Class 091021
In class we observed a steel ball rolling down an incline with slope approximately .03, colliding with a stationary marble, with both objects then falling 90 cm to the floor.
The steel ball, when unimpeded by the marble, traveled 8.5 cm while falling.
The marble (which was initially stationary), after being hit by the ball, traveled 16 cm while falling.
The steel ball after hitting the marble traveled 5.5 cm while falling.
You should be able to figure out the time of fall, assuming the initial vertical velocity to be close to zero.
Approximating the time of fall by .4 sec we conclude that the ball reached the end of the ramp at a speed of about 20 cm/s, the marble traveled at 40 cm/s after being struck, and the steel ball was moving at about 12 cm/s after striking the marble.
`q001. Using these figures:
What is the change in the ball's velocity between the instant before and the instant after it strikes the marble?
What is the change in the marble's velocity the instant before and the instant after the collision?
What do you therefore conjecture to be the ratio of the ball's mass to that of the marble?
`q002. Using your own data for the same experiment, what are the unimpeded steel ball's velocity at the end of the ramp, the velocity of the glass marble the instant after being struck by the ball, and the velocity of the ball immediately after striking the marble?
What therefore should be the ratio of the ball's mass to that of the marble?
Be sure to include your raw data and explain how you get from the raw data to your conclusions.
`q003. Let's assume that we can neglect forces exerted by the ramp on the glass marble and the steel ball, and that we can in fact neglect all forces except the forces exerted by each ball on the other. We can therefore regard the two balls, during the time of collision, as a closed system. We have draw our conclusions about the velocities involved in the collision from the projectile behavior of the objects after collision.
From F_net `dt = `dp and the fact that the forces exerted by the two objects are equal and opposite, we can draw the conclusion that the momentum change `dp of one object should be equal and opposite to that of the other. (recall that p = m v, so `dp = p_f - p_0 = m vf - m v0)
Since momentum changes are equal and opposite, we conclude that the total momentum of the system is the same immediately after collision as it was immediately before.
Let's let m1 and m2 be the two masses, let v1 and v2 be their velocities before the collision, and let u1 and u2 be their velocities after collision.
What are the values of v1, v2, u1 and u2 for your trial?
`q004. Using the symbols m1, m2, v1, v2, u1 and u2, give the symbolic expression for each of the following:
The momentum of the steel ball before collision.
The momentum of the glass marble before collision.
The momentum of the steel ball after collision.
The momentum of the glass marble after collision.
The total momentum of the two objects before collision.
The total momentum of the two objects after collision.
`q005. Write the symbolic equation which states that the total momentum of the two objects before collision is equal to their total momentum after collision. (your equation will set your answers the last two expressions in the preceding question equal to one another).
`q006. Substitute the values you observed for v1, v2, u1 and u2 into your equation. Your equation will still have the symbols m1 and m2, but the other symbols will be replaced by the various velocities you observed.
`q007. Rearrange your equation so that all the quantities containing m1 are on the left-hand side and all the quantities containing m2 are on the right-hand side.
Factor m1 out of the left-hand side and m2 out of the right-hand side.
Solve the equation for m1. The left-hand side should end up just as m1; the right-hand side will include the symbol m2.
Give the steps of your solution:
The foam wheel was observed to rotate through 8 revolutions, coming to rest after about 20 seconds. We want to find its acceleration. We follow the usual procedure for this situation: find the average velocity, use this along with the final velocity to to determine the initial velocity, then find the rate of change of velocity with respect to clock time.
Its position is measured in revolutions.
The average rate of change of position with respect to clock time is therefore
ave rate = change in position / change in clock time = 8 rev / 20 sec = .4 rev / sec.
Revolutions are a measure of angle or angular displacement. The 8 revolutions therefore represent a change in angular position. The rate of change of angular position will be called angular velocity. Angular velocity is represented by the symbol omega (the Greek symbol that looks something like a W but should never be called 'double-yew' or written as 'w'; write it as 'omega').
So we say that
ave angular velocity = .4 rev / sec, or
omega_Ave = .4 rev / sec.
This is the 'average velocity'. To find the acceleration we have to find the initial velocity.
A linear velocity vs. t trapezoid tells us that since final vel is zero, the initial vel is double the average.
Our conclusion:
the initial angular velocity is .8 rev / sec, double the average angular velocity
the average rate of change of angular velocity with respect to clock time is therefore
ave rate = (0 rev / sec - .8 rev/sec) / (20 sec) = -.4 rev / sec^2.
This is what we call acceleration. Since it's the average rate of change of angular velocity, we call this the angular acceleration. Angular acceleration is represented by the symbol alpha.
Thus our conclusion is that
angular acceleration = alpha = `dOmega / `dt = -.04 rev/s^2.
Our conclusions:
omega_Ave = .4 rev / sec
omega_f = .8 rev / sec
alpha = -.04 rev / s^2.
