Team Force and Motion In previous labs, you used a motion sensor to measure the position, velocity, and acceleration of moving objects. You were not concerned about the mechanism that caused the object to move, i.e. the forces that acted on the object. In other words, you answered the kinematic question How do objects move? Now it is time to answer the dynamic question Why do objects move? From your experiences, you know that force and motion are related. To open a door, you must push or pull on the handle. To stop a car, you must step on the brakes. The force of gravity causes the earth to move around the sun. The force of electricity causes the electron in an atom to move around the proton. The strong quantum force glues the protons (quarks) inside the tiny nucleus of an atom. Newton s celebrated equation of motion, F = ma, describes the precise connection between force (F) and motion (a). You have used a motion sensor (accelerometer) to measure a. You will now use a force-ometer to measure F. By measuring the left side of F = ma with a force meter and the right side with a motion meter, you will discover the deep relation between F and ma firsthand in the laboratory. Thought Experiment A 2 kg mass and a 3 kg mass are connected by a massless cord and move on a horizontal frictionless surface. The 3 kg mass is pulled to the right with a force of 20 N. The tension in the cord is 8 N. Fill in the F and ma blanks. 2 kg 8 N 3 kg 20 N Force: Net Force on 2 kg mass = N. Net Force on 3 kg mass = = N. Motion: Mass x Acceleration of 2 kg mass = x = kg m/s 2. Mass x Acceleration of 3 kg mass = x = kg m/s 2.
Part I. Forceometers The Spring Scale The spring scale is the prototype forceometer. For centuries, springs have been used to measure force. The force required to stretch a spring is proportional to the amount of stretch. A spring scale is a spring that has been calibrated to convert the amount of stretch (meters) into the corresponding value of the force (Newtons). Examine the spring scale. Note that there is a large fixed hook at one end and a small hook at the other end. The small hook is attached to a sliding rod, which is connected to the spring. Pull on the small hook so that the spring is stretched several centimeters (half its maximum stretch) and the reading on the force scale is 0.5 N. Double the stretch of the spring so that the force is 1.0 N (maximum scale reading). You already know what a one pound force feels like. But what does a one Newton force feel like? Pull on the spring scale until you have a good kinesthetic sense of a one Newton force. Zero the spring scale as follows. Hold the spring scale in the vertical position with the small hook downward. With nothing attached to the hook, make sure the scale reads zero. If the reading is not 0, then try another spring scale or consult your instructor. Once the scale is zeroed, hang a 50 gram mass from the small hook. What is the value of the force F recorded by the spring scale? Spring Scale Reading: F = N. Since the 50 gram mass is at rest, the upward pull of the spring on the mass is equal to the downward pull of the earth on the mass. The pull of the earth the force of gravity on an object is called the Weight of the object. Calculate the theoretical value of the weight of a 50 gram mass. Calculated Weight of 50 gram mass: W = N. % diff between Measured Weight (spring scale reading) and Calculated Weight is %. The Force Sensor Examine the force sensor. The switch on the force sensor should be set to the 10 N range. The force sensor is an electronic strain gauge. Any force (push or pull) on the hook causes a metal block inside the sensor to bend slightly. The strain in the block is converted into an electrical signal (voltage) which is then converted, via the interface box (ULI), into the corresponding value of the mechanical force. So like the spring scale, the force sensor converts a distance (strain in meters) into a force (Newtons). 2
The force sensor measures F(t) = Force on the hook as a function of time. To activate the force sensor or force probe, start Logger Pro and open the file ForceProbe. Click on Collect. Push and pull on the hook and note the digital values of F(t) that are displayed in the data table and the corresponding F(t) curve that is traced out in the graph window. Zero the force sensor as follows. While holding the force sensor in the vertical position with the hook downward and nothing attached to the hook, click on the Zero button (to the right of the Collect button). Once the sensor is zeroed, hang a 50 gram mass from the hook and click on Collect. Average the F(t) data. What is the value of the force F recorded by the force sensor? Force Sensor Reading: F = N. % diff between Measured Weight (force sensor reading) and Calculated Weight is %. Part II. Apply F. Measure ma. Prove the Law F = ma. Measure F with the Spring Scale First make sure that the track is level using the leveler and the steel ball test. The spring scale will be used to apply the force F. Make sure that the spring scale reads zero when it is held in the horizontal position and nothing is attached to the small hook. Attach the small hook of the spring scale to the cart. Pull the cart along the track with a constant force of 0.10 N. Keep the spring scale tube horizontal (do not tilt the tube). It may be a bit tricky to keep the force constant. In experimental physics, nothing is exactly constant. It is natural for the force to fluctuate around 0.10 N. This error in keeping the force constant should have little affect on your results. Sometimes you pull too hard, sometimes you pull too soft, but on average you pull just right! Practice pulling the cart until you get a feel for how to keep the force scale reading at 0.10 N. While you are pulling, carefully note the amount by which your applied force F varies above and below 0.10 N. This up and down variation in F determines the uncertainty in F. For example, if the spring scale reading varies between 0.070 N and 0.13 N during the pull, then you would report the experimental value of the force as F = 0.10 N ± 0.03 N. If your pull is steadier, then the uncertainty in the force will be smaller. Based on your pulling trials and observed range of force values, report the measured value of your applied force: F = 0.10 N ± N. Significant Figures: The measurement of force (with the spring scale) and the measurement of time (with the stopwatch) are accurate to about two, perhaps three, significant figures. All results that you report in this lab should reflect this two-sig-fig accuracy. 3
Measure ma with a Stopwatch An accurate way to determine the constant acceleration a of your cart is to measure time and space. Use a stopwatch to measure the time t it takes for your cart starting from rest and pulled with the force 0.10 N to cover the distance d = 1.0 m along the track. Use the kinematic relation d = ½ at 2 to compute a from your measured values of t and d. Repeat the run five times. Record your five measured values of t in the motion table below. For each run, compute the values of a and ma. Measure the mass m of your cart using the mass scale. m = kg. Motion Table d (m) t (s) a (m/s 2 ) ma (N) 1.0 1.0 1.0 1.0 1.0 Compute the average value of ma. Estimate the uncertainty from the spread in your five values of ma : Uncertainty = (ma max ma min ) / 2. ma = ± N. Compare ma and F You have measured the force quantity F using a spring scale. In a completely independent measurement, you have measured the motion quantity ma using a stopwatch (and meter stick and mass scale). Here is the BIG question: Is F equal to ma? Remember, since an experimental value (number ± uncertainty) is really a range of numbers, comparing two values really means comparing two ranges. Display your results for F and ma on the following range diagram. Remember that an experimental value, such as 0.10 ± 0.03 N, is displayed on the range diagram as a horizontal line segment from 0.07 N to 0.13 N. Write the numerical values on the axis labeled Newtons. F ma Newtons Use the results displayed on your experimental range diagram to rigorously answer the deep question: Is F equal to ma? 4
Double the Force Use the spring scale to pull the cart with the contant force F = 0.20 N. Use the stopwatch to measure the time t it takes for the cart, starting from rest, to traverse the distance d = 1.0 m along the track. Repeat the run one more time. Average your two values of t. Compute the value of a. t = s. a = m/s 2. When you doubled the force on the cart (F 2F), what happened to the acceleration of the cart? % difference between F = 0.20 N_ and ma = N is %. Double the Mass Place the long bar weight on top of your cart. Measure the mass of the loaded cart. m = kg. Pull the cart with the constant force F = 0.10 N. Measure the time t it takes for the cart, starting from rest, to cover the distance d = 1.0 m along the track. Repeat the run one more time. Average your two values of t. Compute the value of a. t = s. a = m/s 2. When you doubled the mass of the cart (m 2m and applied the same 0.10 N force), what happened to the acceleration of the cart? % difference between F = 0.10 N_ and ma = N is %. 5
