24-Sept-2016: Modeling Friction Forces

Apparatus: A block, a flat board, a pulley, some different  masses, a force sensor, a motion sensor, and a computer.

Purpose: through the experiments and building the model to involve ftiction.

Model: μs = fstatic, max/N,  μk = fkinetic/N, F =   fkinetic, a = sinθg - cosθμkg,

sinθ = cosθμS, μk = (a(M+m)+mg)/Mg

μs is the coefficient of the static friction, μk is the coefficient of the kinetic friction, f is friction, N is normal force, and θ is the angle of the borad.

Process: Part 1: Static Friction

    We place the block on the flat board and use a string which one side tids on the block, and the other side hang a mass and cross through the pulley.

    We weigh the block to get M, and then we keep add mass on the other side of the string until the the block just begin to move. We record the hanging mass. Then, we add a 100g mass on the block and keep add mass on the hanging side until the block just begin moving. Repeat this process five time, record the M and hanging mass m.

    Then, we get the datas:

                      M          m

                     0.18             0.056  

                     0.28             0.085

                     0.38             0.160

                     0.48             0.235

                     0.58             0.320

    We plug these datas into LabPro and use the liner fit. Accoring to the model:  

                                                               μs = fstatic, max/N

    The slope of the line is μs. We get the μs = 0.48.

    

Part 2: Kinetic Friction:

    In this experiment, we use LabPro and force sensor to collect data F. First, we calibrate the force sensor by using 0 g and 1000g. Then, we weight the block to get the block's mass M. We put the block on the flat board, and then we pull it with a  constant speed. We use the Excel to get the average force F. Then, we add 100g on the block and pull it with a constant speed. Repeat it five times and record the datas.

    We get the datas:

                                                        M              F

                                                     0.18                    0.52

                                                     0.28                    0.79

                                                     0.38                    0.94

                                                     0.48                    1.24

                                                     0.58                    1.47

    We plug these datas into LabPro and use the liner fit. Accoring to the model:  

                                              μk = fkinetic/N, F =   fkinetic

    The slope of the line is μkg. We get the μk = 0.263.

Part 3: Static Friction From A Sloped Surface:

   We place a block on a horizontal flat board, slowly raise one end of the board, tilt it until the block starts to slip. Then, we use our phone to measure the angle of the board θ. θ = 22°

    By using the model:

                              sinθ = cosθμS

    We get the μS = tanθ = 0.4

Part 4: Kinetic Friction From Sliding A Block Down An Incline:

    We put a motion detector at the top of the incline which steep enough that a block will accelerate down the incline. We measure the angle of the incline θ and the acceleration of the block a by using motion detector.

  We get a = 0.82 m/s^2, θ = 25°

    By using the model:

                                 a = sinθg - cosθμkg

   We get μk = 0.37

Part 5: Predicting The Acceleration of a Two-Mass System:
    We place the block on the flat board and use a string which one side tids on the block, and the other side hang a mass and cross through the pulley. Then, we hang some mass which are enough to let the block accelerate. We use the motion detector to measure the accleration of the block a. We weight the block and the hanging mass M and m.
    We get a = 2.65m/s^2, M = 280g, and m = 250g.
    By using the model, and the μk from last experiment: 
                          μk = (a(M+m)+mg)/Mg, μk = 0.37
    We can calculate the ac = 2.71 m/s^2.
    Conpare a and ac, we get two very similar result.

20-Sept-2016: Measuring the Density of Metal Cylinders and Propagated Uncertainty in Measurements

Apparatus: Vernier caliper, a aluminum cylinder, and a tin cylinder.


Purpose: To find out the cylinder's density, and calculate the propagated uncertainty

Model: ρ = m/v

Process: We use the vernier caliper to measure the height and diameter of aluminum cylinder and tin cylinder. Then, we weigh the cylinders.

 We get the datas which shows on the white board and calculation is showed on the white board:

Therefore,we get these density of the cylinders and the propagated uncertainty of these density.

22-Sept-2016: Trajectories

Apparatus: 
    Two aluminum "v-channel", a steel ball, a board, a ring stand, a clamp, some paper, a carbon paper.

Purpose: 
     To use your understanding of projectile motion to predict the impact point of a ball on an inclined board.

Model: 
    V0*sinθ1 = Vx, V0*cosθ1 =Vy, Δx1 = Vx*t1, Vy*t1+(gt1^2)/2 = h, tanθ2 =Δy2/Δx2
    V0 is the initial velocity of the ball when the ball left the channel, Vx is the velocity in x-axis, Vy is the velocity in y-axis, t1 is the time thai the ball hit the ground, Δx is the distant between the end of the channel and the hit point on the ground in x-axis, h is the heigh of the end of the channel, and θ2 is the angle of the board.

