PHY 221 Lab 3 Vectors and Motion in 1 and 2 Dimensions

Transcription

1 PHY 221 Lab 3 Vectors and Motion in 1 and 2 Dimensions Print Your Name Print Your Partners' Names Instructions Before lab, read the Introduction, and answer the Pre-Lab Questions on the last page of this handout. Hand in your answers as you enter the general physics lab. You will return this handout to the instructor at the end of the lab period. Goals: In this lab, you will extend your experience with 1-dimensional motion and gain experience with 2-dimensional motion. With 2D motion, you will ponder the dramatic difference between motion in the vertical and horizontal directions, even when vertical and horizontal motions are happening at the same time. In order to describe motion in more than one dimension, the concept of a vector is very useful, so you will get some practice with vectors. Materials: Metal angle bracket PC with LabQuest interface for measuring instruments Motion Sensor PASCO cart on aluminum tracks Meter stick (Activity 1). Rectangular weights (Activity 2). WEB camera and computer Graph paper PHY221 Lab 3 Page 1

2 1. Measuring the gravitational acceleration g. Open Logger Pro Reset up or use the tilted track inclination from the constant acceleration experiment of last week so that the cart may fall freely along the track without pushing once it is started. Set up the computer screen to show the position, velocity, and acceleration curves with time as you did in the last lab. You may already know that all objects including yourself, close to the Earth surface, fall towards the ground with the acceleration g - called gravitational (or free-fall) acceleration, as a result of the gravitational force between the Earth and the object. The cart in our experiments is not free falling. It is rolling on the track. Neglecting friction, the cart should be moving with the acceleration a = g sin, where is the inclination angle of the track (the angle between the horizontal direction and the track). Intuitively, this formula can be understood as the projection of the gravitational acceleration, which is pointing vertically down, onto the direction of the track. Therefore, we should be able to determine g from the a, which we measured in the last laboratory, if we also measure Using a meter stick, determine from two length measurements and a trigonometric function. Sketch a graph to illustrate which length measurements you made. Indicate on your graph. If you are not sure how to do it, talk with the instructor. Now rerun the previous lab s constant acceleration experiment, the last one activity 7 in which you pushed the cart up the track initially and let it roll down. For your fitting, be certain to select just the portion of the recording which, looking at the acceleration plot, shows constant acceleration throughout. Then calculate and report the value of sin sin = PHY221 Lab 3 Page 2

3 Copy the value of the acceleration as determined via the quadratic fit to the new position data in the same way as you did last week: a fit = Put these two numbers together to determine g: g = Is your result close to the nominal value of g = 9.8 m/s 2? If there was sufficient kinetic friction (to be studied in more detail in a later lab) then the apparent acceleration of gravity would be too low as it would be a force opposing the force of gravity. Is this the case? If you assume that the value of g is 9.8 m/s 2, and the apparent value of g determined from a measured without consideration of friction is less than g, then you can make an estimate for the kinetic friction. The coefficient of kinetic friction in our problem is the force of kinetic friction divided by the mass of the cart times the acceleration of gravity ( F / mg, where m is the mass of the PHY221 Lab 3 Page 3 kf cart). The force of gravity is mg sinθ. Including the kinetic friction, the net force on the cart is then mg sin kf mg ma fit or kf sin afit / gwhere g is 9.8 m/s 2. Hence, the estimated coefficient of kinetic friction is 2. Does gravity depend on mass? Put two rectangular weights on top of the cart. Before you do it, feel how heavy the total weight is in comparison with the cart by holding them in different hands at the same time. Repeat the measurement of a as in Activity 1. PRINT the three graphs on one sheet as done in previous labs. Then convert a to the value of g as in the previous activity assuming no kinetic friction: g = The gravitational acceleration does not depend on mass. Because of the measurement errors you are not likely to obtain exactly the same value of g here kf

4 as in the previous activity. However, the values should be reasonably close to each other. Are they? Determine their percentage difference to provide a quantitative answer (divide their difference by their sum and multiply by 200). If g is lower than anticipated you can make use of our estimated value of the coefficient of kinetic friction to correct for estimated in section 1. Including the effect of kinetic friction, g a /(sin ). fit kf The corrected g = Is this closer to the nominal value of 9.8 m/s 2? 3. Recording ballistic motion of a ball (2D motion). Ballistic means like a cannon ball, and refers specifically to the path of objects thrown more-or-less upwards and allowed to move after under the influence of gravity. An individual ballistic motion is two dimensional ( 2D ) if subject to only a single time invariant force (do you know what this means?). In general, multiple ballistic motions can be three dimensional if subject to multiple time varying forces or other induced changes in direction. The ballistic motion happens simultaneously in the horizontal (x) and vertical (y) directions. To study 2D motion we will use a WEB cam. By stepping through the recorded movie one frame at a time, we will measure x(t) and y(t) for the ball. The frames are separated by ~1/15 second. With the video analysis tools on your computer you will create x-y, x-t, and y-t graphs for objects tossed through the air. The video is acquired through the USB webcam attached to your computer and collected in LoggerPro PHY221 Lab 3 Page 4

