EXPERIMENT 7: ANGULAR KINEMATICS

Transcription

1 TA name Lab section Date TA Initials (on completion) Name UW Student ID # Lab Partner(s) EXPERIMENT 7: ANGULAR KINEMATICS This lab is a test of a new type of lab that the Physics Department is considering. There are three parts: 1. A Prelab exercise that intends to help you get used to the ideas covered in the experiment. 2. The Experiment itself, during which you will be asked to make predictions before you measure or calculate. 3. A Postlab assignment that intends to measure your understanding of the experiment. (Eventually the Prelab and Postlab parts would be done outside of class. For now, you should complete them during the lab period.) As a part of this test, we ask you to record the time that you start each section. This will help us gauge the amount of work required for each part. If you have time, we would also appreciate your answers to a few questions at the end of the lab. This lab will be graded credit/no-credit. SYNOPSIS The purpose of this lab is to teach you about motion about a fixed point: angular kinematics. When you have completed the lab you should be able to Describe the motion of a point on a rigid body rotating about a fixed axis in terms of the angular variables θ, ω, and α, and the distance R between the point and the axis. Express the linear variables x, y, v x, and v y in terms of these quantities. Understand the relationships among the position, velocity, and acceleration of a point that is rotating about a fixed axis. PRELAB Start time: Angular Variables When describing the rotation of a rigid body (not a point particle), it is convenient to use coordinates other than the rectangular (x,y,z) set we are used to. The simple cases of rotational motion you will see in these labs involve rotations either about an axis fixed in space (position and direction), or about an axis that if translating, like for an object rolling down an inclined plane, keeps its orientation fixed. Further simplifications occur when rotating objects are symmetric about the axis of rotation, in which case the axis of rotation goes through the Center of Mass of the object. For these simple cases, we will use (r,θ) coordinates as defined in Fig. 1. Fig. 1. The relationship between the (x,y) rectangular coordinates and the (r,θ) angular coordinates. Physics 121Z, Test Rotational Kinematics Lab Copyright 2008, Department of Physics, U. of Washington Summer 2008

2 The relationship between the two sets of coordinates is given by: x = r cosθ y = r sin θ The position vector r of the point P is given by r = x ˆi + yˆj = r cos θˆi + r sin θ ˆj. (1) The important feature of a rigid-body rotation is that the radial length r remains constant; what changes with time is the angle θ. In this case, we can say that θ is a function of time, which we express by writing θ(t). About that angle θ: One of the first things we learn about the number π is that it is the ratio of any circle s circumference C to its diameter d: C d = π. Since the radius of a circle r is half the diameter, we could also writec = 2 πr. A circle cut in half would enclose an arc of length of πr, that is, halfway around the circle; likewise, one quarter of a circle would enclose an arc of length ( π 2)r. Another way to interpret this relationship is to specify the angle subtending any arc in terms of the ratio of the arc length s to the corresponding radius r: s θ =. (2) r This way of measuring angles is called a radian measure. Radians are the standard way of expressing angles in physics and mathematics. You should get used to thinking of a right angle as π 2 radians rather than 90 ; there are 2π radians in a full circle. Prelab Question 1: What value(s) of θ (in radians) make(s) x minimum? What value(s) of θ make(s) y maximum? What values of θ make x = y? Physics 121Z, Test Rotational Kinematics Lab 7-2 Summer 2008

3 With the definition of radians in mind, consider the situation in Fig. 2. A point P is fixed on a wheel rotating counterclockwise about the origin O. Initially, the radius OP makes angle θ 1 with the horizontal axis; after a short time, the point has moved so that the angle is θ 2. The distance that the point travels in this time is the small arc s, which, by the definition of radian measure, must be equal to r( θ2 θ1 ) = r θ. The average speed during this motion is v = s. Thus, s θ v = = r = rω, wherein the last equation we have defined Fig. 2 a new variable called the angular speed ω (Greek letter omega) which is the rate of change of the angle θ. Strictly, this relationship should be taken to the limit of small, that is, dθ ω. By a similar argument, we can define an angular acceleration α as the rate of change dt dω of the angular speed: α. dt There is a direct analogy between the relationships among linear variables and the relationships among angular variables when describing motion with either zero acceleration or constant acceleration. You will recall, for motion along a straight line with x denoting position, v denoting velocity, and a denoting acceleration, that x( t) = x x( t) = x v( t) = v vt + v t at 1 2 at 2 (for constant velocity, and zero acceleration) (for constant acceleration) (for constant acceleration) where x 0 and v 0 are the initial position and velocity, respectively. Likewise, for angular variables, we can write θ( t) = θ θ( t) = θ 0 0 ω( t) = ω 0 + ωt + ω t αt 1 2 αt 2 (for constant angular velocity, and zero angular acceleration) (for constant angular acceleration) (for constant angular acceleration) where θ 0 and ω 0 are the initial angular position and velocity. Physics 121Z, Test Rotational Kinematics Lab 7-3 Summer 2008

