rubber band chain contains the points (6 cm, 3 Newtons) and (8 cm, 9 Newtons). Assuming that the graph is linear, what is the force constant of this rubber band chain?

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SHM: If the net force on a mass m at position x is - k x, then the mass will either remain in its equilibrium position or oscillate in a manner modeled by the projection on a line through the origin of a reference point moving around a circle with angular frequency omega. If A is the radius of the circle then and 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.

`q019. A mass of .8 kg is suspended from a rubber band chain having force constant 400 Newtons / meter.

What is its angular frequency omega?

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How long would it take for the reference point to move once around its circle?

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If at a certain instant the mass is 10 cm from the equilibrium position x = 0, what is the PE at that point (relative to the x = 0 point)?

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If at that point the object is moving at 3 m/s, what its total mechanical energy?

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