5 To use Video Open LoggerPro and click Insert -> Video capture then click Logitech. Add Point - Click once and you can add points to your graph by clicking on the video Sync to graph button allows clicked data points to be plotted on a graph (x-t, y-t, y-x) Set origin allows you set the origin Set Scale allows you define a distance, i.e. to calibrate the system Play, stop, rewind, back one frame, and advance one frame Enable/disable video analysis when the triangle points to the left, clicking disables the analysis buttons shown along the right and they disappear. Clicking again enables analysis and the buttons reappear buttons A good plan is to do a few trial runs to get a good video of a ball throw and try all these instructions before settling on a thorough analysis. A good video would make maximum use of the area recorded by the WEBcam, i.e. go from bottom to near the top, and from the left nearly all the way to the right. Also the distance from the WEBcam to the ball should be approximately constant through the ball s flight, because you are going to have to calibrate the image to a single distance. Start capture. When you have a satisfactory video you can begin analysis. Start by closing the Video Capture window; your movie is likely behind this window. Click to sync your movie to the graph. Click to enable or disable video analysis. When video analysis is enabled, you will see icons on the right side of the video window. Make sure the Enable video analysis button is on. The buttons should appear on the right side of the window. You may set the origin using the appropriate button as the initial point of the ball in free flight. PHY221 Lab 3 Page 5

6 Click (Add Point button) to add points to your graph thus allowing your object path to be graphed. Now scroll through your movie by clicking (the Advance Frame button) until the object starts in motion. Once the ball leaves the hand, click on the ball. This a) places the x and y coordinates on the graph, b) the x and y values are plotted against time on the graph, and c) the movie advances by one frame approximately every 1/15 s. Continue clicking on the ball frame by frame until the end of the motion. Do your graphs make sense? Does the value for the acceleration of gravity agree even approximately with the value you have memorized? Possibly not. Perhaps you need to calibrate the program parameters. You should place a meter stick somewhere in the plane of motion. Calibrate the system clicking the set scale button. Then click and drag the cursor across the meter stick to draw a green line. Set that distance according to the dialog box. Now redo the ball toss being very careful to throw in the plane of your scale and once you are happy with the recorded result, record the x and y values and the exact time from the Video Analysis Table from the computer into the table below. Frame number Time x y v x v y s m m m/s m/s PHY221 Lab 3 Page 6

7 Fit a linear curve to the appropriate computer plot and then a quadratic curve to the appropriate plot and PRINT showing your fits on the printout of x (horizontal motion) as a function to t (time) and y (vertical motion) as a function of t (time). This was very easy, but I also want you to continue the analysis with graph paper. 4. Analysis of motion in horizontal (x) direction. A wonder of multi-dimensional motion is that motion in each direction can be analyzed separately. Let us first analyze the ball motion in horizontal direction. Looking at your computer graph of x vs. t, what kind of simple functional form would describe x(t)? Calculate average velocities in horizontal direction (v x ) between each WEB cam frame (the difference in distance between the next frame and the current frame divided by the interval of time). Do it for each frame (except for the last one) and store your results in the table on the previous page. Since the time interval between each frame is very short (~1/15 s), these average velocities are very close to instantaneous velocities at each point. Make a graph of v x vs t using one-half of the graph space. What can you say about v x (t)? Does motion in horizontal direction fall into the category of uniform motion (we studied different types of uniform motion in last week s lab)? What kind? Without making any calculations, what can you say about acceleration in horizontal direction? a x = m/s 2 PHY221 Lab 3 Page 7