4 Velocity and acceleration in terms of angular variables It would be tempting to say that if the position vector from the origin to the moving point is r, then the velocity vector of that point is v = rω and the acceleration vector is a = rα, but this would be VERY, VERY WRONG! The following exercises should help you get to the correct relationships. The velocity vector 1. The circle at the right represents the path of a point on a rotating object traveling at constant speed. Make a dot on the circle representing the point s position at t 1, and label this point P 1. Make a second dot on the circle representing the point at a time t 2 shortly after t 1 (see Fig. 2 for an example of good dot placement.), and label this point P Draw two position vectors: the tails of each should be at the center of the circle O, and the head of each should be at point P 1 and P 2. Label these vectors r 1 and r Construct a vector r that is the difference between vectors r 1 and r 2. Its tail should lie on point P 1 and its head should touch P 2. In other words, the vectors should obey the vector addition equation r 2 = r 1 + r. 4. The velocity vector for the moving point is, by definition, v r in the limit that becomes small. You should see that the velocity vector is perpendicular to the position vector at any point on the circle, and in fact, points in the direction of motion, which is tangent to the circular path. 5. Also, in the limit of small, the magnitude of the vector r becomes equal to the arc length s. Hence, r s θ v = = = r = rω. So, you see that the velocity vector has the magnitude rω, but, like in the case of projectile motion, it points in a direction that is always tangent to the path of motion. Because the velocity vector is always perpendicular to the position vector (when the origin is at the axis), we can write a similar set of equations to equation (1) to give the velocity in terms of angular coordinates: v = v ˆi + v ˆj = rωsin θˆi + rωcosθˆ x y j. (3) Notice that the sine and cosine functions are in different places between Eqs. (1) and (3). This insures that the directions of these vectors will always be perpendicular to each other. Physics 121Z, Test Rotational Kinematics Lab 7-4 Summer 2008

5 The acceleration vector Even on a wheel spinning at a constant speed, a point on the rim is accelerating. Let s see why. 1. As in the case above, the circle at the right represents the path of a point on a rotating object traveling at constant speed. Make two dots on the circle, as before, representing the point s position at two different times t 1 and t 2, and label them P 1 and P At each point carefully draw two vectors, v 1 and v 2, representing the point s instantaneous velocity at each time t 1 and t 2. The length of these vectors should be the same, representing the same speed v. (Use a ruler, and choose some convenient length, like 2 cm.) The tail of each vector should touch the point P 1 or P 2, as appropriate. Make sure that these are carefully drawn to show both magnitude and direction accurately. 3. Draw another velocity vector at P 2 that is identical to the velocity vector at P 1. Important: Keep the same size and direction for the newly drawn vector as the original one at P 1, but this new vector should have its tail on P To find the acceleration, you are interested in the change in velocity v between the two velocity vectors. Add a new vector v between the tips of the two velocity vectors at P 2, so that v represents the vector that must be added to v 1 to get the resultant vector v 2. It has magnitude v. 5. Now draw two lines: one from the center of the circle to P 1, the other from the center of the circle to P 2. Then draw a third line between P 1 and P 2. The first two lines have length r, and the third has length v, the amount of distance traveled (approximately) by the point during the small time interval. 6. The triangle drawn in step 5 is similar to the triangle drawn in steps 3 and 4: they are both isosceles triangles with the same apex angle. Thus, the ratio of v to v is the same as the ratio of r to v. Use this fact to write an equation expressing the equality of these two ratios. 7. Solve the equation for the magnitude of the acceleration a = v in terms of r and v. Then use the result v = rω to express a in terms of r and ω. 8. Carefully transpose the vector you drew in step 4 (the v vector) so that its tail is at the midpoint of the line you drew between points P 1 and P 2. That is, draw another vector at this midpoint exactly like v. (Again, use a ruler. It is tricky to draw correctly by eye.) 9. Note the direction of the change-in-velocity vector v you just drew. In the limit of small it would be perpendicular to the velocity vector v, and directly opposite the position vector r. In other words, the acceleration of a point moving with constant speed around a fixed axis is always toward the axis. The acceleration you have just found is called the centripetal (Latin: center seeking ) acceleration a c. In the case of constant speed, the velocity vector does not change length, just direction. But if the wheel were speeding up or slowing down, there would be another component to the acceleration to account for the change in length of the velocity vector. This Physics 121Z, Test Rotational Kinematics Lab 7-5 Summer 2008