Part III. Two Body Dynamics. Net Force. In the previous part, you studied a system consisting of one mass (body). Here you will study a mechanical system composed of two connected masses. In physics, the two-body problem refers to any dynamical problem involving two interacting objects. Solving the equations of motion for the sun-earth-moon system is a famous three-body problem in theoretical physics. You will also discover how the net force the sum of all the individual forces acting on a body is the crucial dynamical quantity that determines the motion of the body. The all-encompassing goal of classical mechanics: FIND THE NET FORCE! Two-Cart Train Build the following two-body mechanical system. Body 1 consists of cart 1 plus spring scale 1. Body 2 consists of cart 2 plus spring scale 2. Make sure the spring scales are Zeroed. Both scales should read 0 when held in the horizontal position and nothing is attached to the small hook. Note the orientation of the scales: large hook on left, small hook on right. Use two rubber bands (widely spaced) to secure each scale to its cart. Make sure that the rubber bands do not rub against the wheels underneath the cart. scale 1 scale 2 Cart 1 Cart 2 The experiment consists of pulling the system to the right with a contant force. Here is the Newtonian schematic the textbook picture of this real-world mechanical system: F 12 m 1 m 2 F pull System parameters: m 1 = mass of body 1 (cart 1 + scale 1). m 2 = mass of body 2 (cart 2 + scale 2). F 12 = Connecting Force (Force on 1 due to 2), recorded by spring scale 1. F pull = Pulling Force, recorded by spring scale 2. Note that the Net Force on m 2 is F pull F 12. 6
Measure Mass Use the mass scale to measure the mass of body 1 and the mass of body 2. Remember that each body consists of one cart plus one spring scale. m 1 = kg. m 2 = kg. Measure Force The connecting force F 12 is the force of interaction between body 1 and body 2. It is the force exerted on body 1 due to body 2. By Newton s Third Law, it is equal and opposite to the force on body 2 due to body 1: F 12 = F 21. Examine this force of interaction in your actual two-cart system. Note how the sliding rod of spring scale 1 provides the physical connection between the carts. The stretched spring between cart 1 and cart 2 determines the value of F 12. The bigger the stretch, the larger the force. Place your two-cart system directly on the table surface, not on the metal track. This will allow you to pull the carts over a longer distance. Hold the small hook of spring scale 2 and pull so that the scale always reads 0.2 N. While you are pulling, your team member will constantly monitor the reading on scale 1, thereby measuring the force of interaction F 12. Repeat this pulling/monitoring until you are confident (to within 10%) in your measured value of force. Report your value of F 12 : F pull = 0.20 N. F 12 = N. Measure Acceleration Find the acceleration a of the two-cart system as you pull with the constant force F pull = 0.2 N. Use the stopwatch to measure the time t it takes the system, starting from rest, to cover a distance d = 1.0 m along the table surface. Lay a meter stick (x-axis) on the table surface and let the twocart system move parallel to the stick from one end of the stick to the other end. To minimize friction, do not let the carts rub against the stick. Repeat one or two times or until you are confident in the values of time. Average your values of t. Compute the value of a from your measured value of t (and d = 1.0 m). t = s. a = m/s 2. Experimental Proof of F net = ma Draw the free body diagrams for m 1 and m 2 in the space below. Only include the forces that point in the horizontal direction (the vertical motion is trivial). Write the numerical value of the force next to each force vector. Draw the length of each vector to scale (approximately), i.e. if v = 2 and u = 6, then draw the length of vector u to be three times the length of vector v. 7
Free Body 1 Free Body 2 Sum the forces to find the net force F net on each body. Multiply the mass and the acceleration to find the motion quantity ma for each body. F net 1 = N. F net 2 = N. m 1 a 1 = kg m/s 2. m 2 a 2 = kg m/s 2. Conclusion Do your measured values of F net and ma for each body provide an experimental proof of Newton s Second Law of Motion to within a 10 % margin of experimental error? Explain. Changing the Mass. Testing the Law F net = ma. Add the long bar weight to the top of cart 1. Measure the mass of body 1 (cart + scale + weight). m 1 = kg. Pull the two-body system with a constant force of 0.30 N over a distance of d = 1.0 m. Measure the values of F 12 and a as before by monitoring scale 1 and using a stopwatch. F pull = 0.30 N. F 12 = N. t = s. a = m/s 2. Find the net force on each body and the mass-times-acceleration of each body. F net 1 =. F net 2 =. m 1 a 1 = x. m 2 a 2 = x. Are your experimental results consistent with Newton s Second Law of Motion? % difference between F net 1 = and m 1 a 1 = is %. % difference between F net 2 = and m 2 a 2 = is %. 8