8 5. Analysis of motion in vertical (y) direction. Complete the table by calculating v y for each frame. Pay attention to the sign of v y. Make graph of v y vs t on the other half of the sheet of graph paper. Looking at the computer graph of y(t), what kind of simple functional form could describe y(t)? What kind of simple functional form could describe v y (t)? The measurement errors are large. Try to see a simple trend even if the points fluctuate up and down. Does motion in vertical direction fall into category of uniform motion? What kind? Explain. Calculate average acceleration in vertical direction between the first and the next to last WEB cam frame stored in your table. Pay attention to the sign of a y. a y = m/s 2 What value could be expected for a y? Did your measurement get close to it? PHY221 Lab 3 Page 8

9 6.Position and Displacement Vectors. Make a graph of y vs x on a sheet of graph paper. Include x = 0, y = 0 point within the range of your graph. Each WEB cam frame corresponds to a point on the graph. Draw a smooth curve through your points. Unlike the other graphs you have made so far, this graph does reproduce what we see with our own eyes when the ball moves. The curve gives the two-dimensional path of the ball. For motion in one dimension, position is specified just by one number, x. For motion in two dimensions we need to specify two numbers, x and y. To stress that these two numbers belong to each other we can put them in a bracket: (x, y). In two dimensions a pair of numbers can be interpreted as a two-dimensional vector. The first number gives the length of the vector in x-direction. The second number gives the vector length in y-direction. If any of these numbers is negative, then the vector is pointing in the opposite direction relatively to the arrow of the coordinate axis. Instead of using two numbers in a bracket, we can also use an abbreviated notation in which we simply write, r : r (x, y) Thus, position in two dimensions is given by a two-dimensional position vector. Extension to three dimensions is obvious: r (x, y, z). For this experiment of an object moving in a plane, it is sufficient to represent the motion in a twodimensional space. Position vector can be visualized in our y vs x graph. Pick a frame corresponding to the ball in its highest position. Draw a straight line from the origin of the coordinates, i.e. point (0, 0), to the (x, y) of the highest point. Put an arrow at the (x, y) end. Let us denote this vector r h. Now draw a position vector for the ball position captured by the next frame, right after the highest ball position. We will call this vector r h+1. Displacement is a vector too: r h = r h+1 r h. In bracket notation, r ( x, y). Graphically, the displacement vector starts at the initial position, and ends at the final position. Draw r h on your y vs x graph. PHY221 Lab 3 Page 9

10 7. Velocity and acceleration vectors. It is usual to depict velocity and acceleration vectors in y vs x graph as well. Since units of velocity (m/s) and of acceleration (m/s 2 ) are not the same as the units of position (m) used to set up the coordinate axes in the y vs x graph, such graphical illustration intends to show directions of these vectors rather than their magnitudes. If more than one velocity or acceleration vector is shown on the same graph, their relative lengths should reflect their relative magnitudes. Average velocity is defined as: v r / t. Since t is a number not a vector, the direction of velocity vector v is the same as of the displacement vector r. You already depicted the displacement vector on your graph at the highest ball position. You can consider this vector to illustrate the velocity vector as well. Instantaneous velocity v dr /dt is parallel to infinitely small displacement vector dr, thus it is always tangent to the curve illustrating the object s path. Pick a point on your curve, half way between the initial and the highest ball position. Choose it in between two points representing your measurements. Draw instantaneous velocity vector at that point. Acceleration vector is defined as: a dv /dt. Therefore, its direction reflects change of velocity dv rather than change in position. Draw acceleration vector at the highest ball position. Use the ball position as the starting point of the vector. Draw the line and arrow to reflect direction of the acceleration (the length of the vector is arbitrary as explained above). Hint: a (a x,a y ); use a x and a y determined in activities 4 and 5 (is the acceleration vector constant?). Does the acceleration vector point in direction of motion? Draw position, velocity and acceleration vectors for the next to the last frame included in your y vs x graph. PHY221 Lab 3 Page 10

11 Print Your Name Pre-Lab Questions Read the Introduction to this handout, and answer the following questions before you come to General Physics Lab. Write your answers directly on this page. When you enter the lab, tear off this page and hand it in. 1. A ball is tossed with an initial velocity v 0. For each of the following initial velocities v 0, sketch the initial x and y components of the velocity (v 0x and v 0y ) as well as v 0. For each case, state what v 0x and v 0y are doing at the instant the ball leaves the tosser s hand. a) v = 5 m/s, θ = 0 b) v = 10 m/s, θ = 30 c) v = 7 m/s, θ = 110 d) v = 12 m/s, θ = 180 e) v = 5 m/s, θ = 225 f) v = 7 m/s, θ = 270 g) v = 5 m/s, θ = 335 h) v = 7 m/s, θ = -25 a b c d e f g h PHY221 Lab 3 Page 11