6 component would lie along the direction of v, and is known as the tangential acceleration a t. The tangential acceleration gives the change in speed of the point: ( rω) ω a t = = r = rα. The total acceleration of a point on the rotating body is the vector sum of the centripetal and tangential accelerations. If we define the unit vectors rˆ and vˆ (note the hats) as pointing in the directions of r and v (like the vectors î, ĵ, and kˆ that only indicate direction along the x, y and z axes), then we can write the vector for the acceleration a in terms of the centripetal and tangential components, a c and a t, as 2 a = a rˆ + a vˆ = rω rˆ + rαvˆ c t. (4) Prelab Question 2: What is the meaning of the minus sign in front of the first term in the above equation? Prelab Question 3: What is the magnitude of the acceleration vector a? Physics 121Z, Test Rotational Kinematics Lab 7-6 Summer 2008

7 EXPERIMENT Start time: The experiment has two parts. In the first part, you will study the motion of a spot on the rim of a bicycle wheel by tracking its position over time using video analysis software. The goal of this first part will be to clarify the relationship between the (x,y) and (r,θ) coordinates in a mainly qualitative way. The second part uses a turntable with a high-resolution angular sensor that will give accurate measurements of θ and ω and allow you to test quantitative predictions concerning velocity and acceleration in rotational motion. A. MOTION OF A BICYCLE WHEEL Prediction 1: A spot on the rim of a bicycle wheel is measured as the wheel turns at a constant rate. What is the value of the x coordinate of the spot when the velocity in the x direction v x is maximum? What is the direction of the velocity vector v when the y coordinate is minimum? You may assume that the rotation is counterclockwise from your reference point. The TA will demonstrate how a movie has been made of a freely rotating bicycle wheel, and show you how to access that video on the computer. You will also need to measure the wheel directly with a ruler. A place on the rim of the wheel will have a piece of tape (black or gray marker) so it is easy to track that spot on the video. There is also a half-meter stick in the picture and close to the wheel that will allow you to set the scale of the measurements. You will download the video file and use the VideoPoint software to measure a spot on the wheel as the movie runs from frame to frame. This will produce a table of (x,y) position data as a function of time. You can then make graphs and analyze the motion of the taped dot. 1. Open the program entitled VideoPoint 2.5 in the 117/121Z folder. Download to your computer the file specified by the TA for uniform circular motion following steps below. Analyze the data as indicated from step 2 on. VideoPoint 2.5 Software Directions (a) If the lab software is not already running, double-click on VideoPoint 2.5 on the desktop icon. Even if it is running, you may wish to restart the software to be sure no settings have been altered. (b) After the About Video Point screen appears, click to close it. Physics 121Z, Test Rotational Kinematics Lab 7-7 Summer 2008

