Worksheet for Exploration 6.1: An Operational Definition of Work This Exploration allows you to discover how work causes changes in kinetic energy. Restart. Drag "handy" to the front and/or the back of the cart to impart a force (position is given in meters and time is given in seconds). Look at the force cos(θ) vs. position graph as well as the final velocity. a. What can you say about the relationship between the force applied and the work done? i. Start the animation, push play, and then apply a force to the right, keeping the force large until the object leaves the screen. Right click on the plot to clone it for later examination and comparison. Do again but keep the force small (look at the arrow) until the object is off the screen. Clone the plot and compare to i. Discuss how work relates to what you see on the plots. b. How does the application of the force change the kinetic energy? i. Apply a force to the right to get the object moving slowly. Consider the work you have done and the kinetic energy. Now consider how to apply a force (just a bump) to increase the kinetic energy. KE i = KE f = How must you push in order to increase the KE (that is which way relative to motion)?
i Again get the cart moving slowly to the right with an initial bump (or impulse) of force. iv. Instead of increasing the kinetic energy, now what must be done to decrease it? KE i = KE f = How must you push in order to decrease the KE? v. Try this again but get the cart moving to the left to start with and then increase or decrease the kinetic energy. vi. What is the significance of the blue vs. yellow regions on the plots. v Can you make the Kinetic Energy negative; discuss?
c. What happens when the mass changes? i. Try several different masses and push to give the cart the most kinetic energy that you are able to. Try a few times for each mass. Write down your results. m= m= m= KE trial 1 KE trial 2 KE trial 3 KE trial 4 Just look at the largest kinetic energy you were able to give to each mass. Discuss whether the mass had an effect on this. Explain your answer.
Worksheets for Exploration 6.2: The Two-Block Push Two blocks are pushed by identical forces (position is given in meters and time is given in seconds), each block starting at rest at the first vertical rectangle (start). The top block is twice the mass of the bottom block, m 1 = 2m 2. Restart. The graphs and tables are initially blank. Animation without Graphs and Tables. a. Which object has the greater kinetic energy when it reaches the second vertical rectangle (finish)? Why? (Use the animation without graphs). i. Consider the definition of work. Is there a measurement you can make to test your answer? i Make the appropriate measurements to predict KE2 25m /KE1 25m.
b. Once you have answered the above question, click Animation with Graphs and Tables, to see if you were correct. Consider both the graphs and the tables. i. How does the blue area on the graph relate to work and kinetic energy? At the end rectangle the masses are not traveling the same speeds. Can you predict the speed of one mass compared to the other here? (given the ratio of masses). Remember the objects are pushed with the same force through the same distance. c. If you were incorrect in your answer (your prediction for the ratio of Kinetic Energies), can you figure out why you answered incorrectly? What is the correct rationale you should have used to answer this question? Use the graphs and tables where appropriate.
Worksheet for Exploration 6.3: The Gravitational Force and Work A 1-kg ball is subjected to the force of gravity as shown in the animation (position is given in meters and time is given in seconds). The ball starts at x = 0 m and y = 0 m. You can vary the ball's initial velocity and view how this affects the motion of the ball and the ball's kinetic energy. Also shown are the graphs of force cos(θ) vs. position and kinetic energy vs. position. Restart. With v 0x = 0 m/s and v 0y = 0 m/s: a. In what direction does the net force on the ball point? i. Draw a sketch showing the ball, the direction of the net force and the direction of the acceleration on the ball. b. How would you describe the KE vs. position graph? i. You should consider initial kinetic energy, final, slope etc. KE i = KE f = Slope= With v 0x = 10 m/s and v 0y = 0 m/s: c. What is the minimum amount of kinetic energy the ball has? KE min = i. Where is the ball when the kinetic energy is minimum? What is different about the plot now than in part b above? You may want to run each simulation and clone the plots to compare (right mouseclick).
With v 0x = 10 m/s and v 0y = 10 m/s: d. How would you describe the KE vs. position graph? Be explicit about what happens to the kinetic energy. i. KE i = KE f = Slope= e. What is the condition for the work done by gravity to be zero? i. Examine the F cos(θ) plot to determine where the W gravity is zero (that is work done since the initial launch time). Predict when the ball is at this position. f. What is the ball's minimum kinetic energy? i. Where and when does this occur? With v 0x = -10 m/s and v 0y = 10 m/s: g. How would you describe the KE vs. position graph? Be explicit about what happens to the kinetic energy. h. What is the condition for the work done by gravity to be zero? i. What is the minimum amount of kinetic energy the ball has?
