Author: Rhonda MacDonald

Work-Energy Theorem

  1. Take your Smart Cart out of the box.

  2. Turn it on and open your choice of software: SPARKvue or Capstone.

  3. Wirelessly connect to the Smart Cart.

  4. Change the sample rate of the Smart Cart Position and Force sensors to 40 Hz.

  5. Make a graph of Force vs. Position and another graph of Velocity vs. Time.

  6. Install the hook on the Smart Cart’s force sensor. Without anything touching the force sensor, zero the force sensor in the software.

  7. Put a rubber band on the force sensor hook. Start recording and while one person holds the rubber band in place, the other person slowly pulls the cart back, stretching the rubber band. Then hold the cart in place with the rubber band stretched and stop recording. Do not let go of the cart or rubber band.

  8. Start recording again. Let go of the cart and move the hand holding the rubber band out of the way. Let the cart go up to its maximum speed and then stop recording.

Analysis

  1. Determine the work done in stretching the rubber band by finding the area under the Force vs. Position curve.

  2. Determine the work done as the stretched rubber band pulls the cart by finding the area under the Force vs. Position curve.

  3. On the Velocity vs. Time graph, determine the maximum velocity. Calculate the kinetic energy of the cart and compare to the work done to accelerate the cart.

  4. Why isn’t the work done to stretch the rubber band equal to the work done to accelerate the cart?

Sample Data

The work done loading the rubber band is -1.91 Nm. The work done unloading (when the cart is launched) the rubber band is 0.77 Nm. The resulting kinetic energy of the cart is

KE = ½ mv2 = ½ (0.252 kg)(2.34 m/s)2 = 0.69 J. This is 10% less than the energy available in the stretched rubber band.

The energy stored in the rubber band is less than the work done to stretch the rubber band. Some of that energy goes into heating the rubber band and making the rubber band move.

Static and Kinetic Friction

 

  1. Take your Smart Cart out of the box.

  2. Turn it on and open your choice of software: SPARKvue or Capstone.

  3. Wirelessly connect to the Smart Cart.

  4. Make a graph of Force vs. Position.

  5. Make sure the Smart Cart Force sensor (with the magnetic bumper on it) is not touching anything and then zero the Force sensor in the software.

  6. Set the cart bumper against the book. Start recording. Very slowly push the cart until the book breaks loose and then push it steadily across the table. Stop recording.

  7. Take another run, pushing it at a faster speed once it breaks loose.

  8. Add a second book on top of the first book and repeat.

Analysis

  1. For each run, record the maximum force before the book moved. This is an indication of the static friction. If you can find the mass of the book, you can calculate the static coefficient of friction for the book on the table.

  2. For each run, record the average force while the book was moving. This is an indication of the kinetic friction. If you can find the mass of the book, you can calculate the kinetic coefficient of friction for the book on the table.

  3. What effect does speed have on the kinetic friction?

  4. What changes when the extra book is added? Do the coefficients of static and kinetic friction change?

Sample Data

The average kinetic friction for two books is 6.92 N.

The average kinetic friction for one book going slower vs. faster was 3.35 N compared to 4.19 N. This indicates that the speed does influence the kinetic friction slightly.

Mass of First Book = 1.56 kg

Mass of Second Book = 1.58 kg

For one book:

μs = F/mg = 3.09N/(1.56)(9.8) = 0.2

μk = F/mg = 3.17N/(1.56)(9.8) = 0.2

For two books:

μs = F/mg = 7.11 N/(1.56+1.58)(9.8) = 0.2

μk = F/mg = 6.59N/(1.56+1.58)(9.8) = 0.2

Hooke’s Law

Hooke’s Law states:
where F is the force of the spring, k is the spring constant, and x is the distance the spring has been stretched.

  1. Take your Smart Cart out of the box.

  2. Turn it on and open your choice of software: SPARKvue or Capstone.

  3. Wirelessly connect to the Smart Cart.

  4. Make a graph of Force vs. Position.

  5. Install the hook on the Smart Cart Force Sensor. Make sure the Smart Cart Force sensor is not touching anything and then zero the Force sensor in the software.

  6. Put one end of a spring on the hook and hold the other end stationary with your hand. Move the cart slightly to put a little tension on the spring.

     

  7. Start recording and pull the Smart Cart away from the fixed end of the spring until the spring is stretched out. Then stop recording.

Analysis

  1. On the Force vs. Position graph, apply a linear fit to the straight-line part of the graph.

  2. Determine the spring constant from the slope of the linear fit.

Sample Data


The slope of the graph indicates the spring constant is 6.77 N/m.

Impulse Demonstration

Equipment:

Smart Cart

Accessory Rubber Bumper

Learning Outcome:

A force acting on an object for a period of time imparts an impulse to that object which is defined as a change in momentum.