Process: 
    First, we set up the apparatus. We use the rind stand and a v-channel to make a incline and connect it to a horizontial channel to make a "launch trajectory" of the ball. We try to launch the ball once in order to find out the possible landing point, and then we put a carbon paper on the gound around the predicted landing point and cover a piece of paper on it.

    We release the ball on the incline at a same heigh five times. By determining the distant between the end of the channel and the hit point on the ground in x-axis, we get  Δx1 = 0.605m. By detemining the heigh of the end of the channel, we get h = 0.932m.
    By using the equation and calculation:
                                                          Vy*t1+(gt1^2)/2 = h
                                                          Δx1 = Vx*t1
                                                                             V0*sinθ1 = Vx
                                                                             V0*cosθ1 =Vy
                                                                              h = 0.932m
                                                          Δx1 = 0.605m
     We get:                                                                      
                                                                   t1 = 0.4365s, V0 = 1.387m/s

    Then, we put a board at the end of the channel to make anothor incline, and we use our cellphone to determind the angle of the board, θ2 = 44°.  We place a carbon paper and a paper on the incline, and then we releave the ball at the same position five times. We determine the distant between the end of the channel and the landing point d = 0.535m.
    By using the equation and calculation:
                                                          tanθ2 =Δy2/Δx2
                                                                             V0*sinθ1 = Vx
                                                          V0*cosθ1 =Vy
                                                          Δx2 = Vx*t2
                                                          Vy*t2+(gt2^2)/2 = Δy2
                                                                             D = Δx2/sinθ2
                                                              t1 = 0.4365s, V0 = 1.387m/s ,  θ2 = 44°
    We get:
                                                          D = 0.527

    We compare the result D that got by calculation and the result d that got by determining. We get two very similar number. We conside that there are some error when we determine. We think our experiment is success.                                                                                              

19-Sept-2016: Modeling the fall of an object falling with air resistance

Apparatus: Some coffee filters, a meter stick, a camera, and a computer

Purpose: Build a model about the fall of an object falling with air resistance.

Model: Fresistance = kv^n, a =(mg-Fresistance)/m
Fresistance is the force of the air resistance, k is a contant number which takes into account the shape and area of the object.

Process: We go to the Design Technology building, and our professor will drop the coffee filters from the second floor. Before dropping the filters, professor put a black cloth on the railing with a meter stick on it in order to let us know the scale and see the coffee filters better. Then professor starts dropping the filters. In order to change the mass without changing the k value, professor simply overlays the coffee filter together. Professor drops one filter, two filters,....and six filters separately. We use the computer's camera to record it.

Part 1
    We use the LabPro to open these vedio. First, we use the meter stick to find out the scale. We change the number of frame to ten frames per second. Then, we mark the position of the filter for every frame. Therefore, we get the graph about position vs. time. We use the liner fit to find out the constant speed for every vedio. 

    V1=1.4555m/s     V2=1.809m/s       V3=2.379m/s 
    V4=2.922m/s       V5=3.827m/s       V6=3.737m/s  
    
    By weighing, we can get the mass of the filter.
    m=0.00089494kg
    By the formula                      a =(mg-Fresistance)/m 
                                               a=0
                                               (mg-kv^n)/m=0
                                               m=(k/g)v^n

    we put the mass as the y-axis and the constant speed as x-axis, and then we use the curve fit-power to find the value of k and n.

    y=Ax^B
    A=(k/g)
    B=n
    k=0.001226
    n=1.688

Part 2
    ΔV=aΔt
    a=(mg-kv^n)/m
    v1=v0+Δv
    
    By plug in these equation into Excel, and fill down.

Now, we get the model of the falling object with air resistance.

18-Sept-2016: Non-Constant acceleration problem/Activity

Purpose: To figure out the motion of the an object with a non-constant acceleration and the way to calculate the non-acceleration problem.

Model: P(x)=∫ V(x) dx, V(x)=∫ a(x) dx, and a(x)=Fnet / m

    P(x) is displacement of the object, V(x) is the velocity of the object, a(x) is the acceleration of the object, Fnet is the net force of the object, and m is the mass of the object.

The problem: A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a 1500-kg rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion. The mass of the rocket changes with time ( due to burning the fuel at a rate of 20 kg/s) so that the m(t)=1500 kg - 20 kg/s*t.

Process: 

    Put appropriate values in for Vo adn Xo.

    Set Δt to be 1 second. Put in the other appropriate values in cell B1 through B4. (Recell that teh force is negative here, since it points in the negative x-direction.)

    Input a formula into cell A9 that will calculate the appropriate time, and that you can fill down. Use $B$5 in our formula, so the cell doesn't change as I fill down.

    Input a formula into cell B8 that will let me calculate the acceleration at any time. Fill down

    In cell C9 calculate the average acceleration for that first Δt interval.

    In cell D9 calculate the change in velocity for that first time interval.

    In cell E9 calculate the speed at the end of that time interval.

    In cell F9 calculate the average speed at the end of  that time interval.