Worksheet for Exploration 6.4: Change the Direction of the Force Applied A 20-kg ball has a hole with a rod passing through. The rod exerts a force as needed that constrains the ball to move along the rod. An applied force is now added (the pulling force) so the ball is pulled as shown (position is given in meters and time is given in seconds). The pulling force vector is shown as a red arrow, and makes an angle θ with the horizontal. The velocity is given in meters/second. You may adjust the angle and/or the magnitude of the pulling force (F < 7 N). Restart. a. How does the work done by the applied (pulling) force change as you vary the pulling force for a constant angle? i. Select a force magnitude and angle. F pull = θ= Make a sketch indicating the forces acting on the ball (vectors) and the direction of motion. i Calculate the work done by the F pull from 0 to say 20m. Make sure your calculation agrees with the measurements made using the animation. iv. Now try changing the magnitude of the pulling force and repeat i,ii, and i F pull = θ=
v. Consider your vector sketch with other force vectors on it (gravitational or the rod). Why didn t we need to consider those forces? b. How does the work done by the applied force change as you vary the angle for a constant pulling force? i. Select a force, angle and displacement, then calculate the work done by the pulling force. F pull = θ= x= W pull = Select a new angle then calculate the work done by the pulling force. F pull = θ= x= W pull = c. Combine your answers above to a general mathematical formula for the work done on the ball due to an arbitrary applied force. d. Determine the general mathematical formula for the work done by the normal force the rod exerts on the ball when an arbitrary force is applied to the ball.
Worksheet for Exploration 6.5: Circular Motion and Work A 1-kg black ball is constrained to move in a circle as shown in the animation (position is given in meters, time is given in seconds, and energy on the bar graph is given in joules). In the animation the wire is vertical and the ball is subjected to gravity (downward as usual), as well as the force of the wire. You may set the initial velocity and then play the animation. The blue arrow represents the net force acting on the mass, while the bar graph displays its kinetic energy in joules. Restart. a. Set the speed fast enough to get the ball over the top. Then restart and examine forces at positions near say -45 o and 45 o (very roughly). (hanging straight down is 90 o ). Label your forces as F g (gravity), F wire, and F net. i. Sketch a free body diagram and then a head to tail vector diagram indicating the two forces acting on the ball. Near -45 o (Measure the position x=, y= ) Again. Near 45 o. i Do again for 90 o, 0 o angles.
b. Given your force diagrams there are positions where the speed of the ball is changing more rapidly than others. Take each of the positions you considered and rank them from highest tangential acceleration to lowest. Rank: i. Discuss what must happen for the ball to speed up or slow down. c. Assume that the ball can get to y =10 m. How much kinetic energy does the ball lose in going from y = -10 m to y = 10 m? Is this independent of v 0x initial? i. KE -10 =, KE +10 = KE= d. What is the work done by gravity when the ball goes from y= -10 m to y = 10 m? i. To calculate the work done by gravity first show using symbols only, then determine a numerical value (ie plug numbers into your formula). W g = (symbols) W g = (number) How does your answer for work relate to the change in kinetic energy. e. Determine the minimum speed that the ball must have (initially at the bottom) to go over the top. Once you have an answer check it using the animation. v bottom min =
Worksheet for Exploration 6.6: Forces, Path Integrals, and Work Move your cursor into the animation, then click-drag the crosshair cursor with the mouse. The bar graph on the right displays the work done by the force along the path. For your reference, there are circles every 10 m that form a coordinate grid (position is given in meters and the result of the integral is given on the bar graph in joules). Use the "reset integral" button to re-zero the work calculation between paths. Restart. For each force, answer the following questions: For paths a through f place answers in the tables below.\ a. Starting at the origin (the center, x = 0 m and y = 0 m) and moving to x = 0 m and y = 10 m, what is the work done by the force? b. Starting at x = 0 m and y = 10 m and moving to x = 0 m and y = 0 m, what is the work done by the force? c. Starting at the origin (the center, x = 0 m and y = 0 m) and moving to x = 0 m and y = -10 m, what is the work done by the force? d. Starting at the origin (the center, x = 0 m and y = 0 m) and moving to x = 10 m and y = 0 m, what is the work done by the force? e. Starting at the origin (the center, x = 0 m and y = 0 m) and moving to x = -10 m and y = 0 m, what is the work done by the force? a. Starting at the origin (the center, x = 0 m and y = 0 m), choosing your own path around the For each path described in (a) through (f), measure the work done by the force using the animation. Path a Path b Path c Path d Path e Path f Fx = 0 Fy = -9.8 Fx = 0 Fy = -x Fx = x Fy = y Fx = y Fy = y*x Fx = x*x/4 Fy = y*y/4
Additional Questions For several examples in your table above (you select some) show how to calculate the work done by the force for a given path. Hint: break your calculation into integrals along x and along y. i. Example 1. Example 2. i Example 3. iv. For each force determine if the work done by that force depends on a particular path or not. That is, select some points A and B (you pick) and decide if the work done to get from A to B is the same or different when you select a different path or route from A to B. Answer: Path Dependent or Path Independent. Answer: Fx = 0 Fy = -9.8 Fx = 0 Fy = -x Fx = x Fy = y Fx = y Fy = y*x Fx = x*x/4 Fy = y*y/4