Class Notes 100830

One standard assignment is to come to class with three good questions, ready to turn in.

You should try to do the questions below before class on Wednesday.  However some of you got these questions late and won't be able to do so.  If that's the case, at least try to think about them, which should better prepare you for some of what we'll be doing in class.

Questions for everyone:

1.  What was your count for the pendulum bouncing off the bracket, and how many seconds did this take?  What therefore is the time in seconds between collisions with the bracket?  What was the length of your pendulum?

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2.  What was the period of your pendulum when it was swinging freely?  Give your data and briefly explain how you used it to find the period.  How does your result compare with the time between 'hits' in the first question?

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3.  Give your data for the ball rolling down the ramp, using the bracket pendulum as your timer.  Assuming the ball traveled 30 cm each time, what are the resulting average velocities of the ball for each number of dominoes?

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4.  How did your results change when you allowed the ball to fall to the floor?  What do you conclude about the time required for the ball to fall to the floor?

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5.  Look at the marks made on the paper during the last class, when the ball rolled off the ramp and onto the paper.  Assuming that the ball required the same time to reach the floor in each case (which is nearly but not quite the case), did the ball's end-of-ramp speed increase by more as a result of the second added domino, or as a result of the third?  Explain.

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6.  A ball rolls from rest down a ramp.  Place the following in order:  v0, vf, vAve, `dv, v_mid_t and v_mid_x, where the quantities describe various aspects of the velocity of the ball.  Specifically:

Explain your reasoning.

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7.  A ball rolls from one ramp to another, then down the second ramp, as demonstrated in class.  Place the following in order, assuming that v0 is relatively small:  v0, vf, vAve, `dv, v_mid_t and v_mid_x .

Place the same quantities in order assuming that v0 is relatively large.

Which of these quantities will larger when v0 gets larger?  Which will get smaller when v0 gets larger?  Which will be unchanged if v0 gets larger?

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8.  If the ball requires 1.2 seconds to travel 30 cm down the ramp from rest:

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University Physics questions:

Question 1 below concerns the equations of uniformly accelerated motion.

Questions 2-4 are progressive in difficulty.  You should be able to make a good attempt at the first one and probably the second.  The third is pretty challenging and you might not be able to do it on your first attempt:

1.  If we integrate with respect to clock time t the acceleration function a(t) = a, where a is a constant, we get the velocity function v(t) = v0 + a t, where v0 is the velocity at t = 0.  If we integrate this velocity function we get position function x(t) = x0 + v0 t + 1/2 a t^2, where x0 is the position at t = 0.

If we abbreviate v(t) as just v, and x(t) as just x, our equations are

and

Answer the following:

2.  A ball rolls down a certain incline, starting from rest at the x position 23 cm and accelerating at 20 cm/s^2. 

3.  If the ball in #2 starts from the same position and accelerates at the same rate, but starting with velocity 15 cm/s, what will be its position when its velocity reaches 40 cm/s?

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4.  A ball is rolled down a ramp, starting from rest at x position 30 cm.  Its acceleration on this ramp is 30 cm/s^2.  This ramp is parallel to the ramp in #3, right next to that ramp, and the same meter stick is used to measure positions on both ramps.  If the ball is released 1 second after the ball in #3:

 

Class notes:

How quickly does velocity change?

If a car has velocities 0, 15 mph and 30 mph at clock times 0, 3 and 8 seconds, then on the first interval its velocity changes by 15 mph in 3 seconds.  The average rate of change of velocity with respect to clock time is therefore

On the second interval its velocity changes by 15 mph and the clock time changes by 5 seconds, so the average rate of change of its velocity with respect to clock time is

Thus we have a measure of how quickly velocity changes on each interval.

Note that mph / sec can be converted to meters / sec.  You will be expected to be able to do this more accurately, but a ballpark approximation is that 1 m/s = 2.3 mph.

Ball down ramp to floor, landing position vs. ramp slope and horizontal velocity vs. ramp slope

We assumed that a ball rolled down a 'one-domino' ramp with slope .03 and off the end reached the floor after traveling 12 cm in the horizontal direction.  The same ball rolled down a 'two-domino' ramp with slope .06 traveled 18 cm, and when rolled down the '3-domino' ramp with slope .09 traveled 23 cm, both in the horizontal direction.