    In cell G9 calculate the change in position of the elephant during that time internal.

    In the H9 calculate the position of the elephant.

    Then when the velocith between zero, we get the time is between 19s and 20s, and the displacement of the elephant is between 222.309 m and 223.454 m.

    However, because the Δt is not small enough, the displacement is not accurate enough.

    Therefore, we change the time interval from 1s to 0.1s. Then we get:

    Then when the velocith between zero, we get the time is between 19.6s and 19.7s, and the displacement of the elephant is between 246.119 m and 246.127 m.

    However, because the Δt is not small enough, the displacement is not accurate enough.

    Therefore, we change the time interval from 0.1s to 0.05s. Then we get:

    Then when the velocith between zero, we get the time is between 19.65s and 19.7s, and the displacement of the elephant is between 247.409 m and 244.411 m.

    By the calculation of mathemstic, the x(t)=248.7m. If our x is close enough to it, it means Δt is small enough.

11-Sept-2016： Free Fall Lab--determination of g (and learning a bit about Excel) and some statistics for analyzing data

Apparatus: A free fall equament with a spark generator, a paper tape, and ruler.

Purpose: To examine the validity of the statement: In the absence of all other external forces except gravity, a falling body will accelerate at 9.8 m/s^2

Model: d=1/2gt^2  (d is distance, g is gravity, and t is time)

Process: Professor pulls a piece of paper tape between the vertical wire and the vertical post of the device. He clips it with a weight to keep the paper "tight". Then, he turns the dial hooked up to the electromagnet up a bit and hangs the wooden cylinder with the metal rin around it on the electromagnet. He turns on the power on sparket thing and holds down the spark button on the sparker box. He turns the electromagnet off so that the thing falls, and then he turns off the sparker thing. Finally, he tears off the paper strip.
    We take sixteen consecutive points. We let the first point as the “zero”, and then we measure the distance between each point and the zero point. 
    This sparker work at 60 Hz. Therefore, the time between each point is 1/60 seconds.
    We puts the data (time, distance, Δx, Mid-interval time, and Mid-interval speed) into computer on Excal, and uses the Excal to get the graph.

In cell A1 enter TIME.
In cell A2 enter 0.
In cell A3 enter =A2+1/60

In cell B1 enter DISTANCE
In cell B2 enter 0
In cell B3 enter the date from the second point to the last point.

In cell C1 enter Δx
In cell C2 enter =(B3-B2)
Fill down

In cell D1 enter Mid-interval time
In cell D2 enter =A2+1/120
Fill down

In cell E1 enter Mid-interval speed
In cell E2 enter =C2/(1/60)
Fill down

Use the Chart-Chart Layout

    Let the Mid-interval speed be the y-axis and the Mid-interval time be the x-axis. Fit the points into the linear equation. Then we can get the relationship between speed and time. According to v=gt (v is Mid-interval speed, g is gravity, and t is Mid-interval time).
    By the data that Excal figure out, the gravity basing on our data is 9.50 m/s^2


Part 2: Error and Uncertainty--Physics 4A

    We collect the "gravity" that every group get.

    Model:  
    (N is the number of the "gravity", μ is the average "gravity", and Xi is the each "gravity")

In cell A1 enter Value of g
In cell A2 though A11, enter tehr class' values for g from the spark tape free fall experiment.
In cell A12 enter =average(a2:a11)

In cell B1 enter dev form mean
In cell B2 enter =a2-$a$12
Fill down

In cell C1 enter dev^2
In cell C2 enter =sum(A2:A11)
Fill down

    The Standard Deviation of the Mean of the class' data is 28.27.

29-Aug-2016: Finding a relationship between mass and period for a inertial balance

Apparatus: C-clamp, inertial balance, photogate, computer, and some masses.

Theory: Build a model of the relationship between mass and period and place the model in test.

Model: T=A(m+Mtray)^n

Process: We secure the inertial balance on the tabletop by using a C-clamp and then place a thin tape on the end of the inertial balance. We set up a photogate which can let the tape stay in the center of the photogate.

We set up the LabPro and use it to collect the datas. We place the 0g, 100g, 200g, 300g, 400g, 500g, 600g, 700g, and 800g mass separately on the inertial balance. Make sure that the period is reasonable by timing for some number of oscillations and comparing what you get with what the computer gets.
We record the period with no mass in the tray.

Using the LabPro makes a plot of  LnT  vs.  Ln(m+Mtray). And then we guess and adjust the value of the parameter Mtray until the graph gives us a straight line which means the correlation coefficient of 0.9998 to 0.9999.

We find out two values of  Mtray which make the correlation coefficient of 0.9998 and bound the actual number in a area. And then we find two equations of the relationship between mass and period accroding to calculation.

We find two other objects with unknown mass. We collect their periods, figure out their masses by our equation, and weight these two objects. And then we compare the datas to test our equation.