8 (c) Click on Open Movie. Double click on the directory with the Lab 7 file. In the file selection list, double click on the appropriate data file. (d) Check that the number of objects to be tracked is 1, and then click OK. This insures that the location of a single point in each frame will be recorded. (e) Maximize the screen with the movie to make it easy to see. (f) Now you need to set the scale. Click on the ruler icon (6 th from the top) on the left tool bar. On the Scale Movie screen that appears, you should see 1.00 m, <Origin 1> and Fixed selected. CHANGE the 1.00m to 0.50m. Click Continue. (g) Click the target cursor carefully on both ends of the meter stick in the video. (h) The video has more frames than you need to follow the motion, so you should increase the step size. From the menu, choose Movie>Set Step Size, and when the dialog box opens, select Step by: Frames in the upper box, and Size (frames): 2 in the lower box, and Accept the change. (i) Use the mouse to position the cursor and click on the tape spot in every frame, until the video ends. The image may blur in some frames, but try to click on a consistent part of the image. You can track your progress by the slider underneath the movie window, or by watching the small yellow frame indicator box in the upper corner. (j) By default, the origin of the coordinate system is near the lower left corner, but you want the origin to be at the center of the wheel. Transform the origin as follows: Click on Pointer Arrow (second from the top on left margin). Drag origin from the bottom-left corner to the center of the bicycle wheel. This step renormalizes all your readings to x,y and R,θ as measured from the center of the wheel. This concludes the data taking from the video. 2. Click on the plot icon (8 th from the top) on the left toolbar. Horizontal axis should read Time, and Vertical axis should read Point S1 and x-component. Then click OK to create the graph. This is a graph of x vs. t; print a copy for each partner. Repeat these steps to create a graph of y vs. t for you and your partner(s). 3. If things have gone well, your x vs. t and y vs. t graphs should have a sinusoidal shape. Why do the graphs have this shape? What form of θ(t) would produce this shape? Discuss, and write an equation that could describe x(t) in terms of the variables r, θ 0 and ω. Physics 121Z, Test Rotational Kinematics Lab 7-8 Summer 2008

9 4. Measure the radius of the bicycle wheel with a ruler or tape measure and write it down. R = ( ) [We will not estimate uncertainties, but you still need units.] Does the measured radius correspond with your printout at the maximum and minimum of the x and y coordinates? If this does not seem right, check with TA! 5. Make graphs of v x vs. t and v y vs. t by selecting the velocity components from the plot dialog box. Print copies for each partner. These graphs now may have (likely) some scatter, but they should have a familiar look, corresponding to the respective x vs. t and y vs. t graphs. Draw a smooth line through the velocity points in each of the graphs, averaging by eye as you go along. 6. Check your prediction 1: From your v x vs. t graph, locate the approximate time when the velocity in the x direction is maximum. Then find the x coordinate at that same time from the x vs. t graph. Record these values below: v x (max) = ( ) t at v x (max) = ( ) x at v x (max) = ( ) Also, locate the time when the y coordinate is minimum, and from the values of v x and v y at that time, construct a velocity vector. Record the values below, and then sketch the velocity vector on a circle representing the trajectory of the spot in the space below, so that the sketch shows the direction of the velocity and the corresponding position of the spot at that time. y(min) = ( ) t at y(min) = ( ) v x at y(min) = ( ) v y at y(min) = ( ) Sketch of velocity vector on trajectory circle: Physics 121Z, Test Rotational Kinematics Lab 7-9 Summer 2008

10 7. From your results above, calculate the speed of the spot on the rim, that is, find the magnitude of the velocity vector. Show your work. v = ( ) Prediction 2: What is the angular speed ω of the bicycle wheel? You can estimate it from the data you have taken so far in two ways: (1) when the wheel returns to the same point, it has turned through 2π radians. How long does this take? (2) You now should have a fairly good value of both r and v; these can be used to find ω. Show enough work so that your reasoning is clear. 8. For a careful measure of ω, you can use the plot icon in the VideoPoint software to generate a graph of angle θ vs. t, which you can then analyze. Are the angular measurements in this plot in radians or degrees? Print a copy of this graph for each partner. 9. Write a few sentences describing the graph you just copied. Is the shape of the curve a constant, a line with a constant slope, or a more complex curve? How does your observation of the motion of the wheel correspond to the shape of the curve? 10. Write an equation for θ(t) that describes the graph of θ vs. t. Define the variables in your equation. 11. Comment on the following statement: If I had chosen a point half way between the rim and the axis of rotation, the graph of θ vs. t would look exactly the same. (True or false, and explain why!) Physics 121Z, Test Rotational Kinematics Lab 7-10 Summer 2008