We wished to find the rate of change of the ball's horizontal displacement with respect to ramp slope.  For the first two ramps we found that the change in horizontal displacement was 18 cm - 12 cm = 6 cm and the change in slope was .06 - .03 = .03 units of ramp slope (ramp slope is really unitless, but in order to understand meanings it's helpful to call it something).  So the average rate of change of horizontal displacement with respect to ramp slope was

Similarly between the second and third ramps we get

We next ask how the horizontal velocity of the ball changes with changing ramp slope, assuming that the time of fall is about .40 seconds in every case.  The following are worth noting:

Now, if the time of fall is .40 seconds, a ball which travels 12 cm in the horizontal direction has an average horizontal velocity of 12 cm / (.40 second) = 30 cm / second.  For the cases where the ball travels 18 cm and 23 cm, the respective horizontal velocities are 18 cm / (.40 sec) = 45 cm/s, and 23 cm / (.40 sec) = 58 cm/s. 

Thus between the first ramp and the second, the average rate of change of horizontal speed with respect to ramp slope is

Between the second and third ramps the result is

 

During the lab period everyone constructed a 'bracket pendulum' and estimated the time between 'strikes' of the pendulum on the bracket.  The time was estimated by setting a count of 8 using the pendulum, then continuing at this rate for three additional counts of 8, while watching the second hand of a clock.  We also allowed the pendulum to swing freely and determined the number of cycles in 30 seconds.

Everyone then used the pendulum to time the ball down the track for 1-, 2- and 3-domino systems, then timed the ball as it rolled down the track and fell to the floor.

Answers to questions.

Of general interest but not relevant to this semester:  An ohm is the resistance of a circuit element which, when under potential difference 1 volt, allows 1 amp of current to flow.  For a given potential difference the current that flows is inversely proportional to the resistance (as an example if the resistance is 2 ohms, twice as great, then only half as much current would flow in response to a 1 volt potential difference.)  For a given resistance the current that flows is directly proportional to the voltage.  The first statement is the definition of the ohm.  The last two statements are summarized in Ohm's Law I = V / R.  This question isn't related to anything we will do first semester.

Internal vs. external force (an idea we will be developing later but which is worth thinking about now): Whether the normal force between the beam and the supporting wedge on the buoyant balance is internal or external depends on your definition of the system.  If the system consists of, say, two beams with fixed ends, and a wedge between them, then the normal forces on the wedges are internal to the system.  If your system consists of just one of the beams, then the normal force is external.

Notes related to the concept of Vectors

Speed is just a number with units of length / time (university physics students:  that makes it a scalar quantity).  Speed is equal to the magnitude of the velocity, which is a vector quantity.  In a sketch showing velocity the length of the vector represents the speed, and the direction of the vector represents the direction of motion.

Given a vector A in a plane, you can choose how to orient your x and y axes.  Then A_x is the component of the vector in the x direction, and A_y is the component in the y direction.  We will be doing some sketching exercises shortly, which should begin to clarify how this scheme is used.

A displacement vector is a vector from one point to another.  Its magnitude is the distance between the points, and its direction is the direction from the first point to the second.  For example the vector from (3, 6) to (9, 14) corresponds to the hypotenuse of a right triangle with legs 9 - 3 = 5 in the x direction and 14 - 6 = 8 in the y direction.  Its magnitude is sqrt( 5^2 + 8^2 ) = sqrt(89).  Its direction is arcTangent( 8 / 5 ).  We will study this more specifically soon.

If you know only the magnitudes of two displacement vectors S and T , you can't find their difference, which depends not only on their magnitudes but on the angle between them, and on their directions.

Two vectors whose x and y components are equal and opposite will have the same tangent.  If you know the sign of x you can determine whether the vector is in the first or fourth quadrant (positive x) or the second or third (negative x).

This note is relevant mostly to those taking multivariable calculus.  It goes beyond the scope of the University Physics course: 

If you are moving along a circle, then your velocity vector is tangent to the circle. 

If you are moving at a constant speed on the circle, then your acceleration vector is centripetal--i.e., it points toward the center of the circle, in the direction perpendicular to the velocity. 

If you are speeding up or slowing down as you move on the circle, your acceleration still has its centripetal component, but it also has a tangential component, a component either in the direction or opposite the direction of the velocity. 

The cross product of the acceleration and velocity vectors is perpendicular to the circle, and this defines the orientation of the plane of the circle.  If you want to move to a circle within a different plane, you're going to have to accelerate in the direction perpendicular to that plane. 

For any smooth motion in 3 dimensions we define three vectors that follow the moving point:  the unit tangent vector (in the direction of the velocity), the unit normal vector (perpendicular to the velocity, in the direction of the component of the acceleration which is perpendicular to the velocity), and the unit binormal vector which is perpendicular to both (in the direction of the cross product).