Experimental Setup:

  1. Take your Smart Cart out of the box

  2. Attach the rubber bumper accessory (included with Smart Cart) to the force sensor on the Smart Cart.

  3. Press the power button on the side of the Smart Cart to turn it on.

  4. In SPARKvue or Capstone, pair the Smart Cart to your computer or device. Here are a couple short videos to help you pair in either software:

    1. SPARKvue: https://www.youtube.com/watch?v=tsdHWu4quNo

    2. Capstone: https://www.youtube.com/watch?v=JGy-UG245lY

  5. In the software, you will need to create a graph of Force vs. Acceleration.

    1. In SPARKvue:

      1. Under “Quick Start Experiments” choose: Impulse

      2. Increase the sampling rate of the Force sensor to 1KHz

    2. In Capstone:

      1. Create two graph displays

      2. Graph 1: [Force] vs. Time

      3. Graph 2: [Velocity] vs. Time

      4. Increase the sampling rate of the Force sensor to 1KHz

You will push the cart into a barrier such that the rubber bumper will collide and bounce the cart off the barrier. A wall, book or other solid vertical surface will work.

Data Collection:

  1. Zero the force sensor

  2. Press the record data button

  3. With the rubber bumper facing towards the barrier, give the Smart Cart a push.

  4. After the Smart Cart has reversed direction, stop data collection.

 

Data Analysis:

  1. On the Force vs. Time graph, use the Area tool to measure the area under the curve. This is the impulse that the Smart Cart experienced.

  2. On the Velocity vs. Time graph, use the Coordinate tool to find the velocity just before the impact of the Smart Cart against the barrier and record this value. This is the Smart Cart’s initial velocity.

  3. Next, using the Coordinate tool find the velocity after the collision with the barrier. Record this value. This is the Smart Cart’s final velocity.

  4. Weigh the cart without any bumper and record the mass. You may also estimate the mass of the Smart Cart to be around 0.250 kg.

Calculate the change in momentum of the Smart Cart: pf – pi

Compare your calculated value to the area under the Force vs. Time graph.

Sample Data:

This data was created with a Smart Cart that measured 0.246 kg, for an error around 1.5%.

Centripetal Acceleration and Force

  1. Take your Smart Cart out of the box.

  2. Turn it on and open your choice of software: SPARKvue or Capstone.

  3. Wirelessly connect to the Smart Cart.

  4. Make a graph of Acceleration-x (from the Smart Cart Acceleration Sensor) vs. Angular Velocity-y (from the Smart Cart Gyro Sensor). Add a second plot area with the Force vs. Angular Velocity-y.

  5. Install the rubber bumper on the Smart Cart Force Sensor. With the cart sitting still, with nothing touching the rubber bumper on the Force Sensor, zero the Acceleration-x, Angular Velocity-y, and the Force in the software.

  6. Set up a board or track on a rotatable chair as shown in the picture. Set the end stop near the end of the track and place the cart’s rubber bumper (Force Sensor end) against the end stop.

    Post

  7. Spin the chair and start recording. Let the chair spin down to a stop and then stop recording.

  8. Apply a curve fit to the data to determine how the centripetal acceleration and force are related to the angular velocity. For the quadratic fit, open the curve fit editor at right in Capstone and lock the coefficient B = 0.

 This forces the fit to Aω2 + C. From the curve fit, what is the radius?

  1. In which direction are the centripetal acceleration and the centripetal force?

Further Study

  1. Move the end stop 5 cm closer to the center of rotation. Repeat the experiment.

  2. Continue to move the end stop closer to the center in 5 cm increments.

  3. How does the centripetal force depend on the radius?

Sample Data

Both the centripetal acceleration and the centripetal force are pointing toward the center of the circle (they are negative) and are proportional to the square of the angular velocity.

a = -0.383ω2 – 0.0530

F = -0.0966ω2 – 0.00596

m = 0.25 kg

F = ma = 0.25(-0.383ω2 – 0.0530) = -0.096ω2 – 0.013

The radius is 0.383 m because a = rω2.

Newton’s Second Law Demonstration

Equipment:

  • Smart Cart

  • Accessory Hook

Learning Outcome:

Forces and Accelerations of objects have a linear relationship that relates the mass of an object being accelerated to an unbalanced force acting on it.

