And the first 8 minutes of Doc Schuster’s video on simple harmonic motion:

The lab that we did was identical to what I did in the last high school physics class so I’m just going to link to that post. Physics 12 Hooke’s Law. That post also has a few extra videos on this topic. Because this is such a straight forward lab its a good one to have the students write up as a lab report.

This is going to be a short post because its a very busy week with multiple events and college visits on the schedule. You can check out the post on momentum from the last time I taught physics for a more detailed discussion. We basically did the same experiments, the only thing different is I asked the students to come up with their own experiment to show the conservation of momentum and did not give them any handouts. Some students used the air track and gliders, a few used the hover soccer disks and one group used a Newton’s Cradle.

Here’s two photos from one experiment. The first photo is a screen shot before the collision, with glider 1 coming in with a constant velocity from the left and glider 2 sitting at rest in the middle of the airtrack. Both gliders have velcro on them so when they collide they will stick together. The second photo shows the two gliders after the collision when they are stuck together and now moving at a slower velocity. The red dots in the photos show the position of glider 1 throughout the experiment. Before the collision the dots are fairly far apart, but after colliding and sticking to glider 2, effectively doubling the mass of the object in motion, the velocity decreases, indicated by the dots being much closer together. Students were able to determine by analyzing the data that the velocity decreased by a factor of 2 as expected.

We pretty much jumped right into the lab activity today, which was to measure the coefficients of static friction, μ_{s}, and kinetic friction, μ_{k}, for a block on a surface (table cloth, cardboard, rough wood, whatever). There were a couple of ways students could make the measurements. The first involved placing a block on the surface and tying a string from the block to a weight hanging over a pulley. The amount of weight hanging off the string is the pulling force on the block. If the block is not moving (a = 0, therefore net F = 0) then the force of friction must be equal and opposite to the pulling force. Likewise if the block is not accelerating up or down then the normal force is equal and opposite to the force of gravity, the block’s weight. The weight of the block is measured in Newtons on a spring scale. If we had been doing the experiment on a ramp, then the normal force would NOT be equal to the force of gravity, we’d have to take into account the angle of the ramp – but today we just dragged the blocks horizontally. To measure the coefficent of static friction the students had to find the maximum pulling force they could exert on the block WITHOUT making it move. Once the block starts moving the force of friction is now kinetic friction, and is less. We’re trying to find the maximum force of static friction, F_{static} = μ_{s}F_{Normal}, how much force do we have to overcome to make the block move. When they had that force, roughly 1N, they divided it by the normal force of the block to find the coefficient of static friction. The coefficient of frictions, static and kinetic, depend on the two surfaces. If the block is rough like sandpaper and is being dragged over a rough wood board, then the friction is going to be higher than if the block has been sanded smooth and is being dragged on a polished surface.

The coefficient of kinetic friction should be less than static friction, its easier to keep an object moving then to get it moving in the first place. Think about trying to move something heavy like a fridge, at first it won’t budget until you exert a very large force, but once you have it moving its easier and you don’t have to push so hard. To measure the force of kinetic friction, students dragged the block on the same surface at constant speed with a spring scale. If the block is at constant speed (at least roughly constant), then its acceleration and net force must be zero, so the force measured on the spring scale is equal (opposite direction) to the force of friction on the block.

Just as before they can now find the coefficient of kinetic friction, F_{kinetic} = μ_{k}F_{Normal} by dividing the force on the spring scale by the normal force (weight they measured earlier) and indeed it was less than the coefficient of static friction.

The third measurement involved using the Go Direct Force Sensor from Vernier. Its basically a digital spring scale that sends the data directly to your iPad via bluetooth. Using this sensor students collected data while they slowly increased the force on the block until it started moving and then tried to keep it moving at a constant speed.

The graph of the data looks like this:

As the student increases the force on the block the Go Direct sensor records the increasing force, indicated above with the red line, but the block has NOT started moving yet. As the force reaches 0.72 N, the block starts moving and the force drops to 0.52N (green line). From this one data set students can get read off the forces required for both the static and kinetic coefficients of friction.

Students were asked to read 2.1 Seesaws in How Things Work, or Ch 11: Rotation in the Cartoon Guide to Physics and watch the following videos before class. I really enjoy Doc Schuster’s physics videos.

Before we started the lab, I had cut out some very weird shapes out of cardboard and asked the students to find the center of mass of the shapes. Most knew to try to balance the shape on a finger and the center of mass would be the spot above their finger when it was balanced. But some of the shapes didn’t work for this, like the one on the left in the photo below.

Another way to find the center of mass is to hang the object on a nail or tack and then let a string with a weight (plumb line) hang down as well. The center of mass of the object will fall on this line. If you do this from two points (2nd almost horizontal line was found by hanging the shape from the hole on the right), the center of mass can be found at the intersection of the two lines. For this oddball shape below we see the center of mass is not in the object at all, which is why it was impossible to balance it on a finger.

