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11.1.5.2: Explorations

  • Page ID
    34087
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    Exploration 1: Circular Motion

    A puck travels in a circular path on a frictionless table, propelled by a string pulling from the center of the circle (position is shown in meters and time is shown in seconds). You may adjust the mass \((10\text{ g} < m < 500\text{ g})\), the speed \((1\text{ m/s} < v < 50\text{ m/s})\), and/or the radius \((0.5\text{ m} < r < 3.5\text{ m})\). The tension is displayed on the screen. Restart.

    How does the tension in the string depend upon the mass, the speed of the block, and the radius of the circle?

    1. If you only vary the mass, how does the tension change?
    2. If you only vary the velocity, how does the tension change?
    3. If you only vary the radius, how does the tension change?

    Exploration 2: Force an Object Around a Circle

    In this Exploration you are looking down at a black ball on a table top. Drag the crosshair cursor (position is given in meters and time is given in seconds) to within \(5\text{ m}\) of the \(0.2\text{-kg}\) ball. The cursor will then exert a constant force on the black ball. You may choose either an attractive or a repulsive force. In addition, the black ball is constrained to move in a circle by a very long wire. The blue arrow represents the net force acting on the mass, while the bar graph displays its speed in meters/second. Restart.

    For both attraction and repulsion, drag the cursor around to see how the net force varies.

    1. At the beginning of the animation (before you move the cursor), in what direction does the net force point?
    2. With this force, does the ball move?
    3. What type of applied force makes the ball acquire a tangential velocity?
    4. Describe the direction of the force that makes the ball acquire the maximum tangential velocity for the force applied.
    5. When the ball has a tangential velocity, in which direction does the net force point when the cursor is nearby? In what direction does the acceleration point?
    6. With the object moving, drag the cursor far away from the ball. In what direction is the net force now? What is the direction of the acceleration? Why?

    Exploration 3: Spring Force

    The spring-ball system shown in the animation can be stretched by click-dragging the dark blue ball (position is given in meters and time is given in seconds). The black arrow attached to the ball shows the net, i.e., total, force on the ball. The pale blue ball on the left is the free-body diagram for the dark blue ball. The red and green arrows attached to the pale blue ball show the spring and gravitational forces, respectively. The acceleration due to gravity is \(9.8\text{ m/s}^{2}\) in this animation. Restart.

    1. Find the mechanical equilibrium for this system when the spring constant is \(1.0\text{ N/m},\: 2.0\text{ N/m},\) \(3.0\text{ N/m}\), and \(4.0\text{ N/m}\).
    2. Use your equilibrium measurements to find the mass of the ball. Hint: What forces act on the ball?
    3. Use your equilibrium measurements to find the natural length of the spring, that is, the length of the spring without an attached mass.

    Exploration 4: Circular Motion and a Spring Force

    A \(1\text{-kg}\) mass is attached to the end of a spring of spring constant \(k = 10\text{ N/m}\) and natural length \(l_{0} = 5\text{ m}\) (position is shown in meters and time is shown in seconds). You are to set the spring in motion by setting its initial position \((x_{0}, 0)\) and its initial velocity \((0, v_{0y})\). Restart.

    1. Find the \(v_{0y}\) needed for circular motion at a radius of \(10\text{ m}\) (the red circle).
    2. Determine the period of such a motion.

    Exploration 5: Enter a Formula for the Force

    This Exploration allows you to choose initial conditions and forces with damping, and then view how that force affects the red ball. You can right-click on the graph to make a copy at any time. If you check the "strip chart" mode box, the top graph will show data for a time interval that you set. Note that the animation will end when the position of the ball exceeds \(+/-100\text{ m}\) from the origin. Restart.

    Remember to use the proper syntax such as  \(-10+0.5\ast t\), \(-10+0.5\ast t\ast t\), and \(-10+0.5\ast t\wedge 2\). Revisit Exploration 1.3 to refresh your memory.

    For each of the following forces, first describe the force (magnitude and direction) and then predict the motion of the ball. How close were you? Don't forget to determine how the initial position and velocity affect the motion of the ball for each force.

    1. \(F_{x}(x, vx, t) = 1-0.05\ast vx\)
    2. \(F_{x}(x, vx, t) = 1-0.5\ast vx\)
    3. \(F_{x}(x, vx, t) = 1-vx\)
    4. \(F_{x}(x, vx, t) = -9.8-vx\)
    5. \(F_{x}(x, vx, t) = x-vx\)
    6. \(F_{x}(x, vx, t) = \cos(x)-vx\)
    7. \(F_{x}(x, vx, t) = \cos(t)-vx\)

    Exploration 6: Air Friction

    Two identical balls are dropped. The one on the left is in a resistive medium represented by varying shades of blue. The resistive force is represented as \(b v^{n}\), where \(b\) is a constant between \(0\) and \(2\) and \(n\) is an integer between \(0\) and \(2\) (note that as you vary \(n\), the units of \(b\) also change). Restart.

    Select values for \(b\) and \(n\), and then click on a graph link to show the motion and that particular graph. When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

    1. How does your choice of \(n (0, 1, 2)\) affect the unit of \(b\)?
    2. For \(b = 1\), how does your choice of \(n (0, 1, 2)\) affect the position vs. time graph?
    3. For \(b = 1\), how does your choice of \(n (0, 1, 2)\) affect the velocity vs. time graph?
    4. For \(b = 1\), how does your choice of \(n (0, 1, 2)\) affect the acceleration vs. time graph?
    5. For \(b = 1\), how does your choice of \(n (0, 1, 2)\) affect the terminal velocity?

    Exploration 7: Enter a Formula, \(F_{x}\) and \(F_{y}\), for the Force

    This Exploration allows you to choose initial conditions and forces and then view how that force affects the red ball. You can right-click on the graph to make a copy at any time. If you check the "strip chart" mode box, the top graph will show data for a time interval that you set. Note that the animation will end when the position of the ball exceeds \(+/-100\text{ m}\) from the origin. Restart.

    Remember to use the proper syntax such as \(-10+0.5\ast t\), \(-10+0.5\ast t\ast t\), and \(-10+0.5\ast t\wedge 2\). Revisit Exploration 1.3 to refresh your memory.

    For each of the following forces, first describe the force (magnitude and direction) and then predict the motion of the ball. How close were you? Don't forget to determine how the initial position and velocity affect the motion of the ball for each force.

    \(F_{x}\) \(F_{y}\) \(x_{0}\) \(y_{0}\) \(v_{0x}\) \(y_{0x}\)
    \(1\) \(1\) \(0\) \(0\) \(0\) \(0\)
    \(1\) \(1\) \(0\) \(0\) \(0\) \(0\)
    \(-x\) \(-2\ast y\) \(10\) \(10\) \(0\) \(0\)
    \(-0.5\ast vx\) \(-9.8-0.5\ast vy\) \(-20\) \(0\) \(20\) \(20\)

    Table \(\PageIndex{1}\)

    Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.


    This page titled 11.1.5.2: Explorations is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Wolfgang Christian, Mario Belloni, Anne Cox, Melissa H. Dancy, and Aaron Titus, & Thomas M. Colbert.