We could express these quantities using degrees instead of revolutions. Since 1 revolution = 360 degrees, we have
omega_Ave = .4 rev / sec = .4 (360 deg) / sec = 144 deg / sec
alpha = -.04 rev / s^2 = -.04 * 360 deg / s^2 = -14.4 deg/s^2.
Or we could express these quantities in terms of radians. You might not be familiar with radians, and if not you should note and remember that 1 revolution = 2 pi radians. Using this fact
omega_Ave = .4 rev / sec = .4 * 2 pi rad / sec = .8 *pi rad / sec and
alpha = -.04 rev / s^2 = -.04 * 2 pi rad / s^2 = -.08 pi rad / sec.
How would someone who measured the motion in cycles of a 9 cm pendulum calculate these quantities?
We know that for this length 1 cycle = .2 sqrt(9) sec = .6 sec.
We know omega_Ave in rev / sec. To get this quantity in rev/cycle we need to know what to substitute for 'sec'.
If 1 cycle = .6 sec, then 1 sec = 1 / .6 cycle = 1.67 cycle. So we can substitute 1.67 cycle for 'sec', obtaining
omega_Ave = .4 rev / sec = .4 rev / (1.67 cycle) = .24 rev / cycle.
alpha = -.04 rev / sec^2 = -.04 rev / (1.67 cycle)^2 = -.014 rev / cycle^2.
`q008. If someone observed that the wheel went through 10 revolutions in 50 cycles of a pendulum of unspecified length, then in units of revolutions and cycles, reason out the values of omega_Ave and alpha.
`q009. If we later find that the pendulum in the preceding has length .8 cm, then what would be the values of omega_Ave and alpha in terms of revolutions and seconds, in terms of degrees and seconds, and in terms of radians and seconds?
An Atwood machine consists of a 20 kg mass on one side and a 22 kg mass on the other. Let the positive direction be that in which the 20 kg mass descends. The system therefore experiences forces of 20 kg * 9.8 m/s^2 = 200 N (approx.) in the positive direction, and 22 kg * 9.8 m/s^2 = 220 N in the negative direction, so the net force is 200 N - 220 N = -20 N. The total mass is 20 kg + 22 kg = 42 kg, so the acceleration of the system is
a = F_net / m = -20 N / (42 kg) = -.48 m/s^2.
We come to this conclusion without considering the tension in the string connecting the two masses. To figure out this tension we could consider just the 20 kg mass. The forces on this mass are the tension T, acting upward, and the weight (200 N) of the 20 kg mass, acting downward. Letting downward be the positive direction (consistent with our choice in the original problem) we see that
F_net = 200 N - T.
We know that F_net = m a, so we can say
m a = 200 N - T.
We also know that a = -0.48 m/s^2. Of course we know that m = 20 kg. So we can solve the equation for T:
T = 200 N - m a = 200 N - 20 kg * (-0.48 m/s^2) = 200 N - (-10 N) = 210 N.
That is, the tension is 210 N. We have already assumed that the tension acts upward (we did this by saying that the tension force is -T), so the tension is 210 N in the upward direction.
This makes sense. The net force on the 20 kg mass is 200 N - 210 N = -10 N, i.e., 10 N in the upward direction. Its acceleration is therefore -10 N / (20 kg) = -.5 m/s^2, within approximation error of the more accurate -.48 m/s^2.
`q010. We have seen that the acceleration of the system is -.48 m/s^2. Use this fact to find the tension acting on the 22 kg mass. Your reasoning will be similar to that used above, but some of the details will be different.
A 20 kg mass on a 10 degree incline, with the incline running down and to the right, experiences a gravitational force of about 200 N.
If an x-y coordinate system is imposed on this system, with the x axis down and to the right parallel to the incline, then the downward gravitational force will lie at angle 280 deg, measured counterclockwise from the positive x axis.
From the figure on the board we estimated that the y component of the 200 N weight is about -.95% of the weight, while the x component is about +30% of the weight. According to our estimates we thus estimated
Using the sine and cosine we can make these calculations more accurate:
`q011. Using the sine and cosine what do you get for the x and y components of the weight? How far off were our ballpark estimates?
`q012. What are the x and y components of the weight of a 40 kg mass on a 20 degree incline?
Sketch and estimate, give your estimated percents, and the resulting x and y components of the weight.
Use sines and cosines to get the precise values of the components.
`q013. Continuing the preceding, if the normal force is equal to the y component of the weight and the frictional force resisting motion is 15% of the normal force, then what is the net force on the block? What threrefore is its acceleration?
For a 50-gram pendulum held at 10 degrees from vertical by a horizontal thread, what is the tension in the pendulum string, and what is the tension in the thread? What will be the acceleration of the pendulum mass if the thread is cut? What will be its acceleration when the angle is 5 degrees?
Three rubber band chains all have the same force vs. length graph, with a good straight-line approximation
tension = .5 N / cm * (length - 60 cm),
for any length greater than 60 cm. The rubber bands all exert forces on a small paperclip, and the rubber bands are pulled back so that the paperclip is stationary at the origin. The ends of the first and second rubber bands are at points (50 cm, 55 cm) and (-40 cm, 60 cm), respectively. Where is the end of the third rubber band?