11 12. Calculate the angular velocity ω of the wheel by using the curve fitting button F on VideoPoint. Record your result here. ω (from fit) = ( ) 13. Check your Prediction 2: Is the result of the fit reasonably close to the calculations you made? Discuss, and reproduce your numerical results below. (Hint, you are testing the relation v = rω.) 14. With the origin of your coordinate system located at the center of the wheel, the x coordinate should follow the equation x( t) = r cos[ θ( t)], where θ(t) follows the equation you wrote in step 10. Under the M tab on the graph window for x vs. t, you can apply a model equation of the form Acos(Bt + C) + D, where you enter values of A, B, C and D, to your graph. Try this, and see if you can match the data with your model curve. You should be able to estimate the parameters from the data you have taken, with the exception of C. After you have made the best possible match, print a copy of the graph including the model curve, and record your parameter values below. Question to explore: what do parameters C and D do to the curve? A = ( ) B = ( ) C = ( ) D = ( ) B. MOTION OF A TURNTABLE The rotary motion sensor you will use for Part B is pictured at right. A turntable has holes drilled in it that allow objects to be attached, such as metal rings or disks. In the figure the object is a thick steel ring. Underneath the turntable is a pulley (#1) that you can wind a string around. A string is wound around pulley #1, passed over pulley #2 and attached to a hanger similar to the one used in the One- Dimensional Dynamics lab. Releasing the hanger with its weight exerts a torque on the turntable and the object, causing the turntable and the two pulleys to be accelerated. Physics 121Z, Test Rotational Kinematics Lab 7-11 Summer 2008

12 The sensor unit is attached to inputs 1 (yellow plug) and 2 (black plug) of the Pasco Interface. The sensor provides a sample-signal for each ¼ degree of rotation of the shaft (devices of this sort are called shaft encoders ) and the DataStudio software processes these measurements and generates table and graphs of data. Start DataStudio and open Rotational Motion > Angular Kinematics-Summer When the activity opens, you should see two graph windows opened: θ vs. t, and (x,y) vs. t. There are other windows in the activity, but don t open these yet. 15. Measure the distance from the center of the turntable to one of the screw holes on the edge of the rim with a ruler. Call this measurement r. If you are careful, you should be able to get it to within a millimeter uncertainty. r = ( ) 16. Enter the value of r into the DataStudio Calculator window associated with the variable R that is labeled Value and click Accept. Make sure to select the correct units! 17. Place the heavy disk with the alignment pins on top of the turntable, and make sure it is securely seated. The disk increases the inertia of the turntable and reduces the effect of friction from the turntable bearings. 18. Explore how DataStudio takes in data and processes it: Click the Start button and rotate the turntable back and forth with your finger. You do not need to write anything down in this step, but you should notice the following: Look at each graph. How does the value change with time? Is it what you expect? Is the angle θ measured from an absolute reference, or does it start from zero no matter how the turntable is initially set? Is the angle measured in degrees or radians? How about the x and y values? Do they vary between the extremes set by your value of r? Uniform Rotation 19. After you feel comfortable with how the rotary motion sensor and DataStudio system works, prepare to take some data with a uniformly rotating turntable: Clear any old data sets. Then, Start data collection and spin the turntable gently. Stop the data collection after a few rotations. 20. Look first at the θ vs. t graph. How many radians would correspond to 4 full rotations of the table? Don t use your calculator, but instead use a trick, and do it in your head: for rough estimations, π 3, so a full rotation is about 6 radians. How many radians would 4 rotations be? Physics 121Z, Test Rotational Kinematics Lab 7-12 Summer 2008

13 21. Zoom-select a straight-line portion of it that covers about 4 full rotations of the turntable. Use the Fit menu to find the slope of this line, and thus find the angular speed ω. Print copies of this graph for you and your partner. For reference, record the beginning and end times of this zoom-selected graph, along with your fit result: t begin = ( ) t end = ( ) ω = ( ) Prediction 3: The (x, y) vs. t graph shows two sine curves. Based on the θ vs. t graph, what should the period (repeat time) of these curves be? 22. Highlight the (x,y) vs. t graph and Zoom-select the region of it that lies between the times t begin and t end that you used in Step 20. Use the Smart Tool to determine the period of the sine curve corresponding to the x vs. t data set. Print the graph. Hint: Although you could measure the time for one period, it is better to measure the time for a few periods of rotation, and then divide this longer time by the number of periods. This reduces the uncertainty caused by the difficulty in locating the start and end times accurately. 23. Check your prediction 3: Calculate the period of the sine curve from the fit to the θ vs. t graph (step 21) and from the x vs. t graph (step 22), and compare these values by evaluating the percent difference between them. Note that one period occurs when the angle θ changes by 2π radians. In other words, if the period of rotation is T, then 2 π = ωt. T (from θ vs. t) = ( ) T (from x vs. t) = ( ) 24. On the (x,y) vs. t graph, the curve for x vs. t is similar to the curve for y vs. t, but they are not exactly the same. What accounts for the difference? Explain. Physics 121Z, Test Rotational Kinematics Lab 7-13 Summer 2008