I also had a bird balancing toy out and handed out a paper craft from Ellen McHenry’s website, so the students could make their own balancing bird.

The lab for today was one I did before, but I think we got it to work a bit better this time. Students tied a rubber stopper to a string, fed the string through an acrylic straw and then tied a weight to the other end of the string. By holding on to the straw and letting the string slide through the straw, the students could make the stopper go in a circle of radius r, above their heads at constant speed, v (or at least that’s the goal). The weight on the end of the string is the centripetal force and is measured using a spring scale. For uniform circular motion F_{c} = ma_{c} = mv^{2}/r.

We can rearrange that so v^{2} = ma_{c}r. In this lab we measures the velocity of the rubber stopper being swung around at different values of the radius, r. To measure the speed of the stopper we need to know the distance traveled (d = 2 π r x 10) as it completes 10 revolutions and the time, t, it takes to travel that distance: v = d/t. Since its a bit difficult to measure the time of 10 revolutions, students did each measurement 3 times and used the average time to calculate speed. Once they had the speed, v, they foundv^{2} and graphed it as a function of the radius, r. Data should form a line (y= mx + b, v^{2} = y, m(slope) =a_{c}, x = r and b(y-intercept) = 0 ) with a slope equal to the centripetal acceleration. The value they get from their slope should be the same as F_{c} / m=a_{c }when they plug in the force of the weights hanging on the string in Newtons and the mass of the stopper in kg (measured on a triple beam balance).

Since there were a lot of calculations in this lab, I encouraged the students to try using a spreadsheet on their tablets or smart phone. Most had never used a spreadsheet before so its a good learning experience. (I frequently get frustrated using Numbers to make graphs and found this website which explained a problem I always had.) Here’s a graph of one set of data and the best line drawn through the data points and the origin.

The slope of this graph is 53 m/s^{2} which is very close to the value of acceleration found by taking the known force and dividing by the mass of the stopper (54 m/s^{2}).

This lab worked much better this time and I think using the acrylic straws instead of empty pens helped, as did using a larger centripetal force (0.7-1.0 N).

I wanted to make sure the students understood gravitational potential energy and kinetic energy so I had them measure the potential energy and kinetic energy of a pendulum bob at different times during its swing. When the pendulum bob is at its highest point, A, then all its energy is potential energy (PE = mgh) because its velocity is zero (and therefore its KE = 0) for the split second before it falls back down again. At point C, the pendulum is moving at the greatest velocity and therefore has its maximum kinetic energy (1/2 m v^{2}). It also has its lowest potential energy at point C, equal to zero if we choose that position as our origin (x=0,y=0). So at point A all the energy is potential energy and as the pendulum swings through point B it will have both potential and kinetic energy, but by the time it reaches point C all the potential energy has been converted to kinetic energy.

The top graph below shows the horizontal, x (red), and vertical, y (blue), positions as a function of time. The bottom graph shows the velocities in the x (red) and y (blue) directions. I marked the location of points A, B and C on the graph – again you can see at point A that the velocities are zero and the positions are at their maximum/minimum values and at point C the positions are both zero but the velocity in the x direction is at its maximum. The velocity in the y or vertical direction is also zero at point C because the pendulum bob is really only moving horizontally at that moment. At point B, and you can pick any point between A and C, the pendulum has both potential and kinetic energies.

Students made a table in their lab books and recorded the height (y) of the pendulum at the three points, as well as the velocities (x & y). Then they calculated the potential energy at each point (mgh) and the kinetic energy (1/2 m v^{2}), m = mass of the pendulum bob which was 0.05 kg. Finally they calculated the total energy (PE + KE) at each point and hopefully found the numbers to be roughly the same. For the data shown above we got 0.074J for the total energy at each point.

This is a nice lab but the students get really confused looking at the graphs and confusing position for velocity, x for y, etc. Another problem that popped up in a couple of groups was students didn’t put the origin at the bottom of the swing (C), so when the pendulum was at C the graph showed a nonzero y-position. You also need to make sure you mark enough points (Video Physics app) to catch the pendulum turning around (A) and coming back through the lowest point (C).

Without even doing any math, but just looking at the graphs and realizing that the data for the height of the pendulum, y (blue dots), is proportional to the PE, reaches its maximum at the same time the velocities, and therefore KE, both go to zero. As the horizontal velocity (red data in bottom graph) reaches its maximum value, and therefore KE will be its maximum, you can see the height (and therefore PE) is also zero. As one goes up the other goes down. I guess one way to simple the graphs would have been to have them just plot vertical position (y) and horizontal velocity as a function of time and just do points A and C. Something to think about the next time I do this lab.

I asked the students to watch these videos before class.

George Lakoff has retired as Distinguished Professor of Cognitive Science and Linguistics at the University of California at Berkeley. He is now Director of the Center for the Neural Mind & Society (cnms.berkeley.edu).