The nature of a law vs. a theory:  A law needs to be in some sense universal and general.  A theory can be more local and specific.  There is no clear dividing line between laws and theories, but there is pretty broad agreement on what is and is not a law.

Rate of Change definition applied to velocity:

If you know the time down an incline and the distance, you can easily calculate the average velocity, which is the average rate of change of position with respect to clock time.  By definition of average rate the average velocity is

So for a ball down a ramp, the change in position could be the length of the ramp, the change in clock time the time required to roll from one end to the other.  You would therefore divide the length of the ramp by the time required to roll down the ramp.

Equilibrium of the Balance

The balance is in equilibrium with the total of the upward and downward forces acting on it is zero, and when the total of the torques is zero. 

We haven't really defined torque, but think about this:  If we change the position of a paperclip, the upward and downward forces don't change.  The force on the clip changes position.  If we move it further from the balance point the system will tip in the corresponding direction.  It has more leverage. 

Torques are a way of taking account of forces and their leverage.  We will define torque at a later time, so as not to get too confusing now.

The water in which a paper clip is immersed exerts pressure-related forces on all points of the clip, some upward, some downward, and some horizontal.  The pressure increases with depth, with the result that the total of the upward pressure-related forces on the clip is greater than the total of the downward pressure-related forces.  So the total force exerted by the water on the clip is in the upward direction  This is the force we call the buoyant force.

The more of the clip is submerged, the greater the upward buoyant force. 

If we add a mass to the 'clip end' of the beam, the paper clip will descend, increasing the upward buoyant force until it counterbalances the weight of the added mass. 

Conversions requiring only inch-to-cm info

The only conversion factor you theoretically need at this point is 1 inch = 2.54 cm, provided of course you know how many feet in a mile and such, and the meanings of prefixes like 'milli', 'kilo', 'centi', etc..  Of course it takes practice, and we'll be seeing more examples.

You should be able to use the fact that 1 inch = 2.54 cm to derive the conversion 1 mile/hr = 0.447 m/s, using the fact that there are 12 inches in a foot and 5280 feet in a mile.

2.0 has two significant figures.  2.04 has three significant figures.  A zero in the middle of a bunch of significant figures is significant.

 

Generally you should have an assignment completed by the following Monday, and you should spend some time on the course between Monday and Wednesday in order to be prepared for the Wednesday class.

Tension Force exerted by Rubber Band, graph of tension vs. length

If you stretch the rubberband far enough, you can feel it stiffen, and if you stretch it much further it will break.  The graph of force vs. length develops a large slope when the band stiffens.  When it begins to give way, just before breaking, the slope might well decrease (that would depend on how quickly you are changing the length, because beyond a certain point the rubber band undergoes irreversible changes).

If you pull on two opposing rubber band systems, and the two systems remain stationary, then they are exerting equal and opposite forces on one another.  It doesn't depend on how hard you are pulling.

On a graph of rubber band tension vs. length, tension is the vertical or 'y' component and length is the horizontal or 'x' component. 

Miscellaneous answers to questions

If you dive into the water from the side of a pool you hit the water at maybe 12 miles/hour.  If you hit flat it can sting.  If you do the same from a diving board it can really sting.  Do it from the high dive and you can knock the wind out of yourself.  That would be at about 25 miles/hour.  Every time you double your speed your energy increases by a factor of 4.  When you hit the water you dissipate the energy, and much of that occurs when you first strike the surface.  If you hit at 50 mph you will not only knock the wind out of yourself, you will probably bruise your ribs severely.  At 80 mph you're going to break most of your ribs and sustain massive internal injuries.  You probably won't survive.  At 100 mph you don't really have much of a chance.

A Newton is 9.8 / 2.2 pounds.  We haven't yet talked about the definition of a Newton, but it's the net force required to accelerate 1 kg at 9.8 m/s^2, the acceleration of gravity.  We'll be measuring that soon, then we can begin to make sense of the definition of a Newton.

Don't delete anything from a document containing questions and answers.  Insert your answers and/or questions, but don't delete anything that was already there.  You're going to want to see all that information when the document is posted, and in order to efficiently move through your document to review it I need for all the information to be there in a predictable order.

To find a specific position function from a given velocity function you would integrate from some fixed reference time to a general time.  For instance you would integrate from t_0 to t, where t_0 is a fixed (constant) clock time and t is the variable clock time.  However the general integrations we did in the last class will suffice for current applications.

The determinant definition of the vector product is probably the easiest way to remember it.

For the ball falling off the track, or for any other motion, velocity is the rate at which position changes, and acceleration is the rate at which velocity changes.  Of course it takes practice to understand how to apply these definitions.