14 Prediction 4: What is the magnitude of the velocity and the magnitude of the acceleration of the hole in the turntable? 25. Double-click on the icon for the minimized window labeled v x vs. t so that it expands. Zoom-select the same time interval between t begin and t end that you had analyzed above. Do the same for the icon labeled v y vs. t. Print copies of both graphs. From these graphs, determine the magnitude of the velocity of the turntable hole. Record the value, and explain how you came up with it below. v = ( ) 26. To find the acceleration, you need to find v x and v y for sets of points on the v x and v y graphs. Pick two data points (call them a and b ) on the v x vs. t graph, and record the values of the x-velocity v x and the time t for each point. Then look at the same two time points and record the corresponding y-velocities v y for those. t a = ( ) t b = ( ) v x (t a ) = ( ) v x (t b ) = ( ) v y (t a ) = ( ) v y (t b ) = ( ) 27. Form the fractions and from the above, and calculate a x and a y, and finally v x a. Show your work below. v y a = ( ) 28. Check your prediction 4: Compare your predicted values with the ones you found in steps 25 and 27. Physics 121Z, Test Rotational Kinematics Lab 7-14 Summer 2008

15 Accelerated rotation To accelerate the turntable, you will allow a falling weight attached to a string to exert a constant torque on the turntable pulley. This constant torque will cause a constant rate-of-change of ω, that is, a constant α. Prediction 5: In the spaces below, sketch the curves you expect to see for θ vs. t, x vs. t and v x vs. t in the case of constant angular acceleration. θ x v x t t t 29. Explore how the falling weight apparatus works. Obtain a string that is long enough to reach the floor from the pulley under the turntable (if it is not already attached). Attach the approximately 15g hanger, without any additional weight, (the hanger alone is heavy enough) to the end of the string and place the string over pulley #2. Adjust the level of pulley #2 (it can be pivoted up and down without unclamping the pulley from the sensor body) until the string from pulley #1 is level. Orient pulley #2 by twisting it sideways while clamped so that the string is in line with the pulley groove as it unwinds. If the string and pulleys are placed as described, the string can be wound easily on the lower sheave by turning the turntable clockwise with your finger. A finger on the turntable will hold it in place. Check that the heavy disk is well-seated on the turntable, and turn the turntable clockwise to wind the string around the lower (larger diameter) pulley sheave. When the turntable is released, it will rotate counterclockwise (as viewed from above). Try this once or twice to see how it works. Rewind the string, Start data collection and release the turntable. Try it a couple of times until you can reliably get a good-looking data set. 30. When you are ready to take real data, rewind the string, press Start, and release the turntable. Stop the data collection after the string has extended completely. On the (x,y) vs. t graph, Zoom-select the portion of the data set that begins at the time of release, and extends to 4 full rotations of the turntable. Zoom-select the same time periods on the θ vs. t and v x vs. t graphs. Print these three graphs. Physics 121Z, Test Rotational Kinematics Lab 7-15 Summer 2008

16 31. Check your prediction 5: Do these graphs have the same general shape that you drew above? Discuss your predictions and the measurements, and compare their features. You should look at the following: The shape of the θ vs. t graph: is it straight or curved? If it is curved, what kind of curve does it look like? The shape of the x vs. t graph: Does the amplitude (maximum and minimum values of x) increase, decrease or remain constant? Why? Does the period (time between successive x = 0points) increase, decrease or remain the same? Why? The shape of the v x vs. t graph: Does the amplitude increase, decrease or remain constant? Why? How about the period (time between successive v = 0 points) of this curve? x 32. Confirm that the angular acceleration α is indeed constant: Double-click on the icon for the ω vs. t graph so that it expands. Zoom-select the same time region that you used above, and look at the shape of the curve. What shape should it have if the angular acceleration is constant? Does it have that shape? Use the Fit menu to fit the curve to obtain α. Print the graph, and record your value of α below. α = ( ) 33. Is the centripetal acceleration a c of the hole constant in magnitude? Is the tangential acceleration a t constant in magnitude? Explain, and justify your answers by referring to qualities of the ω vs. t graph. Physics 121Z, Test Rotational Kinematics Lab 7-16 Summer 2008