Experimental Setup:

  1. Take your Smart Cart out of the box.

  2. Attach the hook accessory (included with Smart Cart) to the force sensor on the Smart Cart.

  3. Press the power button on the side of the Smart Cart to turn it on.

  4. In SPARKvue or Capstone, pair the Smart Cart to your computer or device. Here are a couple short videos to help you pair in either software:

    1. SPARKvue: https://www.youtube.com/watch?v=tsdHWu4quNo

    2. Capstone: https://www.youtube.com/watch?v=JGy-UG245lY

  5. In the software, you will need to create a graph of Force vs. Acceleration.

    1. SPARKvue: Under “Quick Start Experiments” choose: Newton’s Second Law

    2. Capstone:

      1. Create a Graph Display

      2. Select measurement of [Force] for the y-axis

      3. Select measurement of [Acceleration – x] for the x-axis

      4. In the sampling control panel, press the “Zero Sensor” button

Before you collect data, practice rolling the cart in a forwards and backwards motion by only holding on to the hook. You want to apply a force along the cart’s x-axis, and have the cart roll only along this direction. This is made easier using a PASCO track to keep the cart moving in one direction, but not necessary for the demonstration. (Hint: Try not to wiggle or knock the Smart Cart hook as this will result in extraneous data points.) 

Data Collection:

  1. Press the record data button

  2. Holding only the hook, roll the Smart Cart forwards and backwards in the x-direction.

  3. Repeat this motion a few times to generate enough data points to see the graphical relationship.

  4. Stop data collection

Data Analysis:

  1. Turn on the ‘Linear Fit’ tool

  2. This relationship shows that there is a proportionality constant between the unbalanced force, and the Smart Cart’s resulting acceleration.

  3. The proportionality constant is the mass of the cart.

Add mass to the Smart Cart and repeat data collection for the new system mass.

Sample Data:

This data was created with a Smart Cart that measured .246 kg, for an error around 2%.

Average and Instantaneous Velocity and Speed

  1. Take your Smart Cart out of the box.
  2. Turn it on and open your choice of software: SPARKvue or Capstone.
  3. Wirelessly connect to the Smart Cart. Change the sample rate of the Position Sensor to 40 Hz.
  4. Open the calculator in the software and make the following calculation:
‎speed‎=abs([Velocity, Red (m/s)‎])       with units of m/s
  1. Create a graph of Velocity vs. Time and add a second plot area of speed vs. Time and add a third plot area of Position vs. Time.
  2. Mark a starting point with a piece of tape.
  3. Start recording. Push the cart about 20 cm out and back, ending at the same point where you started.
Analysis
  1. On the Velocity vs. Time graph, find the maximum positive velocity.
  2. What is the instantaneous velocity at the point where you reversed the cart?
  3. What is the average velocity over the entire motion of the cart? Highlight the area of the Velocity vs. Time graph during the time of the motion and turn on the mean statistic.
  4. What is the average speed over the entire motion of the cart? Highlight the area of the speed vs. Time graph during the time of the motion and turn on the mean statistic.
  5. What is the difference between speed and velocity?
Sample Data
The instantaneous velocity when the cart reversed was zero.
The average velocity over the whole trip was zero because we started and stopped in the same place.
The average speed over the whole trip was 0.36 m/s.
Speed is a scalar that is the magnitude of the velocity. Velocity is a vector and has both magnitude (speed) and direction.

The Differences Between Velocity and Acceleration

  1. Take your Smart Cart out of the box.

  2. Turn it on and open your choice of software: SPARKvue or Capstone.

  3. Wirelessly connect to the Smart Cart.

  4. Change the sample rate of the Smart Cart Position sensor to 40 Hz.

  5. Set up a graph of Velocity vs. Time and Acceleration vs. Time using the Position sensor’s Velocity and Acceleration.

  6. Make an inclined plane by placing the top edge of one textbook on top of a second textbook.

  7. Put the Smart Cart at the bottom of the incline, with its force sensor end oriented up the incline.

  8. Start recording and push the cart so it just barely reaches the top of the incline and then rolls back down. Stop recording when it gets back down.

  9. Examine the graphs and determine where the cart is:

    1. going up the incline.

    2. going down the incline.

    3. at the top of the incline.

For each of these cases, is the velocity positive, negative, zero, and/or constant? Is the acceleration positive, negative, zero, and/or constant?

  1. When the cart is going up the incline, which direction is the velocity? Which direction is the acceleration? Is the cart accelerating or decelerating?

  2. When the cart is at the top of the incline, the velocity is zero. Which direction is the acceleration? Is the cart accelerating or decelerating?

  3. When the cart is going down the incline, which direction is the velocity? Which direction is the acceleration? Is the cart accelerating or decelerating?

  4. On the Velocity vs. Time graph, find the slope of the straight-line portion. Compare this to the acceleration on the Acceleration vs. Time graph.

Sample Data

When the cart is going up the incline, the velocity is positive (up the incline) while the acceleration is constant and negative (down the incline). The cart is decelerating.

When the cart is at the top of the incline, the velocity is zero while the acceleration is constant and negative (down the incline). The cart is accelerating.

When the cart is going down the incline, the velocity is negative (down the incline) while the acceleration is constant and negative (down the incline).

The slope of the Velocity vs. Time graph is -1.57 m/s2. The average acceleration from the Acceleration vs. Time graph is -1.577 m/s2, which is 0.6% different from the slope.

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