... pendulum at 40 deg held by a thread inclined 30 deg above horizontal ...
... bird on a wire ...
... ball down one track onto another at small angle from straight; change in velocity nearly perpendicular to each path; force apparent by motion of 2d track, greater for greater velocity, or for greater change in angle; impulsive force, proportional to v^2, inversely proportional to radius of curvature
... subtract two velocities (equal speeds, small angle) and show that difference nearly perpendicular to both
... two velocities of magnets on rotating strap, change nearly perpendicular, fn of omega, or of v and r, leads to centrip accel, centrip force
address 121 vs. 201 (focus on labs and problem sets)
atwood from last time; friction; move into analyzing frictional forces on incline
what would you feel if you were riding the Atwood (elevator problem)
text ch 2, 4, 6, 7; find problems ...
Knowledge to date:
unif accel motion (definitions, equations)
Newton's 2d law
definitions of energy, work, impulse, momentum
work-KE thm, work-energy thm in terms of `dW_NC_ON, `dPE, `dKE, conservation of energy in isolated systems, special case `dW_NC = 0
impulse-momentum thm, conservation of momentum in isolated systems
vector calculations
representing forces, velocities, etc., by vectors
all basic systems to date
introduce collisions
lab instruments: proj to measure velocity, pend timer, rb force (& energy source), etc.
basic systems: 1 & 2 inclines, pendulum, incline-projectile, pend-projectile, rb & block, atwood, 3 rbs, rotating strap
automatic techniques: motion from or to rest with unif accel; time of fall from init vertical velocity zero; slope of incline (reconciling differing units of measurement when necessary), resolving gravitational force at given slope or angle, resolving pendulum tension
do intro collision expt (system isn't quite isolated)
rotating strap (keep verbal count, record last number heard at each half-rotation)
begin to think about motion on circle, centripetal acceleration, centripetal force
accel of domino across smooth tabletop, on inclines
... angles, forces for 3-domino ramp; for 24 cm pendulum pulled back 4 cm; for initial velocity of ball off end of 4-domino incline; for a 3- and a 5-domino rubber band opposed to another of given (linear) calibration graph
... energy stored in rubber bands
... const-vel ramps, accel in opposite direction -> force of rolling friction
... do we have a rich experiential base yet? can we create one with D? pictures and story construction, bringing multiple skills to bear on a complex problem
on experiments: ability to get data quickly touted as excuse for excessive use of computer-interfaced instruments; another option is to use really simple instruments, which in reasonable combination with interfaces etc. are more than sufficient to the task of learning physics and experimental technique
do expt with 3 rb, then sketch and estimate three vectors corresponding to claimed forces (see how long with your chosen # of dominoes (3-7), 'war' one against 2 others at 90 deg to find equilibrant, analyze
3 dominoes instead of 1 on 2-ramp expt; does apparent coeff of rolling friction change and if so by how much; or accel on const-vel ramp .... use screw mechanism
labs: establish basic instruments and their properties:
revisit F vs. x graphs for rbs
Atwood graph for neg accel; effect of friction on graph. Either counterbalance both ways, graph a vs. wt_net, discover friction; or accel in neg direction etc..
`dt for intercepting ball at perpendicular, at angles (collision cs changes with angle); why not just let the thing go across the floor and check out the reaction? optimize angle of ramps (head-on not good because of uncertainties in path, perpendicular probably not good because of lesser cross-section), steepness of 'missile' ramp (to some extent steeper means less time delay between release and result; however too fast reduces chance of target hitting 'missile' as opposed to reverse ... )
revisit v vs. t graph for ball down ramp
resolve forces on incline and for pendulum
init velocity of ball off inclined ramp (how much error to assume horiz velocity is uniform?)
is that 90% cm horiz range really possible for 24 cm pendulum?
some might work better from equations ... provide later in this document ... interpret this equation in terms of our lab experiences
qa's 15-18 (impulse-mom, vectors; 19 is on vector quantities)
20 on forces, inclines, friction
21 more projectiles
22 motion in a force field
Don't remove or overwrite **** or &&&&. You and I both need those marks to be able to separate your answers from the question. Your work would get less scrutiny if these marks aren't both present, and it will be harder for you when you want to review it. If any of these marks are missing I will likely ask you to reinsert any missing marks and resubmit.
if rb has tension equal to weight of dominoes, what happens
if double; if fourple, etc.
rb force * dist, dist of slide
draw vectors representing init vel of ball off ramp, etc. etc.
assign intro set 5; briefly introduce vectors in context of incline to show what they're good for
Homework:
Your label for this assignment:
ic_class_091021
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.
Read Chapter 2 of the text and take notes on things you do and do not understand. I might ask for a synopsis next time.
Do next two qa's.
Intro prob sets 3 and 4