17 34. In the spaces below, sketch curves representing the magnitudes of a c vs. t and a t vs. t for a point on a turntable that is experiencing a constant angular acceleration, assuming that the turntable starts from rest at t = 0. a c a t t t 35. You can extract the centripetal and tangential acceleration (for selected time points) directly from your v x vs. t graph. Referring to Eq. (3) on page 7-4, the velocity in the x direction is given by = rωsin θ, where now both θ and ω change with time. Notice: v x 2 When sin θ 0, then sin θ θ and v x rω( θ ) = rω = ac. In other words, when the velocity in the x direction is close to zero, the rate of change of the x velocity is the centripetal acceleration. Thus, the centripetal acceleration is equal to the slope of the v x vs. t graph at those points where v x crosses zero. The factor rω in front of the sine function defines an envelope around the oscillating part. Since r is fixed, the rate of change of this envelope is r ( ω ) = rα = at. Thus, the tangential acceleration is equal to the slope of the envelope. Obtain the centripetal acceleration at two time points, and the tangential acceleration overall, from your v x vs. t graph as follows. Use a ruler to draw three lines on your printout of this graph: (1) a line tangent to the curve at the point that v x first crosses zero in the positive direction (about 1 second after the release of the weight), (2) a line tangent to the curve at the point where v x crosses zero three rotations later, and (3) a line that connects each positive peak of the v x vs. t curve this represents the envelope of the sine function. On each of these lines make two dots at points where the line comes close to a grid intersection (the dots should be far apart), and calculate the slope of the line from the coordinates of these dots. The slopes should be the centripetal acceleration at the two times where lines (1) and (2) cross zero, and the tangential acceleration overall, from line (3). Show your work below and on the graph. Line (1): t 1 = ( ) a c = ( ) Line (2): t 2 = ( ) a c = ( ) Line (3): a t = ( ) Physics 121Z, Test Rotational Kinematics Lab 7-17 Summer 2008

18 36. Of course, you can calculate the centripetal and tangential acceleration much more easily from the ω vs. t graph. At the same two time points that you used above calculate the magnitude of centripetal and tangential acceleration from the value of ω at those points, and the slope of the line overall. Time point 1: a c = ( ) Time point 2: a c = ( ) a t = ( ) 37. Do your calculations using the two different graphs agree within a reasonable uncertainty? Do these results support your answers to Steps 33 and 34? Discuss. Physics 121Z, Test Rotational Kinematics Lab 7-18 Summer 2008

19 POSTLAB Start time: Test your understanding 1. Two beads are fixed to a rod that is rotating at constant angular speed about a pivot at its left end, as indicated in the figure. (a) Which bead has the greater speed? Explain your reasoning. (b) Which bead has a greater magnitude of acceleration? Explain. 2. The graph below gives the position x vs. t for the outer bead in Question 1. (a) On the same graph, sketch a curve that could represent the velocity v x vs. t for the outer bead. Label it clearly. (b) On the same graph, sketch a curve that could represent the position of x vs. t for the inner bead. Label it clearly. Physics 121Z, Test Rotational Kinematics Lab 7-19 Summer 2008

20 3. From the graph, determine the following quantities associated with the outer bead. Show enough work/comments so that your reasoning is clear. (a) The angular speed ω. (b) The speed v. (c) The centripetal acceleration a c. (d) The angular acceleration α. (e) The tangential acceleration a t. Physics 121Z, Test Rotational Kinematics Lab 7-20 Summer 2008

21 FOLLOW-UP QUESTIONS (OPTIONAL) If you have time, we would like to know what you think of this lab, by the answers to the following questions. 1. Making the predictions was, for me, Fairly easy I always got them right without much effort. Interesting I usually got them right, but I really had to think. Challenging I sometimes got them wrong. Hopeless I rarely knew what to do. 2. I found the pre-lab assignment to be Helpful Not helpful Other: 3. I found the post-lab assignment to be A good way to measure what I learned Not very relevant to what I learned Other: 4. I would prefer to have A two-hour lab period with pre- and post-lab work done outside of class on paper. A two-hour lab period with pre- and post-lab work done outside of class online (i.e., a web page). A three-hour lab, with all parts done in class. Reason for your preference: 5. Compared to the other labs in this course, did you like the style of the lab better or worse? Why? 6. If you were the instructor of the course, what would you change? Physics 121Z, Test Rotational Kinematics Lab 7-21 Summer 2008