Remove the bar weight from cart 1 and add it to the top of cart 2. Measure the mass of body 2 (cart + scale + weight). m 2 = kg. Pull the two-body system with a constant force of 0.3 N over a distance of d = 1.0 m. Measure the values of F 12 and a as before by monitoring scale 1 and using a stopwatch. F pull = 0.30 N. F 12 = N. t = s. a = m/s 2. Find the net force on each body and the mass-times-acceleration of each body. F net 1 =. F net 2 =. m 1 a 1 = x. m 2 a 2 = x. Are your experimental results consistent with Newton s Second Law of Motion? % difference between F net 1 = and m 1 a 1 = is %. % difference between F net 2 = and m 2 a 2 = is %. Summary Exercise The following three pictures summarize your three experiments, but with different values for the forces and masses. The pictures show two masses connected by a massless cord. The system is pulled with a force of 6 N on a horizontal frictionless surface. Based on pure theory, find the values of the tension (N) in the cord and the acceleration (m/s 2 ) of the system. Write your results directly on the pictures. 1 kg N 1 kg 6 N m/s 2 2 kg N 1 kg 6 N m/s 2 1 kg N 2 kg 6 N m/s 2 9
Part IV. Creating a Physics Sculpture Title: Motionless Bodies on a Tilted Beam. Gravity and Tension in Perfect Harmony Cart Track Box Weight (150 gram) Depending on the tilt of the track, the cart will accelerate up or down. However, there exists one special tilt angle such that the two-mass system is in Equilibrium a special state where the masses do not move at all! In this state of zero acceleration, the net force on each mass is equal to zero. In the sculpture above, the forces of gravity and tension are in perfect balance. Design Goal: Where should you place the box so that the masses are motionless? Work out the theory behind this mechanical structure before you actually build the structure. The Theory Here is a schematic of the system showing the system parameters m 1, m 2, L, H. θ m 1 L H m 2 Note that the variable L specifies the location of the box of height H. More specifically, the upper left corner of the box touches the track at a point that is a distance L away from the point where the track touches the table. Note the tilt relation sin θ = H/L. Answer the following questions using only symbols. No numbers allowed. 1. Draw two free body diagrams, one for m 1 and one for m 2. m 1 m 2 10
2. Set up Newton s equation of motion F net 1 = m 1 a for body 1 and Newton s equation of motion F net 2 = m 2 a for body 2. 3. Simplify the motion equations using the motionless (equilibrium) condition a = 0 and the tilt relation sin θ = H/L. 4. Solve Newton s equations for L as a function of m 1, m 2, and H. L = *Have your instructor check your theoretical work before your build your sculpture. Measure the values of H and m 1. The value of m 2 is given to be 150 grams. H = m. m 1 = kg. m 2 = 0.150 kg. Plug the values of these system parameters into your theoretical formula for L. L (theory) = m. 11
Building the Sculpture 1. Tighten the leveling screw underneath the left end of the track so that the left end of the track touches the table. You want the track to pivot from the aluminum edge of the track, not the leveling screw. 2. Remove the bracket at the right end of the track. 3. Make sure that the string is parallel to the track. You can pivot the pulley up or down to adjust the angle of the string. 4. Place the box at the location predicted by your value of L (theory) calculated above. Check to see if the system is in the special state of equilibrium, where gravity and tension are in perfect harmony. Release the cart. Perhaps give it a nudge. Observe the motion. If the masses move, then adjust the location of the box until all motion is eliminated! Once you have found equilibrium, measure the value of L with the meter stick. Do not use the scale on the track. Remember to measure L from the point where the track touches the table to the point where the track touches the box. Note: In your experimental search for this state of equilibrium, you will find a small range of L values slightly greater than and slightly less than the perfect value for which the system remains motionless. Welcome to the real world of experimental errors and uncertainty! What ingredient(s) did you leave out of your theoretical analysis that could account for this experimental range of L values? Report your experimental value of L (including the uncertainty) for which the system is in equilibrium. L (experiment) = ± m. Compare the Theoretical Design with the Experimental Structure % difference between L (theory) and L (experiment) is %. 12